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Remark on the Daugavet property for complex Banach spaces

  • Han Ju Lee and Hyung-Joon Tag EMAIL logo
Published/Copyright: August 13, 2024
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Demonstratio Mathematica
From the journal Demonstratio Mathematica Volume 57 Issue 1

Abstract

In this article, we study the Daugavet property and the diametral diameter two properties (DD2Ps) in complex Banach spaces. The characterizations for both Daugavet and Δ -points are revisited in the context of complex Banach spaces. We also provide relationships between some variants of alternative convexity and smoothness, nonsquareness, and the Daugavet property. As a consequence, every strongly locally uniformly alternatively convex or smooth (sluacs) Banach space does not contain Δ -points from the fact that such spaces are locally uniformly nonsquare. We also study the convex diametral local diameter two property and the polynomial Daugavet property in the vector-valued function space A ( K , X ) . From an explicit computation of the polynomial Daugavetian index of A ( K , X ) , we show that the space A ( K , X ) has the polynomial Daugavet property if and only if either the base algebra A or the range space X has the polynomial Daugavet property. Consequently, we obtain that the polynomial Daugavet property, Daugavet property, DD2Ps, and property ( D ) are equivalent for infinite-dimensional uniform algebras.

Keywords: Daugavet points; Δ-points; alternative convexity or smoothness; nonsquareness; polynomial Daugavet property
MSC 2010: Primary 46B20; Secondary 46B04, 46E40, 46J10

1 Introduction

In the theory of Banach spaces, various properties that are related to certain behaviors of vector measures and bounded linear operators have been studied from a geometrical point of view. We focus on Daugavet property and diametral diameter two properties (DD2Ps) in this article. Let X be a Banach space on F = R or C . For a Banach space X , we denote the unit ball and the unit sphere of X by B X and S X , respectively. The dual space of X is denoted by X * . For ε > 0 , x * ∈ S X * , and x ∈ S X , a slice S ( x * , ε ) of the unit ball B X is defined by S ( x * , ε ) = { x ∈ B X : Re x * x > 1 − ε } and a weak * -slice S ( x , ε ) of B X * by S ( x , ε ) = { x * ∈ B X * : Re x * x > 1 − ε } .

Many results on Banach spaces with the Radon-Nikodým property have been obtained from this perspective. Even though its definition was originally given in terms of vector measures, it is now well-known that the Banach spaces with the Radon-Nikodým property can be characterized by the existence of slices with arbitrarily small diameter as well as the existence of denting points. For more details on the geometrical aspect of the Radon-Nikodým property and its application in other research topics in Banach spaces, we refer to [1].

A Banach space X is said to have the Daugavet property if every rank-one operator T : X → X satisfies the following equation:

‖ I + T ‖ = 1 + ‖ T ‖ .

We call this equation the Daugavet equation. The spaces C ( K ) , where K is a compact Hausdorff space without isolated points, L 1 ( μ ) , and L ∞ ( μ ) with a nonatomic measure μ are classical examples of the Daugavet property. The infinite-dimensional uniform algebras also have the Daugavet property if and only if their Shilov boundaries do not have isolated points [2,3]. Moreover, the Daugavet property in Musielak-Orlicz spaces [4], Lipschitz-free spaces [5], and rearrangement invariant Banach function lattices [6,7] has been examined. It is well-known that every slice of B X has a diameter of 2, if X has the Daugavet property, which tells us that Banach spaces with this property are on the opposite spectrum to the Radon-Nikodým property.

The following characterization allows us to study the Daugavet property with slices.

Lemma 1.1

[8, Lemma 2.2] The following are equivalent:

  1. A Banach space ( X , ‖ ⋅ ‖ ) has the Daugavet property.

  2. For every slice S = S ( x * , ε ) and x ∈ S X , there exists y ∈ S X ∩ S such that ‖ x + y ‖ > 2 − ε .

  3. For every weak * -slice S * = S ( x , ε ) and every x * ∈ S X * , there exists y * ∈ S X * ∩ S * such that ‖ x * + y * ‖ > 2 − ε .

Later, the DD2Ps and the property ( D ) have gained attention [9,10] from many researchers. They are known to be weaker than the Daugavet property.

Definition 1.2

  1. A Banach space X has property ( D ) if for every rank-one, norm-one projection P : X → X satisfies ‖ I − P ‖ = 2 .

  2. A Banach space X has the diametral local diameter two property (DLD2P) if for every slice S of the unit ball, every x ∈ S ∩ S X , and every ε > 0 there exists y ∈ S such that ‖ x − y ‖ ≥ 2 − ε .

  3. A Banach space X has the DD2P if for every nonempty weakly open subset W of the unit ball, every x ∈ W ∩ S X , and every ε > 0 , there exists y ∈ W such that ‖ x − y ‖ ≥ 2 − ε .

The first known example that possesses the property ( D ) is a certain subspace of L 1 constructed with martingales [11]. Later on, this space was shown to have the Daugavet property [12]. In view of [13], every rank-one projection P on a Banach space X with the DLD2P satisfies ‖ I − P ‖ ≥ 2 , and so the DLD2P implies the property ( D ). As a matter of fact, the property ( D ) was thought to be equivalent to the DLD2P. However, since a scalar multiple of a projection is not a projection, the validity of the equivalence is not clear up to now [9]. The implication (iii) ⇒ (ii) holds because every slice is a weakly open subset of the unit ball.

The DLD2P and the Daugavet property can also be considered from a local perspective by using Δ -points and Daugavet points. Let Δ ε ( x ) = { y ∈ B X : ‖ x − y ‖ ≥ 2 − ε } , where x ∈ S X and ε > 0 .

Definition 1.3

  1. A point x ∈ S X is a Δ -point, if x ∈ conv ¯ Δ ε ( x ) for every ε > 0 .

  2. A point x ∈ S X is a Daugavet point, if B X = conv ¯ Δ ε ( x ) for every ε > 0 .

Note that the set Δ ε ( x ) is defined independent of the scalar fields F = R or C on Banach spaces. Hence, we may use the same definitions for Δ -points and Daugavet points for complex Banach spaces. We denote the set of all Δ -points of X by Δ X . There is also a weaker property than the Daugavet property described by the set Δ X .

Definition 1.4

A Banach space X has the convex diametral local diameter two property (convex-DLD2P) if conv ¯ Δ X = B X .

It is well-known that a real Banach space X has the Daugavet property (resp. the DLD2P), if and only if every point on the unit sphere is a Daugavet point (resp. a Δ -point).

We mention that many recent results on the Daugavet property, the DD2Ps, Daugavet points, and Δ -points have mostly revolved around real Banach spaces. But there are several results concerning these concepts in complex Banach spaces [14,15]. For a real Banach space X , it is well-known that Δ -points are connected to certain behaviors of slices of the unit ball and of rank-one projections. The notation x * ⊗ x stands for a linear operator from X to itself defined by ( x * ⊗ x ) ( y ) = x * y ⋅ x , where x , y ∈ X and x * ∈ X * .

Theorem 1.5

[9] Let X be a real Banach space. Then, the following statements are equivalent:

  1. x ∈ S X is a Δ -point.

  2. For every slice S of B X with x ∈ S ∩ S X and ε > 0 , there exists y ∈ S such that ‖ x − y ‖ ≥ 2 − ε .

  3. For every rank-one projection P = x * ⊗ x with x * x = 1 , we have ‖ I − P ‖ ≥ 2 .

Even though the complex analogue of this relationship may be well-known to the specialists, we will state and prove it for completeness. In addition, while the Daugavet property for complex Banach spaces can be examined through rank-one real-linear operators [16], it has not been known whether we can examine the DLD2P in a similar spirit with rank-one real-projections. We also study this here.

Since a denting point is always contained in a slice of arbitrarily small diameter, such a point is neither a Δ -point nor a Daugavet point. This implies that any (locally) uniformly rotund real Banach spaces cannot have Δ -points. Recently, identifying the Banach spaces that do not contain these points has been an active research topic. For example, it is shown in [17] that every uniformly nonsquare (UNSQ) real Banach space does not have Δ -points. Furthermore, a locally uniformly nonsquare (LUNSQ) real Banach space does not have Δ -points [18]. We will examine strongly locally uniformly alternatively convex or smooth (sluacs) Banach spaces in this article. We mention that alternative convexity or smoothness is related to the anti-Daugavet property and the nonsquareness property [19,20].

Banach spaces that satisfy the Daugavet equation for weakly compact polynomials are also studied in [21–23]. For Banach spaces X , Y , let ℒ ( X k ; Y ) be the space of bounded k -linear mappings from X to Y and let Ψ k : X → X k be a diagonal mapping defined by

Ψ k ( x ) = ( x , x , … , x ) ︸ k times .

A mapping is called a bounded k-homogeneous polynomial from X to Y if it is the composition of Ψ k with an element in ℒ ( X k ; Y ) . We denote the set of all bounded k -homogenous polynomials from X to Y by P ( X k ; Y ) . A polynomial is a finite sum of bounded homogenous polynomials from X to Y . We also denote the set of all polynomials from X to Y by P ( X ; Y ) and the set of all scalar-valued continuous polynomials by P ( X ) . The space P ( X ; Y ) is a normed space endowed with the norm ‖ P ‖ = sup x ∈ B X ‖ P x ‖ X and P ( X k ; Y ) is a closed subspace of P ( X ; Y ) .

We say a polynomial P ∈ P ( X ; Y ) is weakly compact if P ( B X ) is a relatively weakly compact subset of Y . A Banach space X is said to have the polynomial Daugavet property if every weakly compact polynomial P ∈ P ( X ; X ) satisfies

‖ I + P ‖ = 1 + ‖ P ‖ .

If X has the polynomial Daugavet property, then the space also has the Daugavet property. It is also well-known that the polynomial Daugavet property can be described in terms of scalar-valued polynomials. Here the notation p ⊗ x stands for a polynomial from X to itself defined by ( p ⊗ x ) ( y ) = p ( y ) ⋅ x , where x , y ∈ X and p ∈ P ( X ) .

Theorem 1.6

[21, Corollary 2.2] Let X be a real or complex Banach space. Then, the following statements are equivalent:

  1. X has the polynomial Daugavet property.

  2. For every p ∈ P ( X ) with ‖ p ‖ = 1 , every x 0 ∈ S X , and every ε > 0 , there exist ω ∈ S C and y ∈ B X such that Re ω p ( y ) > 1 − ε and ‖ x 0 + ω y ‖ > 2 − ε .

  3. For every p ∈ P ( X ) and every x 0 ∈ X , the polynomial p ⊗ x 0 satisfies the Daugavet equation.

In this article, we will look at the polynomial Daugavet property in a function space A ( K , X ) over the base algebra A , which will be defined later. This class of function spaces includes uniform algebras and the space of Banach space-valued continuous functions on a compact Hausdorff space. The Daugavet property and the DD2Ps of the vector-valued function spaces A ( K , X ) are studied in [2]. From the same article, assuming the uniform convexity of the range space X and A ⊗ X ⊂ A ( K , X ) , it is shown that the space A ( K , X ) has the Daugavet property, if and only if its base algebra also has the Daugavet property. It is also shown in [24] that if X has the Daugavet property, then A ( K , X ) has the Daugavet property. Here we attempt to find a necessary and sufficient condition for A ( K , X ) to have the polynomial Daugavet property.

This article consists of three parts. In Section 2, we revisit well-known facts about Δ -points and Daugavet points in the context of complex Banach spaces. Like the Daugavet property, the DLD2P can also be analyzed by using rank-one real projections (Theorem 2.2). In Section 3, we examine the relationship between alternative convexity or smoothness and nonsquareness. From the fact that strongly locally uniformly alternatively convex or smooth (sluacs) Banach spaces are LUNSQ (Proposition 3.8.(i)), no sluacs Banach space contains Δ -points (Theorem 3.4). In Section 4, we study the polynomial Daugavet property of the space A ( K , X ) . Here we explicitly compute the polynomial Daugavetian index of the space A ( K , X ) (Theorem 4.11). The space A ( K , X ) has a bicontractive projection if the Shilov boundary of the base algebra A has isolated points (Proposition 4.9). As a consequence, we will show that A ( K , X ) has the polynomial Daugavet property if and only if the base algebra A or the range space X has the polynomial Daugavet property (Corollary 4.12).

2 Delta-points and Daugavet points in complex Banach spaces

In this section, we study Δ -points and Daugavet points for complex Banach spaces. Although one may find that a certain portion of the proofs are similar to the real case, we include them in this article for completeness. However, we mention that the complex scalar field C provides something more, namely, a tool to analyze the Daugavet property and the DLD2P for complex Banach spaces through rank-one real-linear operators and rank-one real-projections, respectively. We recall the following useful lemma that holds for both real and complex Banach spaces.

Lemma 2.1

[13, Lemma 2.1] Let x * ∈ S X * , ε > 0 . Then, for every x ∈ S ( x * , ε ) and every δ ∈ ( 0 , ε ) , there exists y * ∈ S X * such that x ∈ S ( y * , δ ) and S ( y * , δ ) ⊂ S ( x * , ε ) .

Theorem 2.2

Let X be a complex Banach space and x ∈ S X . Then, the following statements are equivalent:

  1. x is a Δ -point.

  2. For every slice S of B X with x ∈ S and ε > 0 , there exists y ∈ S such that ‖ x − y ‖ ≥ 2 − ε .

  3. For every rank-one projection P = x * ⊗ x with x * x = 1 , we have ‖ I − P ‖ ≥ 2 .

  4. For every rank-one real-projection P = Re x * ⊗ x with x * x = 1 , we have ‖ I − P ‖ ≥ 2 .

  5. For every rank-one linear operator T = x * ‖ x * ‖ ⊗ x with x * x = 1 , we have ‖ I − T ‖ = 2 .

  6. For every rank-one real-linear operator T = Re x * ‖ x * ‖ ⊗ x with x * x = 1 , we have ‖ I − T ‖ = 2 .

Proof

The implication (i) ⇔ (ii) ⇔ (iv) was established in [13, 25]. Hence, we prove the implications (iii) ⇒ (ii) ⇒ (vi) ⇒ (iv), and (vi) ⇒ (v) ⇒ (iii). We additionally provide a different approach to the proof of (i) ⇔ (ii).

(i) ⇒ (ii): Let ε > 0 and S be a slice containing x ∈ S X . If x is a Δ -point on X , then conv ¯ Δ ε ( x ) ∩ S ≠ ∅ . From the fact that the set B X \ S is a closed convex set, we see that Δ ε ( x ) ⊄ B X \ S . Hence, S ∩ Δ ε ( x ) ≠ ∅ and so there exists y ∈ S such that ‖ x − y ‖ ≥ 2 − ε .

(ii) ⇒ (i): Suppose that x ∉ conv ¯ Δ ε ( x ) for some ε > 0 . Note that a singleton { x } is convex as well as conv ¯ Δ ε ( x ) . Moreover, the set { x } is compact. So by Hahn-Banach separation theorem, there exist x * ∈ S X * and α > 0 such that the slice S = S ( x * , α ) contains x and S ∩ conv ¯ Δ ε ( x ) = ∅ . Hence, we see that S ∩ Δ ε ( x ) = ∅ , and so ‖ x − y ‖ < 2 − ε for every y ∈ S .

(iii) ⇒ (ii): Consider a slice S ( x * , δ ) containing x and ε > 0 . Then, there exist δ 1 > 0 such that 2 δ 1 1 − δ 1 < ε 4 and a bounded linear functional y * ∈ S X * such that x ∈ S ( y * , δ 1 ) ⊂ S ( x * , δ ) .

Consider a rank-one projection P : y ↦ y * y x y * x . Then, by Assumption (iii), for every β < ε 2 , there exists y ∈ S X such that

(1) ‖ y − P y ‖ = y − y * y x y * x = γ y − Re y * ( γ y ) x y * x > 2 − β ,

where γ ∈ S C such that ∣ y * y ∣ = γ y * y = y * ( γ y ) = Re y * ( γ y ) .

Moreover, for β small enough, we also see that γ y ∈ S ( y * , δ 1 ) . Let y ˜ = γ y . Then, by (1), we obtain

‖ y ˜ − x ‖ = γ y − Re y * ( γ y ) x y * x + Re y * ( γ y ) x y * x − x ≥ γ y − Re y * ( γ y ) x y * x − x − Re y * ( γ y ) x y * x > 2 − β − 1 − Re y * ( γ y ) y * x .

Since ∣ y * x ∣ ≥ Re y * x > 1 − δ 1 , we can see that

1 − Re y * ( γ y ) y * x = ∣ y * x − Re y * ( γ y ) ∣ ∣ y * x ∣ ≤ ∣ 1 − y * x ∣ + ( 1 − Re y * ( γ y ) ) 1 − δ 1

and

( Im y * x ) 2 = ∣ y * x ∣ 2 − ( Re y * x ) 2 < 1 − ( 1 − δ 1 ) 2 = 2 δ 1 − δ 1 2 .

Hence, we have

∣ 1 − y * x ∣ = ( 1 − Re y * x ) 2 + ( Im y * x ) 2 < ( 1 − Re y * x ) 2 + 2 δ 1 − δ 1 2 < δ 1 2 + 2 δ 1 − δ 1 2 = 2 δ 1 .

These consequently show that

‖ y ˜ − x ‖ > 2 − β − 2 2 δ 1 1 − δ 1 > 2 − ε .

(ii) ⇒ (vi) : Consider a rank-one linear operator T = Re x * ‖ x * ‖ ⊗ x , where x * ∈ X * satisfies x * x = 1 . Let 0 < ε < 1 10 , α ∈ 0 , ε ( 1 − ε ) ‖ x * ‖ , and S = y ∈ B X : Re x * y ‖ x * ‖ > 1 ‖ x * ‖ − α . Then, by the assumption, there exists y ∈ S such that ‖ x − y ‖ ≥ 2 − ε . Moreover, from the fact that y ‖ y ‖ ∈ S and 1 ‖ y ‖ ≤ 1 1 − ε , we have

x − y ‖ y ‖ > ‖ x − y ‖ − 1 ‖ y ‖ − 1 ≥ 2 − ε − ε 1 − ε > 2 − 2 ε 1 − ε .

Now, define a convex function g ( λ ) ≔ λ x ‖ x * ‖ − ( 1 − λ ) y ‖ y ‖ on [ 0 , 1 ] . Note that g ( 0 ) = 1 , g ( 1 ) = 1 ‖ x * ‖ , and

g 1 2 = 1 2 x ‖ x * ‖ − y ‖ y ‖ = 1 2 x ‖ x * ‖ − x + x − y ‖ y ‖ ≥ 1 2 x − y ‖ y ‖ − 1 − 1 ‖ x * ‖ > 1 2 + 1 2 ‖ x * ‖ − ε 1 − ε .

Since g ( λ ) is convex, we see that, for every λ ∈ [ 0 , 1 2 ) ,

g ( 1 ) − g ( 1 ∕ 2 ) 1 ∕ 2 ≥ g ( 1 ∕ 2 ) − g ( λ ) 1 ∕ 2 − λ > g ( 1 ∕ 2 ) − g ( λ ) 1 ∕ 2 .

Hence,

1 2 ‖ x * ‖ − 1 2 + ε 1 − ε > g ( 1 ) − g 1 2 > g 1 2 − g ( λ ) > 1 2 + 1 2 ‖ x * ‖ − ε 1 − ε − g ( λ ) ,

and so g ( λ ) > 1 − 2 ε ( 1 − ε ) for every λ ∈ [ 0 , 1 2 ) . From the continuity of g ( λ ) , g ( λ ) ≥ 1 − 2 ε 1 − ε for every λ ∈ [ 0 , 1 2 ] . Also, for λ ∈ ( 1 2 , 1 ] , we have

g ( 0 ) − g ( 1 ∕ 2 ) λ − 1 ∕ 2 > g ( 0 ) − g ( 1 ∕ 2 ) 1 ∕ 2 ≥ g ( 1 ∕ 2 ) − g ( λ ) λ − 1 ∕ 2 .

Hence, we see that

1 − 1 + 2 ε 1 − ε ≥ g ( 0 ) − g 1 2 > g 1 2 − g ( λ ) ≥ 1 − 2 ε 1 − ε − g ( λ ) ,

and so g ( λ ) > 1 − 4 ε 1 − ε for every λ ∈ ( 1 2 , 1 ] .

Let us consider two cases. First, assume that Re x * y ‖ y ‖ 1 + Re x * y ‖ y ‖ ≤ 1 2 . Since g ( λ ) ≥ 1 − 2 ε ( 1 − ε ) for every λ ∈ [ 0 , 1 2 ] , we obtain

( I − T ) y ‖ y ‖ = 1 + Re x * y ‖ y ‖ ⋅ g Re x * y ‖ y ‖ 1 + Re x * y ‖ y ‖ ≥ 1 + Re x * y ‖ y ‖ 1 − 2 ε 1 − ε > ( 2 − α ‖ x * ‖ ) 1 − 2 ε 1 − ε > 2 − ε 1 − ε 1 − 2 ε 1 − ε = 2 − ε 1 − ε − 4 ε 1 − ε + 2 ε 1 − ε 2 > 2 − 5 ε 1 − ε .

Now, assume that Re x * y ‖ y ‖ 1 + Re x * y ‖ y ‖ > 1 2 . Since g ( λ ) > 1 − 4 ε ( 1 − ε ) for every λ ∈ ( 1 2 , 1 ] , we have

( I − T ) y ‖ y ‖ = 1 + Re x * y ‖ y ‖ ⋅ g Re x * y ‖ y ‖ 1 + Re x * y ‖ y ‖ > 1 + Re x * y ‖ y ‖ 1 − 4 ε 1 − ε > ( 2 − α ‖ x * ‖ ) 1 − 4 ε 1 − ε > 2 − ε 1 − ε 1 − 4 ε 1 − ε = 2 − ε 1 − ε − 8 ε 1 − ε + 4 ε 1 − ε 2 > 2 − 9 ε ( 1 − ε ) .

Therefore, from the fact that ε > 0 is arbitrary in all cases, we have (ii) ⇒ (vi).

(vi) ⇒ (iv): Let T = Re x * ‖ x * ‖ ⊗ x , where x * ∈ X * satisfies x * x = 1 . Define a function φ ( λ ) = ‖ I − λ T ‖ , where λ ∈ [ 0 , ∞ ) . It is easy to show that φ is a convex function on [ 0 , ∞ ) . Also, φ ( 0 ) = 1 and φ ( 1 ) = ‖ I − T ‖ = 2 from the assumption. Furthermore,

0 < φ ( 1 ) − φ ( 0 ) 1 − 0 ≤ φ ( s ) − φ ( 1 ) s − 1 for all s ≥ 1 .

So, for every s ≥ 1 , we see that φ ( s ) > φ ( 1 ) = 2 . In particular, if s = ‖ x * ‖ ≥ 1 , we obtain φ ( ‖ x * ‖ ) = ‖ I − ‖ x * ‖ ⋅ T ‖ = ‖ I − P ‖ ≥ 2 . Therefore, the implication (vi) ⇒ (iv) holds.

(vi) ⇒ (v) : Let T = x * ‖ x * ‖ ⊗ x , where x * ∈ X * satisfies x * x = 1 . By the assumption, for every ε > 0 , there exists y ∈ S X such that y − Re x * y ‖ x * ‖ x ≥ 2 − ε 2 2 . Note that Re x * y ‖ x * ‖ ≥ 1 − ε 2 2 and so Im x * y ‖ x * ‖ ≤ ε .

Then, we have

y − x * y ‖ x * ‖ x ≥ y − Re x * y ‖ x * ‖ x − x * y ‖ x * ‖ x − Re x * y ‖ x * ‖ x ≥ 2 − ε 2 2 − Im x * y ‖ x * ‖ ≥ 2 − ε 2 2 − ε .

This shows (vi) ⇒ (v). The proof of (v) ⇒ (iii) is identical to (vi) ⇒ (iv).□

Hence, we can verify the relationship between the Δ -points and the DLD2P as well as the spaces with bad projections [13] for complex Banach spaces.

Corollary 2.3

Let X be a complex Banach space. The following statements are equivalent:

  1. The space X has the DLD2P.

  2. Every point on the unit sphere S X is a Δ -point.

  3. Every rank-one projection P has ‖ I − P ‖ ≥ 2 , i.e., X is a space with bad projections.

  4. Every rank-one real-projection P has ‖ I − P ‖ ≥ 2 .

The following statement about Daugavet points is a compilation of well-known results, but we include them for completeness.

Theorem 2.4

Let X be a complex Banach space and x ∈ S X . Then, the following statements are equivalent:

  1. x is a Daugavet point.

  2. For every slice S of B X and ε > 0 , there exists y ∈ S such that ‖ x − y ‖ ≥ 2 − ε .

  3. Every rank-one operator T = x * ⊗ x satisfies ‖ I − T ‖ = 1 + ‖ T ‖ .

  4. Every rank-one operator T = x * ⊗ x with norm-one satisfies ‖ I − T ‖ = 2 .

  5. Every rank-one real-linear operator T = Re x * ⊗ x satisfies ‖ I − T ‖ = 1 + ‖ T ‖ .

  6. Every rank-one real-linear operator T = Re x * ⊗ x with norm-one satisfies ‖ I − T ‖ = 2 .

Proof

One may see the proof for the real case in [25, Proposition 1.4.5].

(i) ⇒ (ii): Let S be a slice of B X and ε > 0 . If x is a Daugavet point, then S ⊂ conv ¯ Δ ε ( x ) = B X . By using a similar reasoning as (i) ⇒ (ii) in Theorem 2.2, we see that S ∩ Δ ε ( x ) ≠ ∅ . Therefore, there exists y ∈ S such that ‖ x − y ‖ ≥ 2 − ε .

(ii) ⇒ (i): Suppose that B X ≠ conv ¯ Δ ε ( x ) for some ε > 0 . A singleton { x } is compact and convex, and the set conv ¯ Δ ε ( x ) is also convex. So, by a similar argument as (ii) ⇒ (i) in Theorem 2.2, there exists a slice S such that S ∩ Δ ε ( x ) = ∅ . Therefore, for every y ∈ S , we have ‖ x − y ‖ < 2 − ε .

(iii) ⇒ (iv) is clear, and (iv) ⇒ (iii) comes from the fact that for Daugavet property considering rank-one operators with norm-one is enough [10]. The equivalence (ii) ⇔ (v) ⇔ (vi) is a well-known result in [9].

(iv) ⇒ (ii): Let x * ∈ S X * , ε > 0 , and S = { y ∈ B X : Re x * y > 1 − ε 2 } . Let T ∈ L ( X ) be the rank-one operator with norm one defined by T ( y ) = x * y ⋅ x . By Assumption (iv), there exists y ∈ S X such that ‖ y − T y ‖ = ‖ y − x * y ⋅ x ‖ > 2 − ε 2 . Note that ∣ x * y ∣ = Re x * ( γ y ) for some γ ∈ S C , and so,

1 + Re x * ( γ y ) ≥ ‖ y − x * y ⋅ x ‖ > 2 − ε 2 .

Hence, γ y ∈ S . Let y ˜ = γ y . Then, we have

‖ y ˜ − x ‖ ≥ ‖ y ˜ − Re x * ( γ y ) ⋅ x ‖ − 1 + Re x * ( γ y ) > 2 − ε .

Therefore, we see that (ii) holds.

(ii) ⇒ (iv): Consider T = x * ⊗ x , where ‖ x * ‖ = 1 . Let ε > 0 and S = { y ∈ B X : Re x * y > 1 − ε 2 } be a slice of B X . By Assumption (ii), there exists y ∈ S such that ‖ x − y ‖ > 2 − ε 2 . Note that 1 ≥ ∣ x * y ∣ 2 = ( Re x * y ) 2 + ( Im x * y ) 2 . Hence, we have

∣ 1 − x * y ∣ = ( 1 − Re x * y ) 2 + ( Im x * y ) 2 < ε 2 4 + 1 − 1 − ε 2 2 = ε .

Moreover,

‖ ( I − T ) y ‖ = ‖ y − x * y x ‖ ≥ ‖ y − x ‖ − ‖ x − x * y ⋅ x ‖ > 2 − ε 2 − ∣ 1 − x * y ∣ > 2 − ε 2 − ε > 2 − ε .

Since ε > 0 is arbitrary, we obtain ‖ I − T ‖ ≥ 2 . Then, (iv) holds immediately from the fact that ‖ I − T ‖ ≤ 1 + ‖ T ‖ = 2 .□

Corollary 2.5

Let X be a complex Banach space. Then, the following statements are equivalent:

  1. The space X has the Daugavet property.

  2. Every point on the unit sphere S X is a Daugavet point.

Now, we make a similar observation on rank-one, norm-one projections.

Proposition 2.6

Let X be a complex Banach space and suppose that there are x * ∈ X * and x ∈ X such that x * x = 1 = ‖ x * ‖ = ‖ x ‖ . Then, the rank-one projection P = x * ⊗ x on X satisfies ‖ I − P ‖ = 2 , if and only if the rank-one real-projection P ′ = Re x * ⊗ x on X satisfies ‖ I − P ′ ‖ = 2 .

Proof

Let P ′ = Re x * ⊗ x . Then, for every ε > 0 , there exists y ∈ S X such that ‖ y − x * y ⋅ x ‖ ≥ 2 − ε . Now, take γ ∈ S C such that ∣ x * y ∣ = γ x * y . This implies that x * ( γ y ) = Re x * ( γ y ) . Hence, we see that

‖ I − P ‖ ≥ ‖ γ y − Re x * ( γ y ) ⋅ x ‖ = ‖ γ ( y − x * y ⋅ x ) ‖ = ‖ y − x * y ⋅ x ‖ ≥ 2 − ε .

Since ε > 0 is arbitrary, we obtain ‖ I − P ‖ = 2 .

Let P = x * ⊗ x . Then, we see that for every ε > 0 , there exists y ∈ S X such that ‖ y − Re x * y ⋅ x ‖ ≥ 2 − ε 2 . Then, we have ∣ Re x * y ∣ ≥ 1 − ε 2 . Note that

( Im x * y ) 2 = ∣ x * y ∣ 2 − ( Re x * y ) 2 ≤ 1 − 1 − ε 2 2 < ε .

Hence, we obtain

‖ I − P ‖ ≥ ‖ y − x * y ⋅ x ‖ ≥ ‖ y − Re x * y ⋅ x ‖ − ∣ Im x * y ∣ > 2 − 2 ε .

Therefore, since ε > 0 is arbitrary, we obtain ‖ I − P ‖ = 2 .□

3 Alternative convexity or smoothness, nonsquareness, and the Daugavet property

In this section, we study the relationship between alternative convexity or smoothness, nonsquareness, and the Daugavet property. First, we recall various nonsquareness properties, in the sense of James [19,20]. The uniform nonsquareness has been examined on both real and complex Banach spaces via Jordan-von Neumann constants [26].

Definition 3.1

  1. A Banach space X is UNSQ, if there exists δ > 0 such that for every x , y ∈ S X , min { ‖ x ± y ‖ } ≤ 2 − δ .

  2. A Banach space X is LUNSQ, if for every x ∈ S X , there exists δ > 0 such that min { ‖ x ± y ‖ } ≤ 2 − δ for every y ∈ S X .

  3. A Banach space X is nonsquare (NSQ), if for every x , y ∈ S X , min { ‖ x ± y ‖ } < 2 .

Here we call each point x ∈ S X in (ii) a LUNSQ point (or uniformly non- ℓ 1 2 point). We have the following implication for these classes:

UNSQ ⇒ LUNSQ ⇒ NSQ .

It has been recently shown that UNSQ real Banach spaces do not have Δ -points [17] at all. We extend this result for the class of LUNSQ Banach spaces on the complex scalar field. Let us start with an improvement of Theorem 2.2. For the proof on the real case, we refer to [5, Lemma 2.1].

Lemma 3.2

Let X be a complex Banach space and x ∈ S X be a Δ -point. For every ε > 0 , α 1 − α < ε , and every slice S = S ( x * , α ) containing x, there exists a slice S ( z * , α 1 ) of B X such that S ( z * , α 1 ) ⊂ S ( x * , α ) and ‖ x − y ‖ > 2 − ε for all y ∈ S ( z * , α 1 ) .

Proof

First, choose η > 0 such that η < min 1 − 1 − α Re x * ( x ) , ε − α 1 − α . Since x ∈ S X is a Δ -point, for the projection P ( y ) = Re x * y Re x * x ⋅ x , we have ‖ I − P ‖ ≥ 2 by Theorem 2.2. Then, there exists y * ∈ S X * such that ‖ y * − P * y * ‖ ≥ 2 − η . Now, define z * = P * y * − y * ‖ P * y * − y * ‖ ∈ S X * and α 1 = 1 − 2 − η ‖ P * y * − y * ‖ , where P * y * = y * x Re x * x ⋅ Re x * . For every y ∈ S ( z * , α 1 ) , note that

Re x * y Re x * x ⋅ Re y * x − Re y * y = Re z * y ⋅ ‖ P * y * − y * ‖ > 2 − η .

Hence, we see that Re x * y Re x * x ⋅ Re y * x > 1 − η . Since Re y * x cannot be zero, without loss of generality, assume that Re y * x > 0 . Then, we have Re x * y > ( 1 − η ) ⋅ Re x * x > 1 − α , which shows that S ( z * , α 1 ) ⊂ S ( x * , α ) . Furthermore, note that Re x * y ≤ ∣ x * y ∣ ≤ 1 , and so,

x Re x * x − y ≥ Re y * x Re x * x − Re y * y ≥ Re x * y Re x * x ⋅ Re y * x − Re y * y > 2 − η .

Therefore, we obtain

□ ‖ x − y ‖ ≥ x Re x * x − y − 1 Re x * x − 1 > ( 2 − η ) − α 1 − α > 2 − ε .

Applying Lemma 2.1 again, we can also show the converse of the previous statement. The same proof in [5, Lemma 2.2] transfers to complex Banach spaces.

Corollary 3.3

Let X be a complex Banach space. Then, x ∈ S X is a Δ -point, if and only if for every ε > 0 and every slice S = S ( x * , α ) containing x , there exists a slice S ( z * , α 1 ) such that S ( z * , α 1 ) ⊂ S ( x * , α ) and  ‖ x − y ‖ > 2 − ε for all y ∈ S ( z * , α 1 ) .

As a consequence, we obtain the relationship between the LUNSQ points and the Δ -points on both real and complex Banach spaces. The real case was shown in [18, Theorem 4.2].

Proposition 3.4

Let X be a real or complex Banach space. A LUNSQ point x ∈ S X is not a Δ -point of X.

Proof

We show that a Δ -point x ∈ S X cannot be a LUNSQ point. Let ε > 0 and η ∈ ( 0 , ε 2 ) . By Lemma 3.2, for every α > 0 where α 1 − α < η and every slice S = S ( x * , α ) containing x , there exists a slice S ( z * , α 1 ) ⊂ S such that ‖ x − y ‖ > 2 − η > 2 − ε for all y ∈ S ( z * , α 1 ) . In particular, we have Re z * y ‖ y ‖ > 1 − α 1 ‖ y ‖ ≥ 1 − α 1 for every y ∈ S ( z * , α 1 ) . Hence, y ′ = y ‖ y ‖ ∈ S ( z * , α 1 ) and ‖ x − y ′ ‖ > 2 − ε for y ∈ S ( z * , α 1 ) .

Moreover, by the fact that α < α 1 − α < η and x , y ∈ S , we have ‖ x + y ′ ‖ ≥ Re x * x + Re x * y ′ > 2 − 2 α > 2 − 2 η > 2 − ε . Thus, for every ε > 0 there exists y ′ ∈ S X such that min { ‖ x + y ′ ‖ , ‖ x − y ′ ‖ } > 2 − ε . This shows that x ∈ S X is not a LUNSQ point.□

Corollary 3.5

Let X be a real or complex Banach space. If X is LUNSQ, then the space does not admit Δ -points. As a consequence, every LUNSQ space does not satisfy the Daugavet property, DD2P, and DLD2P.

A Banach space X is said to have the anti-Daugavet property for a class of operators ℳ if the following equivalence holds:

‖ I + T ‖ = 1 + ‖ T ‖ ⇔ ‖ T ‖ ∈ σ ( T ) ,

where σ ( T ) is the spectrum of T ∈ ℳ . If ℳ = L ( X ) , we simply say the space X satisfies the anti-Daugavet property. We mention that only if the part always holds for any bounded linear operator.

It is well-known that any uniformly rotund or uniformly smooth Banach spaces have the anti-Daugavet property. Moreover, this property is connected to the alternative convexity or smoothness properties that are introduced in [8].

Definition 3.6

  1.   A Banach space X is uniformly alternatively convex or smooth (uacs), if for all sequences ( x n ) , ( y n ) ⊂ S X , ( x n * ) ⊂ S X * , ‖ x n + y n ‖ → 2 , and x n * ( x n ) → 1 implies x n * ( y n ) → 1 .

  2.  A Banach space X is strongly locally uniformly alternatively convex or smooth (sluacs), if for every x ∈ S X , ( x n ) ⊂ S X , ( x n * ) ⊂ S X * , ‖ x n + x ‖ → 2 , and x n * ( x n ) → 1 implies x n * ( x ) → 1 .

  3.   A Banach space X is alternatively convex or smooth (acs), if for all x , y ∈ S X , x * ∈ S X * , ‖ x + y ‖ = 2 , and x * ( x ) = 1 implies x * ( y ) = 1 .

Any uniformly convex (resp. LUR, rotund) and uniformly smooth (resp. uniformly Gateaux-smooth, smooth) Banach spaces are known to be uacs (resp. sluacs, acs) [27].

Even though it is mentioned in [8] that alternative convexity or smoothness for complex Banach spaces can be defined in a similar fashion, any recent investigation on this property also assumes the scalar field to be R . Hence, we provide equivalent definitions that only involve the real part of bounded linear functionals which enable us to consider complex Banach spaces.

Proposition 3.7

  1.   A Banach space X is uacs if and only if for all sequence ( x n ) , ( y n ) ⊂ S X , ( x n * ) ⊂ S X * , ‖ x n + y n ‖ → 2 , and Re x n * x n → 1 implies Re x n * y n → 1 .

  2.   A Banach space X is sluacs if and only if for every x ∈ S X , ( x n ) ⊂ S X , ( x n * ) ⊂ S X * , ‖ x n + x ‖ → 2 , and Re x n * x n → 1 implies Re x n * x → 1 .

  3.   A Banach space X is acs if and only if for all x , y ∈ S X and x * ∈ S X * , ‖ x + y ‖ = 2 , and Re x * ( x ) = 1 implies Re x * ( y ) = 1 .

Proof

We assume that X is a complex Banach space. Since the proofs for (i) and (ii) are similar, we only prove (i).

Suppose that ( x n ) , ( y n ) ⊂ S X , ( x n * ) ⊂ S X * , x n * x n → 1 and ‖ x n + y n ‖ → 2 . Then, for every ε > 0 , there exists N 1 ∈ N such that for every n ≥ N 1 , ∣ x n * x n ∣ > 1 − ε 2 . We see that

1 ≥ ( Re x n * x n ) 2 + ( Im x n * x n ) 2 = ∣ x n * x n ∣ 2 > 1 − ε 2 2 + ( Im x n * x n ) 2 .

Hence, Im x n * x n < ε , which in turn implies that Re x n * x n → 1 . Then, by the assumption, we obtain Re x n * y n → 1 . Thus, for every ε > 0 , there exists N 2 ∈ N such that Re x n * y n > 1 − ε 2 . This implies that

1 ≥ ∣ x n * y n ∣ 2 = ( Re x n * y n ) 2 + ( Im x n * y n ) 2 ≥ 1 − ε 2 2 + ( Im x n * y n ) 2 ,

and so Im x n * y n < ε . Therefore, we see that x n * y n = Re x n * y n + i Im x n * y n → 1 .

For (iii), if for x , y ∈ S X and x * ∈ S X * we have ‖ x + y ‖ = 2 and x * x = Re x * x = 1 , then Re x * y = 1 by the assumption. Hence, we see that

1 ≥ ∣ x * y ∣ 2 = ( Re x * y ) 2 + ( Im x * y ) 2 = 1 + ( Im x * y ) 2 .

Therefore, x * y = Re x * y = 1 .□

Even though every uacs Banach spaces are UNSQ [8,27], there has been no explicit description on the relationship between the sluacs and LUNSQ (resp. acs and NSQ) Banach spaces. As a matter of fact, a similar statement also holds for both sluacs and acs Banach spaces.

Proposition 3.8

  1. Every sluacs space is LUNSQ.

  2. Every acs space is NSQ.

Proof

(i) Suppose that a sluacs space X is not LUNSQ. Then, there exists x ∈ S X such that for every δ > 0 , there exists y ∈ S X such that ‖ x + y ‖ > 2 − δ and ‖ x − y ‖ > 2 − δ . So choose a sequence ( x n ) n = 1 ∞ ⊂ S X such that ‖ x + x n ‖ > 2 − 1 2 n and ‖ x − x n ‖ > 2 − 1 2 n . In view of the Hahn-Banach theorem, we can also find a sequence ( x n * ) n = 1 ∞ ⊂ S X * such that

Re x n * x n − Re x n * x = ‖ x − x n ‖ and Re x n * x n + Re x n * x = ‖ x + x n ‖ .

Then, we see that

2 − 1 2 n < Re x n * x n + Re x n * x ≤ ‖ x ‖ + Re x n * x n = 1 + Re x n * x n ,

and so Re x n * x n → 1 as n → ∞ . Since the space X is assumed to be sluacs, Re x n * x → 1 as n → ∞ . However, by repeating the same argument to ‖ x − x n ‖ , we also obtain that − Re x n * x → 1 . This leads to a contradiction.

(ii) Suppose that an acs space X is not nonsquare. Then, there exist x , y ∈ S X such that ‖ x + y ‖ = ‖ x − y ‖ = 2 . Let x * ∈ S X * such that Re x * ( x ) + Re x * ( y ) = ‖ x + y ‖ . From the fact that

2 = Re x * ( x ) + Re x * ( y ) ≤ Re x * ( x ) + ‖ y ‖ = Re x * ( x ) + 1 ,

we have Re x * x = 1 . This also shows that Re x * y = 1 . However, since ‖ x − y ‖ = 2 and the space X is acs, we have − Re x * x = 1 , which is a contradiction. Therefore, the space X must be nonsquare.□

We mention that any locally uniformly rotund (LUR) Banach space does not have Δ -points. As a matter of fact, we can show further that every sluacs Banach space does not have Δ -points based on our observation.

Corollary 3.9

Let X be a Banach space. Every sluacs Banach space does not contain Δ -points.

Proof

Every sluacs Banach space is LUNSQ by Proposition 3.8.(i). Then, by Proposition 3.4, we see that the space does not contain Δ -points.□

We mention that the analogous statement for NSQ Banach spaces cannot be obtained because there exists a rotund Banach space with the DLD2P [28, Theorem 2.12]. This fact also leads us to ask whether rotund or NSQ Banach spaces can have the Daugavet property, which has been a long standing open problem for decades. While there is a rotund normed space (not complete) with the Daugavet property [29], the existence has not been verified for Banach spaces.

4 Remarks on the Daugavet property of A ( K , X )

Let K be a compact Hausdorff space. The space C ( K ) is the set of all continuous functions over K endowed with the supremum norm ‖ ⋅ ‖ ∞ . A uniform algebra A is a closed subalgebra of C ( K ) that separates points and contains constant functions. For a compact subset K ⊂ C , the space P ( K ) (resp. R ( K ) ) of continuous functions that can be approximated uniformly on K by polynomials in z (resp. by rational functions with poles off K ) and the space A ( K ) of continuous functions that are analytic on the interior of K are well-known examples of uniform algebras. When K = D ¯ , the corresponding uniform algebra A ( K ) = A ( D ¯ ) is the disk algebra. We refer to [30,31] for more details on uniform algebras.

For a complex Banach space X , let C ( K , X ) be the set of all vector-valued continuous functions over K equipped with the supremum norm. We recall the definition of the vector-valued function space A ( K , X ) .

Definition 4.1

Let K be a compact Hausdorff space and X be a Banach space. The space A ( K , X ) is called a function space over the base algebra A if it is a subspace of C ( K , X ) that satisfies:

  1. The base algebra A ≔ { x * ∘ f : x * ∈ X * , f ∈ A ( K , X ) } is a uniform algebra.

  2. A ⊗ X = { f ⊗ x : f ∈ A and x ∈ X } ⊂ A ( K , X ) .

  3. For every g ∈ A and every f ∈ A ( K , X ) , we have g ⋅ f ∈ A ( K , X ) .

If X = F , then the space A ( K , X ) becomes the uniform algebra A on a compact Hausdorff space K . It is clear that C ( K , X ) is a function space over a base algebra C ( K ) . As a nontrivial example, for given Banach spaces X and Y , let A w * u ( B X * , Y ) be the space of all weak * -to-norm uniformly continuous functions on the closed unit ball B X * that are holomorphic on the interior of B X * . It is a closed subspace of C ( B X * ; Y ) , where B X * is the weak * -compact set. Then, A w * u ( B X * ; Y ) is a function space over base algebra A w * u ( B X * ) .

A subset L ⊂ K is said to be a boundary for A if for every f ∈ A , there exists t ∈ L such that f ( t ) = ‖ f ‖ ∞ . The smallest closed boundary for A is called the Shilov boundary denoted by Γ . A point x ∈ K is a strong boundary point for a uniform algebra A if for every open subset U ⊂ K containing x , there exists f ∈ A such that ‖ f ‖ ∞ = ∣ f ( x ) ∣ = 1 and sup t ∈ K \ U ∣ f ( t ) ∣ < 1 . For a compact Hausdorff space K , the set of all strong boundary points on A coincides with the Choquet boundary Γ 0 , i.e., the set of all extreme points on the set K A = { λ ∈ A * : ‖ λ ‖ = λ ( 1 A ) = 1 } [30, Theorem 4.3.5], where 1 A is the unit of A . Moreover, the closure of Γ 0 is Γ in this case [30, Corollary 4.3.7.a]. For instance, the Shilov boundary of the disk algebra A ( D ¯ ) is the unit circle ∂ D ¯ .

To study various geometrical properties of A ( K , X ) and Bishop-Phelps-Bollobás property for Asplund operators whose range space is a uniform algebra, a Urysohn-type lemma has played an important role. Here we use a stronger version of the lemma provided in [32].

Lemma 4.2

[24, Lemma 3.10] Let K be a compact Hausdorff space. If t 0 is a strong boundary point for a uniform algebra A ⊂ C ( K ) , then for every open subset U ⊂ K containing t 0 and ε > 0 , there exists ϕ = ϕ U ∈ A such that ϕ ( t 0 ) = ‖ ϕ ‖ ∞ = 1 , sup K \ U ∣ ϕ ( t ) ∣ < ε and

∣ ϕ ( t ) ∣ + ( 1 − ε ) ∣ 1 − ϕ ( t ) ∣ ≤ 1

for every t ∈ K .

We can also construct a Urysohn-type function at an isolated point in the Shilov boundary.

Lemma 4.3

[2, Lemma 2.5] Let A be a uniform algebra on a compact Hausdorff space K and let t 0 be an isolated point of the Shilov boundary Γ of A. Then there exists a function ϕ ∈ A such that ϕ ( t 0 ) = ‖ ϕ ‖ ∞ = 1 and ϕ ( t ) = 0 for t ∈ Γ \ { t 0 } .

We recall the following lemma that will be useful for later.

Lemma 4.4

[2, Lemma 2.4] Let X be a Banach space. Suppose that L is a closed boundary for A. The space of restrictions of elements of A ( K , X ) to L is denoted by A ( L , X ) and the restrictions of elements of A to L is denoted by A ( L ) . Then, A ( L , X ) is isometrically isomorphic to A ( K , X ) .

The next statement is in the proof of the case (iii) for [2, Theorem 4.2], but we state it explicitly here.

Lemma 4.5

Let K be a compact Hausdorff space and Γ be the Shilov boundary of the base algebra for the space A ( K , X ) . Suppose that Γ has an isolated point t 0 . Then, A ( K , X ) is isometrically isomorphic to X ⊕ ∞ Y where Y is A ( Γ , X ) restricted to Γ \ { t 0 } .

Proof

Let t 0 ∈ Γ be an isolated point. By Lemma 4.3, there exists ϕ ∈ A such that ϕ ( t 0 ) = ‖ ϕ ‖ ∞ = 1 and ϕ ( t ) = 0 for t ∈ Γ \ { t 0 } . Let Γ ˜ = Γ \ { t 0 } . Define a norm-one linear operator P : A ( Γ , X ) → A ( Γ , X ) by P f = ϕ ⋅ f and denote Y as the restriction of A ( Γ , X ) to Γ ˜ . The operator P is a projection because P f ( t ) = 0 for all t ≠ t 0 and P 2 f ( t 0 ) = f ( t 0 ) = P f ( t 0 ) . As a matter of fact, the image P ( A ( Γ , X ) ) is isometrically isomorphic to X . Indeed, define a linear operator Ψ : P ( A ( Γ , X ) ) → X by Ψ ( P f ) = f ( t 0 ) . Then, ‖ Ψ ( P f ) ‖ X = ‖ f ( t 0 ) ‖ X = ‖ P f ‖ . Moreover, we see that for every x ∈ X , there exists f ∈ A ( Γ , X ) such that f ( t 0 ) = x . Hence, Ψ is surjective, which in turn implies that the operator Ψ is an isometric isomorphism on P ( A ( Γ , X ) ) .

Now, we claim that the space A ( Γ , X ) is isometrically isomorphic to X ⊕ ∞ Y . For f ∈ A ( Γ , X ) , define a bounded linear operator Φ : A ( Γ , X ) → X ⊕ ∞ Y by Φ f = ( P f , f ∣ Γ ˜ ) . Then, we see that

‖ Φ f ‖ = max { ‖ P f ‖ , ‖ f ∣ Γ ˜ ‖ } = max { ‖ f ( t 0 ) ‖ X , sup t ∈ Γ ˜ ‖ f ( t ) ‖ X } = ‖ f ‖ .

Note that for a given ( f , g ) ∈ X ⊕ ∞ Y , there exist f 1 , f 2 ∈ A ( Γ , X ) such that f = P f 1 and g = f 2 ∣ Γ ˜ . Let h = P f 1 + f 2 − P f 2 ∈ A ( Γ , X ) . From the fact that h ∣ Γ ˜ = f 2 ∣ Γ ˜ , we have Φ ( h ) = ( P f 1 , f 2 ∣ Γ ˜ ) . Hence, the operator Φ is also surjective, and so it is an isometric isomorphism between A ( Γ , X ) and X ⊕ ∞ Y . Consequently, in view of Lemma 4.4, the space A ( K , X ) is isometrically isomorphic to X ⊕ ∞ Y .□

4.1 Polynomial Daugavet property in A ( K , X )

First we provide a sufficient condition for A ( K , X ) to have the polynomial Daugavet property. We mention that the proof method is inspired by [22, Theorem 2.7].

Theorem 4.6

Let K be a compact Hausdorff space and let Γ be the Shilov boundary of the base algebra A of A ( K , X ) . If Γ does not have isolated points, then A ( K , X ) has the polynomial Daugavet property.

Proof

In view of Theorem 1.6, it suffices to show that for every p ∈ P ( A ( K , X ) ) with ‖ p ‖ = 1 , every f 0 ∈ S A ( K , X ) , and every ε > 0 , there exist α ∈ S C and g ∈ B A ( K , X ) such that

Re α p ( g ) > 1 − ε and ‖ f 0 + α g ‖ > 2 − ε .

Let 0 < ε < 1 , P ∈ P ( A ( K , X ) ) with ‖ P ‖ = 1 , and f 0 ∈ S A ( K , X ) . Choose h ∈ S A ( K , X ) and α ∈ S C such that ∣ P ( h ) ∣ > 1 − ε 2 and Re α P ( h ) > 1 − ε 2 . Also choose t 0 ∈ Γ 0 such that ‖ f 0 ( t 0 ) ‖ X > 1 − ε 8 . Let U = { t ∈ K : ‖ f 0 ( t ) − f 0 ( t 0 ) ‖ X < ε 8 and ‖ h ( t ) − h ( t 0 ) ‖ X < ε 8 } be a nonempty open subset of K . We consider two cases.

Case 1: Suppose that there exists ( t i ) i = 1 ∞ ⊂ U such that ‖ α − 1 f 0 ( t i ) − h ( t i ) ‖ X → 0 . Then, we have

‖ f 0 + α h ‖ ≥ sup i ‖ f 0 ( t i ) + α h ( t i ) ‖ X ≥ sup i ( 2 ‖ f 0 ( t 0 ) ‖ X − 2 ‖ f 0 ( t 0 ) − f 0 ( t i ) ‖ X − ‖ f 0 ( t i ) − α h ( t i ) ‖ X ) ≥ 2 − ε 4 − ε 4 − ε 4 > 2 − ε ,

and we are done with g = h .

Case 2: Now suppose that there exists η > 0 such that ‖ α − 1 f 0 ( t ) − h ( t ) ‖ X > η for every t ∈ U . Since Γ is perfect, we see that the strong boundary point t 0 ∈ U is not an isolated point. Let { U i } i = 1 ∞ be a collection of pairwise disjoint open subsets of U such that ∪ i = 1 ∞ U i ⊂ U and U i ∩ Γ ≠ ∅ . From the fact that the Choquet boundary Γ 0 is dense in Γ , there exist strong boundary points t i ∈ U i for each i ∈ N . Then, by Lemma 4.2, there exists ϕ i ∈ A such that

(2) ϕ i ( t i ) = 1 , sup K \ U i ∣ ϕ i ( t ) ∣ < ε 2 i + 3 , and ∣ ϕ i ( t ) ∣ + 1 − ε 2 i + 3 ∣ 1 − ϕ i ( t ) ∣ ≤ 1 for every t ∈ K .

Let h i = h + ϕ i ( α − 1 f 0 ( t i ) − h ( t i ) ) ∈ A ( K , X ) . Then, for every t ∈ ∪ i = 1 ∞ U i , by (2), we have

‖ h i ( t ) ‖ X = ‖ h ( t ) + ϕ i ( t ) α − 1 f 0 ( t i ) − ϕ i ( t ) h ( t i ) ‖ X ≤ ‖ h ( t ) − h ( t i ) ‖ X + ‖ h ( t i ) − ϕ i ( t ) h ( t i ) ‖ X + ‖ ϕ i ( t ) α − 1 f 0 ( t i ) ‖ X ≤ ‖ h ( t ) − h ( t 0 ) ‖ X + ‖ h ( t 0 ) − h ( t i ) ‖ X + ∣ 1 − ϕ i ( t ) ∣ + ∣ ϕ i ( t ) ∣ ≤ ε 4 + 1 − ε 2 i + 3 ∣ 1 − ϕ ( t ) ∣ + ε 2 i + 3 ∣ 1 − ϕ i ( t ) ∣ + ∣ ϕ i ( t ) ∣ ≤ ε 4 + 1 + ε 2 i + 2 < 1 + ε 2 .

On the other hand, for every t ∈ K \ ∪ i = 1 ∞ U i ,

(3) ‖ h i ( t ) ‖ X ≤ ‖ h ( t ) ‖ X + ∣ ϕ i ( t ) ∣ ‖ α − 1 f 0 ( t i ) − h ( t i ) ‖ X ≤ 1 + ε 2 i + 3 ⋅ 2 < 1 + ε 2 .

Moreover, we see that

(4) ‖ h i ‖ ≥ ‖ h i ( t i ) ‖ X = ‖ f 0 ( t i ) ‖ X ≥ ‖ f 0 ( t 0 ) ‖ − ‖ f 0 ( t 0 ) − f 0 ( t i ) ‖ ≥ 1 − ε 2 .

Now, let g i = h i ‖ h i ‖ . By (4), we obtain

‖ h i − g i ‖ = ∣ 1 − ‖ h i ‖ ∣ < ε 2 .

For every ( β i ) ∈ ℓ ∞ , note that

sup n ∑ i = 1 n β i ϕ i ( α − 1 f 0 ( t i ) − h ( t i ) ) X ≤ sup n sup t ∈ K ∑ i = 1 n ∣ β i ∣ ∣ ϕ i ( t ) ∣ ‖ α − 1 f 0 ( t i ) − h ( t i ) ‖ X ≤ sup n sup t ∈ K ∑ i = 1 n 2 ∣ β i ∣ ∣ ϕ i ( t ) ∣ ≤ 2 sup i ∣ β i ∣ 1 + ε 2 4 + ε 2 5 + ⋯ = 2 1 + ε 8 sup i ∣ β i ∣ .

Hence, by [33, Theorem V.6], the series ∑ i = 1 ∞ β i ϕ i ( α − 1 f 0 ( t i ) − h ( t i ) ) is weakly unconditionally Cauchy. Since we assumed that ‖ α − 1 f 0 ( t ) − h ( t ) ‖ X > η , there exists a basic subsequence ϕ σ ( i ) ( α − 1 f 0 ( t σ ( i ) ) − h ( t σ ( i ) ) ) that is equivalent to the basis ( e i ) in c 0 by using the Bessaga-Pełczynski principle [33, p. 45]. From the fact that a polynomial on a bounded subset of c 0 is weakly continuous [34, Proposition 1.59], we have Re α P ( h σ ( i ) ) → Re α P ( h ) as i → ∞ .

Choose k ∈ N such that Re α P ( h k ) > 1 − ε 2 . Then, we have

Re α P ( g k ) = Re α P ( h k ) ‖ h k ‖ > 1 − ε ∕ 2 1 + ε ∕ 2 ≥ 1 − ε .

Therefore, we finally obtain

‖ f 0 + α g k ‖ ≥ ‖ f 0 + α h k ‖ − ‖ g k − h k ‖ ≥ ‖ f 0 ( t k ) + α h k ( t k ) ‖ X − ε 2 = 2 ‖ f 0 ( t k ) ‖ X − ε 2 ≥ 2 ‖ f 0 ( t 0 ) ‖ X − 2 ‖ f 0 ( t k ) − f 0 ( t 0 ) ‖ X − ε 2 ≥ 2 ‖ f 0 ( t 0 ) ‖ X − 3 ε 4 > 2 − ε ,

and we are done with g = g k .□

Let P K ( X , X ) be the set of all compact polynomials from X to itself. For P ∈ P K ( X , X ) , the numerical range V ( P ) is defined by

V ( P ) = { x * ( P x ) : x * ∈ S X * and x ∈ S X where x * ( x ) = 1 } .

Now, we recall the polynomial Daugavetian index.

Definition 4.7

[35] For an infinite-dimensional complex Banach space X , the polynomial Daugavetian index Daug p ( X ) is defined by

Daug p ( X ) = max { m ≥ 0 : ‖ I + P ‖ ≥ 1 + m ‖ P ‖ , for every P ∈ P K ( X , X ) } = inf { ω ( P ) : P ∈ P K ( X , X ) , ‖ P ‖ = 1 } ,

where ω ( P ) = sup Re V ( P ) .

It is well-known that Daug p ( X ) ∈ [ 0 , 1 ] and Daug p ( X ) ≤ Daug ( X ) , where Daug ( X ) is the Daugavetian index introduced in [36] considering compact linear operators. A complex Banach space X has the polynomial Daugavet property, if and only if Daug p ( X ) = 1 . This comes from the fact that a Banach space X satisfies the Daugavet equation for every rank-one polynomials, if and only if X satisfies the same equation for every weakly compact polynomials (Theorem 1.6). We recall the following lemma that will be useful later.

Lemma 4.8

[35, Proposition 2.2, 2.3] Let { X λ } λ ∈ Λ be a family of infinite-dimensional complex Banach spaces and let Z be the c 0 - or ℓ ∞ -sum of the family. Then,

Daug p ( Z ) = inf { Daug p ( X λ ) : λ ∈ Λ } .

If there exists a finite-rank projection on X such that ‖ P ‖ = ‖ I − P ‖ = 1 , then Daug ( X ) = 0 [36]. Hence, Daug p ( X ) = 0 in this case. Examples of such spaces are C ( K ) where K has isolated points and Banach spaces X with one unconditional basis [36, p. 635]. Similar to the space C ( K ) , we can also construct such a projection for uniform algebras.

Proposition 4.9

Let A be a uniform algebra on a compact Hausdorff space K and let t 0 be an isolated point of the Shilov boundary Γ of A. Then, there exists a projection P : A → A such that ‖ P ‖ = ‖ I − P ‖ = 1 .

Proof

As shown in the proof of Lemma 4.5 with X = C , the projection P : A → A defined by P f = ϕ ⋅ f has norm one. For every f ∈ S A , note that ‖ f − P f ‖ ∞ = sup t ∈ Γ \ { t 0 } ∣ f ( t ) ∣ ≤ 1 .

If we choose a strong boundary point t 1 ∈ Γ \ { t 0 } , there exists g ∈ A such that ‖ g ‖ ∞ = g ( t 1 ) = 1 , and so we have ‖ g − P g ‖ ∞ = sup t ∈ Γ \ { t 0 } ∣ g ( t ) ∣ = 1 . Therefore, ‖ P ‖ = ‖ I − P ‖ = 1 .□

Corollary 4.10

Let K be a compact Hausdorff space and let A be a uniform algebra on K. If the Shilov boundary of A contains an isolated point, then Daug p ( A ) = 0 .

Proof

This is an immediate consequence of Proposition 4.9.□

The following result is inspired by [35, Proposition 2.4].

Theorem 4.11

Let X be a complex Banach space and let K be a compact Hausdorff space. Then,

Daug p ( A ( K , X ) ) = max { Daug p ( A ) , Daug p ( X ) } .

Proof

Let P ∈ P K ( A ( K , X ) ; A ( K , X ) ) . We first show that

‖ I + P ‖ ≥ 1 + Daug p ( X ) ‖ P ‖ .

For a given ε > 0 , there exists f 0 ∈ S A ( K , X ) and t 0 ∈ Γ 0 such that ‖ P ( f 0 ) ( t 0 ) ‖ X ≥ ‖ P ‖ − ε 2 . Since P is continuous at f 0 , there exists δ > 0 such that

(5) ‖ f 0 − g ‖ < δ implies ‖ P ( f 0 ) − P ( g ) ‖ < ε 2 .

Now, consider U = { t ∈ K : ‖ f 0 ( t ) − f 0 ( t 0 ) ‖ X < δ 4 } . Since the set U is a nonempty open subset of K that contains the strong boundary point t 0 , by Lemma 4.2, there exists ϕ ∈ A such that

(6) ϕ ( t 0 ) = 1 , sup K \ U ∣ ϕ ( t ) ∣ < δ 8 , and ∣ ϕ ( t ) ∣ + 1 − δ 8 ∣ 1 − ϕ ( t ) ∣ ≤ 1 for every t ∈ K .

Fix x 0 ∈ S X such that f 0 ( t 0 ) = ‖ f 0 ( t 0 ) ‖ X ⋅ x 0 and define Ψ : C → A ( K , X ) by

Ψ ( z ) = 1 − δ 8 ( 1 − ϕ ) f 0 + ϕ ⋅ x 0 ⋅ z .

Then, we have

Ψ ( ‖ f 0 ( t 0 ) ‖ X ) ( t ) − f 0 ( t ) = 1 − δ 8 ( 1 − ϕ ( t ) ) f 0 ( t ) + ϕ ( t ) f 0 ( t 0 ) − f 0 ( t ) = ϕ ( t ) ( f 0 ( t 0 ) − f 0 ( t ) ) − δ ( 1 − ϕ ( t ) ) f 0 ( t ) 8 .

In view of (6), note that

ϕ ( t ) ( f 0 ( t 0 ) − f 0 ( t ) ) − δ ( 1 − ϕ ( t ) ) f 0 ( t ) 8 X ≤ ‖ ϕ ‖ ∞ ⋅ ‖ ( f 0 ( t 0 ) − f 0 ( t ) ) ‖ X + δ 4 < δ 2

for every t ∈ U and that

ϕ ( t ) ( f 0 ( t 0 ) − f 0 ( t ) ) − δ ( 1 − ϕ ( t ) ) f 0 ( t ) 8 X < 2 ⋅ δ 4 = δ 2

for every t ∈ K \ U . Hence, we can see that ‖ Ψ ( ‖ f 0 ( t 0 ) ‖ X ) − f 0 ‖ < δ , and so,

‖ P ( Ψ ( ‖ f 0 ( t 0 ) ‖ X ) ) ( t 0 ) − P ( f 0 ) ( t 0 ) ‖ X ≤ ‖ P ( Ψ ( ‖ f 0 ( t 0 ) ‖ X ) ) − P ( f 0 ) ‖ < ε 2

by (5). This implies that

‖ P ( Ψ ( ‖ f 0 ( t 0 ) ‖ X ) ) ( t 0 ) ‖ X > ‖ P ( f 0 ) ( t 0 ) ‖ X − ε 2 > ‖ P ‖ − 2 ⋅ ε 2 = ‖ P ‖ − ε .

In view of Hahn-Banach theorem, there exists x 0 * ∈ S X * such that

x 0 * ( P ( Ψ ( ‖ f 0 ( t 0 ) ‖ X ) ) ( t 0 ) ) = ‖ P ( Ψ ( ‖ f 0 ( t 0 ) ‖ X ) ) ( t 0 ) ‖ X > ‖ P ‖ − ε .

Note that the function f ( z ) = x 0 * ( P ( Ψ ( z ) ) ( t 0 ) ) is holomorphic. Hence, by the maximum modulus theorem, there exists z 0 ∈ S C such that

‖ P ( Ψ ( z 0 ) ) ( t 0 ) ‖ X ≥ x 0 * ( P ( Ψ ( ‖ f 0 ( t 0 ) ‖ X ) ) ( t 0 ) ) > ‖ P ‖ − ε .

Take x 1 = z 0 x 0 ∈ S X and let x 1 * ∈ S X * such that x 1 * x 1 = 1 . Define a function Φ : X → A ( K , X ) by

Φ ( x ) = x 1 * x 1 − δ 8 ( 1 − ϕ ) f 0 + ϕ ⋅ x .

We see that ‖ Φ ( x ) ‖ ≤ 1 for every x ∈ B X from (6). In particular, Φ ( x 1 ) = Ψ ( z 0 ) . Hence, ‖ P ( Φ ( x 1 ) ) ( t 0 ) ‖ X > ‖ P ‖ − ε . Consider Q ∈ P K ( X ; X ) defined by Q ( x ) = P ( Φ ( x ) ) ( t 0 ) . Note that

‖ Q ‖ ≥ ‖ Q x 1 ‖ X = ‖ P ( Φ ( x 1 ) ) ( t 0 ) ‖ X > ‖ P ‖ − ε .

This implies that ‖ I + Q ‖ ≥ 1 + Daug p ( X ) ‖ Q ‖ > 1 + Daug p ( X ) ( ‖ P ‖ − ε ) . Now choose x 2 ∈ B X such that ‖ x 2 + Q x 2 ‖ > 1 + Daug p ( X ) ( ‖ P ‖ − ε ) and let g = Φ ( x 2 ) . Then, we obtain

‖ I + P ‖ ≥ ‖ g + P g ‖ ≥ ‖ g ( t 0 ) + P ( g ) ( t 0 ) ‖ X ≥ x 1 * x 2 1 − δ 8 ( 1 − ϕ ( t 0 ) f ( t 0 ) ) + ϕ ( t 0 ) x 2 + Q ( x 2 ) X = ‖ x 2 + Q ( x 2 ) ‖ X > 1 + Daug p ( X ) ( ‖ P ‖ − ε ) .

As ε → 0 , we have ‖ I + P ‖ ≥ 1 + Daug p ( X ) ‖ P ‖ . This consequently shows that Daug p ( A ( K , X ) ) ≥ Daug p ( X ) .

If Γ does not have isolated points, then A ( K , X ) has the polynomial Daugavet property by Theorem 4.6. This implies that

Daug p ( A ( K , X ) ) = Daug p ( A ) = 1 ,

and so we have Daug p ( A ( K , X ) ) = max { Daug p ( A ) , Daug p ( X ) } .

If Γ has isolated points, then A ( K , X ) = X ⊕ ∞ Y by Lemma 4.5 and Daug ( A ) = 0 by Corollary 4.10. From Lemma 4.8, we see that Daug p ( A ( K , X ) ) ≤ Daug p ( X ) . Therefore, we also obtain Daug p ( A ( K , X ) ) = max { Daug p ( A ) , Daug p ( X ) } .□

Corollary 4.12

Let K be a compact Hausdorff space. Then, the space A ( K , X ) has the polynomial Daugavet property, if and only if either the base algebra A or X has the polynomial Daugavet property.

4.2 Remarks on the property ( D ) and the convex-DLD2P in A ( K , X )

Since the equivalence between the property ( D ) and the DLD2P is not clear, it is natural to explore various Banach spaces that potentially distinguish these properties. However, we show that this is not the case for A ( K , X ) . Under the additional assumption that X is uniformly convex, the space A ( K , X ) has the Daugavet property if and only if the Shilov boundary of the base algebra does not have isolated points [2, Theorem 5.6]. Moreover, the Daugavet property of A ( K , X ) is equivalent to all DD2Ps under the same assumption. In fact, carefully inspecting the proof of [2, Theorem 5.4], we see that the rank-one projection constructed in there has norm-one. With the aid of our previous observations, we can see that the DLD2P is also equivalent to the property ( D ) for A ( K , X ) .

Proposition 4.13

[2, Theorem 5.4] Let X be a uniformly convex Banach space, K be a compact Hausdorff space, Γ be the Shilov boundary of the base algebra A of A ( K , X ) , and f ∈ S A ( K , X ) . Then, the following statements are equivalent:

  1. f is a Daugavet point.

  2. f is a Δ -point.

  3. Every rank-one, norm-one projection P = ψ ⊗ f , where ψ ∈ A ( K , X ) * with ψ ( f ) = 1 , satisfies ‖ I − P ‖ = 2 .

  4. There is a limit point t 0 of Γ such that ‖ f ‖ = ‖ f ( t 0 ) ‖ X .

Proof

(i) ⇒ (ii) is clear. The implication (ii) ⇒ (iii) comes from Theorem 2.2. Indeed, for a Banach space Y , a point f ∈ S Y is a Δ -point if and only if every rank-one projection of the form P = ψ ⊗ f , where ψ ∈ Y * with ψ ( f ) = 1 , satisfies ‖ I − P ‖ ≥ 2 . Hence, we immediately have (iii) if this projection P has norm one. (iii) ⇒ (iv) and (iv) ⇒ (i) are identical to the proof of (ii) ⇒ (iii) and (iii) ⇒ (i) in [2, Theorem 5.4], respectively.□

Corollary 4.14

[2, Corollary 5.5] Let K be a compact Hausdorff space, Γ be the Shilov boundary of A ( K ) , and f ∈ S A ( K ) . Then, the following statements are equivalent:

  1. f is a Daugavet point.

  2. f is a Δ -point.

  3. Every rank-one, norm-one projection P = ψ ⊗ f , where ψ ∈ A ( K ) * with ψ ( f ) = 1 , satisfies ‖ I − P ‖ = 2 .

  4. There is a limit point t 0 of Γ such that ‖ f ‖ ∞ = ∣ f ( t 0 ) ∣ .

As a consequence, we obtain the following characterizations for the space A ( K , X ) and infinite-dimensional uniform algebras.

Proposition 4.15

[2, Theorem 5.6] Let X be a uniformly convex Banach space, let K be a compact Hausdorff space, and let Γ be the Shilov boundary of the base algebra A of A ( K , X ) . Then, the following statements are equivalent:

  1. A ( K , X ) has the polynomial Daugavet property.

  2. A ( K , X ) has the Daugavet property.

  3. A ( K , X ) has the DD2P.

  4. A ( K , X ) has the DLD2P.

  5. A ( K , X ) has the property ( D ).

  6. The Shilov boundary Γ does not have isolated points.

Proof

(i) ⇒ (ii) ⇒ (iii) ⇒ (iv) ⇒ (v) is clear from their definitions. Theorem 4.6 shows the implication (vi) ⇒ (i). Showing (v) ⇒ (vi) is also identical to the proof of [2, Theorem 5.6] with Proposition 4.13.□

Corollary 4.16

[2, Corollary 5.7] Let K be a compact Hausdorff space and let Γ be the Shilov boundary of a uniform algebra A ( K ) . Then, the following are equivalent:

  1. A ( K ) has the polynomial Daugavet property.

  2. A ( K ) has the Daugavet property.

  3. A ( K ) has the DD2P.

  4. A ( K ) has the DLD2P.

  5. A ( K ) has the property ( D ).

  6. The Shilov boundary Γ does not have isolated points.

In view of Lemma 4.2, we can also show that the sufficient condition for the convex-DLD2P in [2, Theorem 5.9] can be described with strong boundary points.

Theorem 4.17

Let K be a compact Hausdorff space, X be a uniformly convex Banach space, and let Γ be the Shilov boundary of the base algebra of A ( K , X ) . Denote by Γ ′ the set of limit points of the Shilov boundary. If  Γ ′ ∩ Γ 0 ≠ ∅ , then A ( K , X ) has the convex-DLD2P.

Proof

In view of Lemma 4.4, we assume that K = Γ . Denote the set of all Δ -points of A ( K , X ) by Δ and the base algebra of A ( K , X ) by A . We claim that S A ( K , X ) ⊂ conv ¯ Δ .

Let f ∈ S A ( K , X ) . Choose a point t 0 ∈ Γ ′ ∩ Γ 0 and let λ = 1 + ‖ f ( t 0 ) ‖ X 2 . For ε > 0 , let U be an open subset of K such that ‖ f ( t ) − f ( t 0 ) ‖ X < ε . Then, by Lemma 4.2, there exists ϕ ∈ A such that ‖ ϕ ‖ ∞ = ϕ ( t 0 ) = 1 , sup t ∈ K \ U ∣ ϕ ( t ) ∣ < ε , and

∣ ϕ ( t ) ∣ + ( 1 − ε ) ∣ 1 − ϕ ( t ) ∣ ≤ 1

for every t ∈ K .

Choose a norm-one vector v 0 ∈ X and let

x 0 = f ( t 0 ) ‖ f ( t 0 ) ‖ X if f ( t 0 ) ≠ 0 , v 0 if f ( t 0 ) = 0 .

Now, define

f 1 ( t ) = ( 1 − ε ) ( 1 − ϕ ( t ) ) f ( t ) + ϕ ( t ) x 0 , f 2 ( t ) = ( 1 − ε ) ( 1 − ϕ ( t ) ) f ( t ) − ϕ ( t ) x 0 , t ∈ K .

Note that f 1 , f 2 ∈ A ( K , X ) because A ⊗ X ⊂ A ( K , X ) . Moreover,

‖ f 1 ( t ) ‖ X = ∥ ( 1 − ε ) ( 1 − ϕ ( t ) ) f ( t ) + ϕ ( t ) x 0 ∥ X ≤ ( 1 − ε ) ∣ 1 − ϕ ( t ) ∣ + ∣ ϕ ( t ) ∣ ≤ 1 ,

for every t ∈ K . In particular, we have ‖ f 1 ( t 0 ) ‖ X = 1 , and so, ‖ f 1 ( t 0 ) ‖ X = ‖ f 1 ‖ = 1 . By the same argument, we also have ‖ f 2 ( t 0 ) ‖ X = ‖ f 2 ‖ = 1 . Thus, f 1 , f 2 ∈ Δ by Proposition 4.13. Let g ( t ) = λ f 1 ( t ) + ( 1 − λ ) f 2 ( t ) . We need to consider two cases.

Case 1: Suppose f ( t 0 ) ≠ 0 . Then g ( t ) = ( 1 − ε ) ( 1 − ϕ ( t ) ) f ( t ) + ϕ ( t ) f ( t 0 ) . We see that

‖ g ( t ) − f ( t ) ‖ X = ‖ ( 1 − ε ) ( 1 − ϕ ( t ) ) f ( t ) + ϕ ( t ) f ( t 0 ) − f ( t ) ‖ X = ‖ ( 1 − ε ) ( 1 − ϕ ( t ) ) f ( t ) + ϕ ( t ) f ( t 0 ) − ( 1 − ε ) f ( t ) − ε f ( t ) ‖ X = ‖ ( 1 − ε ) ( − ϕ ( t ) ) f ( t ) + ( 1 − ε ) ϕ ( t ) f ( t 0 ) + ε ϕ ( t ) f ( t 0 ) − ε f ( t ) ‖ X = ‖ ( 1 − ε ) ϕ ( t ) ( f ( t 0 ) − f ( t ) ) + ε ϕ ( t ) f ( t 0 ) − ε f ( t ) ‖ X ≤ ( 1 − ε ) ∣ ϕ ( t ) ∣ ⋅ ‖ f ( t ) − f ( t 0 ) ‖ X + ε ∣ ϕ ( t ) ∣ ⋅ ‖ f ( t 0 ) ‖ X + ε ‖ f ( t ) ‖ X ≤ ( 1 − ε ) ∣ ϕ ( t ) ∣ ⋅ ‖ f ( t ) − f ( t 0 ) ‖ X + 2 ε .

For t ∈ U , we see that ( 1 − ε ) ∣ ϕ ( t ) ∣ ⋅ ‖ f ( t ) − f ( t 0 ) ‖ X ≤ ( 1 − ε ) ε < ε . On the other hand, for t ∈ K \ U , we have ( 1 − ε ) ∣ ϕ ( t ) ∣ ⋅ ‖ f ( t ) − f ( t 0 ) ‖ X ≤ 2 ( 1 − ε ) ε < 2 ε . Hence, ‖ g − f ‖ < 4 ε , and so f ∈ conv ¯ Δ .

Case 2: Now, suppose f ( t 0 ) = 0 . Then, we have ‖ f ( t ) ‖ X < ε for every t ∈ U . Moreover, note that λ = 1 2 and g ( t ) = ( 1 − ε ) ( 1 − ϕ ( t ) ) f ( t ) . This implies that

‖ g ( t ) − f ( t ) ‖ X = ‖ ( 1 − ε ) ( 1 − ϕ ( t ) ) f ( t ) − ( 1 − ε ) f ( t ) − ε f ( t ) ‖ X ≤ ( 1 − ε ) ∣ ϕ ( t ) ∣ ⋅ ‖ f ( t ) ‖ X + ε ‖ f ( t ) ‖ X ≤ ( 1 − ε ) ∣ ϕ ( t ) ∣ ⋅ ‖ f ( t ) ‖ X + ε .

Note that ( 1 − ε ) ∣ ϕ ( t ) ∣ ⋅ ‖ f ( t ) ‖ X ≤ ( 1 − ε ) ε < ε for every t ∈ U . From the fact that ∣ ϕ ( t ) ∣ < ε for t ∈ K \ U , we have ( 1 − ε ) ∣ ϕ ( t ) ∣ ⋅ ‖ f ( t ) ‖ X ≤ ( 1 − ε ) ε < ε . This shows that ‖ g − f ‖ < 2 ε , and so f ∈ conv ¯ Δ .

Since f ∈ S A ( K , X ) is arbitrary, we see that S X ⊂ conv ¯ Δ . Therefore, the space A ( K , X ) has the convex-DLD2P.□

Corollary 4.18

Let K be a compact Hausdorff space and Γ ′ be the set of limit points in the Shilov boundary of a uniform algebra. If Γ ′ ∩ Γ 0 ≠ ∅ , then the uniform algebra has the convex-DLD2P.

5 Conclusion

In this article, we showed that Daugavet property and the DLD2P on complex Banach spaces can also be studied through rank-one real-linear operators and rank-one real-projections, respectively. Also, strongly locally uniformly alternatively convex or smooth (sluacs) Banach spaces are LUNSQ in both real and complex Banach spaces. Hence, sluacs Banach spaces do not contain Δ -points. By using the polynomial Daugavetian index, we also showed that the vector-valued function space A ( K , X ) has the polynomial Daugavet property if and only if either the underlying base algebra A or the range space X has the polynomial Daugavet property. As a consequence, we obtained that the polynomial Daugavet property, the Daugavet property, the DD2Ps, and the property ( D ) are equivalent for uniform algebras.

Acknowledgements

The authors would like to thank the anonymous referees for their valuable comments and suggestions on the article.

  1. Funding information: Han Ju Lee was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology [NRF-2020R1A2C1A01010377]. Hyung-Joon Tag was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology [NRF-2020R1A2C1A01010377].

  2. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and consented to its submission to the journal, reviewed all the results and approved the final version of the manuscript. All authors contributed equally to the content.

  3. Conflict of interest: The authors state no conflict of interest.

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Received: 2023-02-22
Revised: 2024-02-08
Accepted: 2024-03-16
Published Online: 2024-08-13

© 2024 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

  1. Regular Articles
  2. On the p-fractional Schrödinger-Kirchhoff equations with electromagnetic fields and the Hardy-Littlewood-Sobolev nonlinearity
  3. L-Fuzzy fixed point results in ℱ -metric spaces with applications
  4. Solutions of a coupled system of hybrid boundary value problems with Riesz-Caputo derivative
  5. Nonparametric methods of statistical inference for double-censored data with applications
  6. LADM procedure to find the analytical solutions of the nonlinear fractional dynamics of partial integro-differential equations
  7. Existence of projected solutions for quasi-variational hemivariational inequality
  8. Spectral collocation method for convection-diffusion equation
  9. New local fractional Hermite-Hadamard-type and Ostrowski-type inequalities with generalized Mittag-Leffler kernel for generalized h-preinvex functions
  10. On the asymptotics of eigenvalues for a Sturm-Liouville problem with symmetric single-well potential
  11. On exact rate of convergence of row sequences of multipoint Hermite-Padé approximants
  12. The essential norm of bounded diagonal infinite matrices acting on Banach sequence spaces
  13. Decay rate of the solutions to the Cauchy problem of the Lord Shulman thermoelastic Timoshenko model with distributed delay
  14. Enhancing the accuracy and efficiency of two uniformly convergent numerical solvers for singularly perturbed parabolic convection–diffusion–reaction problems with two small parameters
  15. An inertial shrinking projection self-adaptive algorithm for solving split variational inclusion problems and fixed point problems in Banach spaces
  16. An equation for complex fractional diffusion created by the Struve function with a T-symmetric univalent solution
  17. On the existence and Ulam-Hyers stability for implicit fractional differential equation via fractional integral-type boundary conditions
  18. Some properties of a class of holomorphic functions associated with tangent function
  19. The existence of multiple solutions for a class of upper critical Choquard equation in a bounded domain
  20. On the continuity in q of the family of the limit q-Durrmeyer operators
  21. Results on solutions of several systems of the product type complex partial differential difference equations
  22. On Berezin norm and Berezin number inequalities for sum of operators
  23. Geometric invariants properties of osculating curves under conformal transformation in Euclidean space ℝ3
  24. On a generalization of the Opial inequality
  25. A novel numerical approach to solutions of fractional Bagley-Torvik equation fitted with a fractional integral boundary condition
  26. Holomorphic curves into projective spaces with some special hypersurfaces
  27. On Periodic solutions for implicit nonlinear Caputo tempered fractional differential problems
  28. Approximation of complex q-Beta-Baskakov-Szász-Stancu operators in compact disk
  29. Existence and regularity of solutions for non-autonomous integrodifferential evolution equations involving nonlocal conditions
  30. Jordan left derivations in infinite matrix rings
  31. Nonlinear nonlocal elliptic problems in ℝ3: existence results and qualitative properties
  32. Invariant means and lacunary sequence spaces of order (α, β)
  33. Novel results for two families of multivalued dominated mappings satisfying generalized nonlinear contractive inequalities and applications
  34. Global in time well-posedness of a three-dimensional periodic regularized Boussinesq system
  35. Existence of solutions for nonlinear problems involving mixed fractional derivatives with p(x)-Laplacian operator
  36. Some applications and maximum principles for multi-term time-space fractional parabolic Monge-Ampère equation
  37. On three-dimensional q-Riordan arrays
  38. Some aspects of normal curve on smooth surface under isometry
  39. Mittag-Leffler-Hyers-Ulam stability for a first- and second-order nonlinear differential equations using Fourier transform
  40. Topological structure of the solution sets to non-autonomous evolution inclusions driven by measures on the half-line
  41. Remark on the Daugavet property for complex Banach spaces
  42. Decreasing and complete monotonicity of functions defined by derivatives of completely monotonic function involving trigamma function
  43. Uniqueness of meromorphic functions concerning small functions and derivatives-differences
  44. Asymptotic approximations of Apostol-Frobenius-Euler polynomials of order α in terms of hyperbolic functions
  45. Hyers-Ulam stability of Davison functional equation on restricted domains
  46. Involvement of three successive fractional derivatives in a system of pantograph equations and studying the existence solution and MLU stability
  47. Composition of some positive linear integral operators
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  49. Generalized result on the global existence of positive solutions for a parabolic reaction-diffusion model with an m × m diffusion matrix
  50. Online makespan minimization for MapReduce scheduling on multiple parallel machines
  51. The sequential Henstock-Kurzweil delta integral on time scales
  52. On a discrete version of Fejér inequality for α-convex sequences without symmetry condition
  53. Existence of three solutions for two quasilinear Laplacian systems on graphs
  54. Embeddings of anisotropic Sobolev spaces into spaces of anisotropic Hölder-continuous functions
  55. Nilpotent perturbations of m-isometric and m-symmetric tensor products of commuting d-tuples of operators
  56. Characterizations of transcendental entire solutions of trinomial partial differential-difference equations in ℂ2#
  57. Fractional Sturm-Liouville operators on compact star graphs
  58. Exact controllability for nonlinear thermoviscoelastic plate problem
  59. Improved modified gradient-based iterative algorithm and its relaxed version for the complex conjugate and transpose Sylvester matrix equations
  60. Superposition operator problems of Hölder-Lipschitz spaces
  61. A note on λ-analogue of Lah numbers and λ-analogue of r-Lah numbers
  62. Ground state solutions and multiple positive solutions for nonhomogeneous Kirchhoff equation with Berestycki-Lions type conditions
  63. A note on 1-semi-greedy bases in p-Banach spaces with 0 < p ≤ 1
  64. Fixed point results for generalized convex orbital Lipschitz operators
  65. Asymptotic model for the propagation of surface waves on a rotating magnetoelastic half-space
  66. Multiplicity of k-convex solutions for a singular k-Hessian system
  67. Poisson C*-algebra derivations in Poisson C*-algebras
  68. Signal recovery and polynomiographic visualization of modified Noor iteration of operators with property (E)
  69. Approximations to precisely localized supports of solutions for non-linear parabolic p-Laplacian problems
  70. Solving nonlinear fractional differential equations by common fixed point results for a pair of (α, Θ)-type contractions in metric spaces
  71. Pseudo compact almost automorphic solutions to a family of delay differential equations
  72. Periodic measures of fractional stochastic discrete wave equations with nonlinear noise
  73. Asymptotic study of a nonlinear elliptic boundary Steklov problem on a nanostructure
  74. Cramer's rule for a class of coupled Sylvester commutative quaternion matrix equations
  75. Quantitative estimates for perturbed sampling Kantorovich operators in Orlicz spaces
  76. Review Articles
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  78. Differential sandwich theorems for p-valent analytic functions associated with a generalization of the integral operator
  79. Special Issue on Development of Fuzzy Sets and Their Extensions - Part II
  80. Higher-order circular intuitionistic fuzzy time series forecasting methodology: Application of stock change index
  81. Binary relations applied to the fuzzy substructures of quantales under rough environment
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  118. Topological structure of the solution sets to neutral evolution inclusions driven by measures
  119. (α, F)-Geraghty-type generalized F-contractions on non-Archimedean fuzzy metric-unlike spaces
  120. Solvability of infinite system of integral equations of Hammerstein type in three variables in tempering sequence spaces c 0 β and ℓ 1 β
  121. Special Issue on Nonlinear Evolution Equations and Their Applications - Part I
  122. Fuzzy fractional delay integro-differential equation with the generalized Atangana-Baleanu fractional derivative
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  125. Special Issue on Blow-up Phenomena in Nonlinear Equations of Mathematical Physics - Part I
  126. Long time decay of incompressible convective Brinkman-Forchheimer in L2(ℝ3)
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  130. A memory-type thermoelastic laminated beam with structural damping and microtemperature effects: Well-posedness and general decay
  131. The asymptotic behavior for the Navier-Stokes-Voigt-Brinkman-Forchheimer equations with memory and Tresca friction in a thin domain
  132. Absence of global solutions to wave equations with structural damping and nonlinear memory
  133. Special Issue on Differential Equations and Numerical Analysis - Part I
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  135. Limiting dynamics for stochastic complex Ginzburg-Landau systems with time-varying delays on unbounded thin domains
  136. A comparison of two nonconforming finite element methods for linear three-field poroelasticity
https://doi.org/10.1515/dema-2024-0004
Keywords for this article
Daugavet points; Δ-points; alternative convexity or smoothness; nonsquareness; polynomial Daugavet property
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Articles in the same Issue

  1. Regular Articles
  2. On the p-fractional Schrödinger-Kirchhoff equations with electromagnetic fields and the Hardy-Littlewood-Sobolev nonlinearity
  3. L-Fuzzy fixed point results in ℱ -metric spaces with applications
  4. Solutions of a coupled system of hybrid boundary value problems with Riesz-Caputo derivative
  5. Nonparametric methods of statistical inference for double-censored data with applications
  6. LADM procedure to find the analytical solutions of the nonlinear fractional dynamics of partial integro-differential equations
  7. Existence of projected solutions for quasi-variational hemivariational inequality
  8. Spectral collocation method for convection-diffusion equation
  9. New local fractional Hermite-Hadamard-type and Ostrowski-type inequalities with generalized Mittag-Leffler kernel for generalized h-preinvex functions
  10. On the asymptotics of eigenvalues for a Sturm-Liouville problem with symmetric single-well potential
  11. On exact rate of convergence of row sequences of multipoint Hermite-Padé approximants
  12. The essential norm of bounded diagonal infinite matrices acting on Banach sequence spaces
  13. Decay rate of the solutions to the Cauchy problem of the Lord Shulman thermoelastic Timoshenko model with distributed delay
  14. Enhancing the accuracy and efficiency of two uniformly convergent numerical solvers for singularly perturbed parabolic convection–diffusion–reaction problems with two small parameters
  15. An inertial shrinking projection self-adaptive algorithm for solving split variational inclusion problems and fixed point problems in Banach spaces
  16. An equation for complex fractional diffusion created by the Struve function with a T-symmetric univalent solution
  17. On the existence and Ulam-Hyers stability for implicit fractional differential equation via fractional integral-type boundary conditions
  18. Some properties of a class of holomorphic functions associated with tangent function
  19. The existence of multiple solutions for a class of upper critical Choquard equation in a bounded domain
  20. On the continuity in q of the family of the limit q-Durrmeyer operators
  21. Results on solutions of several systems of the product type complex partial differential difference equations
  22. On Berezin norm and Berezin number inequalities for sum of operators
  23. Geometric invariants properties of osculating curves under conformal transformation in Euclidean space ℝ3
  24. On a generalization of the Opial inequality
  25. A novel numerical approach to solutions of fractional Bagley-Torvik equation fitted with a fractional integral boundary condition
  26. Holomorphic curves into projective spaces with some special hypersurfaces
  27. On Periodic solutions for implicit nonlinear Caputo tempered fractional differential problems
  28. Approximation of complex q-Beta-Baskakov-Szász-Stancu operators in compact disk
  29. Existence and regularity of solutions for non-autonomous integrodifferential evolution equations involving nonlocal conditions
  30. Jordan left derivations in infinite matrix rings
  31. Nonlinear nonlocal elliptic problems in ℝ3: existence results and qualitative properties
  32. Invariant means and lacunary sequence spaces of order (α, β)
  33. Novel results for two families of multivalued dominated mappings satisfying generalized nonlinear contractive inequalities and applications
  34. Global in time well-posedness of a three-dimensional periodic regularized Boussinesq system
  35. Existence of solutions for nonlinear problems involving mixed fractional derivatives with p(x)-Laplacian operator
  36. Some applications and maximum principles for multi-term time-space fractional parabolic Monge-Ampère equation
  37. On three-dimensional q-Riordan arrays
  38. Some aspects of normal curve on smooth surface under isometry
  39. Mittag-Leffler-Hyers-Ulam stability for a first- and second-order nonlinear differential equations using Fourier transform
  40. Topological structure of the solution sets to non-autonomous evolution inclusions driven by measures on the half-line
  41. Remark on the Daugavet property for complex Banach spaces
  42. Decreasing and complete monotonicity of functions defined by derivatives of completely monotonic function involving trigamma function
  43. Uniqueness of meromorphic functions concerning small functions and derivatives-differences
  44. Asymptotic approximations of Apostol-Frobenius-Euler polynomials of order α in terms of hyperbolic functions
  45. Hyers-Ulam stability of Davison functional equation on restricted domains
  46. Involvement of three successive fractional derivatives in a system of pantograph equations and studying the existence solution and MLU stability
  47. Composition of some positive linear integral operators
  48. On bivariate fractal interpolation for countable data and associated nonlinear fractal operator
  49. Generalized result on the global existence of positive solutions for a parabolic reaction-diffusion model with an m × m diffusion matrix
  50. Online makespan minimization for MapReduce scheduling on multiple parallel machines
  51. The sequential Henstock-Kurzweil delta integral on time scales
  52. On a discrete version of Fejér inequality for α-convex sequences without symmetry condition
  53. Existence of three solutions for two quasilinear Laplacian systems on graphs
  54. Embeddings of anisotropic Sobolev spaces into spaces of anisotropic Hölder-continuous functions
  55. Nilpotent perturbations of m-isometric and m-symmetric tensor products of commuting d-tuples of operators
  56. Characterizations of transcendental entire solutions of trinomial partial differential-difference equations in ℂ2#
  57. Fractional Sturm-Liouville operators on compact star graphs
  58. Exact controllability for nonlinear thermoviscoelastic plate problem
  59. Improved modified gradient-based iterative algorithm and its relaxed version for the complex conjugate and transpose Sylvester matrix equations
  60. Superposition operator problems of Hölder-Lipschitz spaces
  61. A note on λ-analogue of Lah numbers and λ-analogue of r-Lah numbers
  62. Ground state solutions and multiple positive solutions for nonhomogeneous Kirchhoff equation with Berestycki-Lions type conditions
  63. A note on 1-semi-greedy bases in p-Banach spaces with 0 < p ≤ 1
  64. Fixed point results for generalized convex orbital Lipschitz operators
  65. Asymptotic model for the propagation of surface waves on a rotating magnetoelastic half-space
  66. Multiplicity of k-convex solutions for a singular k-Hessian system
  67. Poisson C*-algebra derivations in Poisson C*-algebras
  68. Signal recovery and polynomiographic visualization of modified Noor iteration of operators with property (E)
  69. Approximations to precisely localized supports of solutions for non-linear parabolic p-Laplacian problems
  70. Solving nonlinear fractional differential equations by common fixed point results for a pair of (α, Θ)-type contractions in metric spaces
  71. Pseudo compact almost automorphic solutions to a family of delay differential equations
  72. Periodic measures of fractional stochastic discrete wave equations with nonlinear noise
  73. Asymptotic study of a nonlinear elliptic boundary Steklov problem on a nanostructure
  74. Cramer's rule for a class of coupled Sylvester commutative quaternion matrix equations
  75. Quantitative estimates for perturbed sampling Kantorovich operators in Orlicz spaces
  76. Review Articles
  77. Penalty method for unilateral contact problem with Coulomb's friction in time-fractional derivatives
  78. Differential sandwich theorems for p-valent analytic functions associated with a generalization of the integral operator
  79. Special Issue on Development of Fuzzy Sets and Their Extensions - Part II
  80. Higher-order circular intuitionistic fuzzy time series forecasting methodology: Application of stock change index
  81. Binary relations applied to the fuzzy substructures of quantales under rough environment
  82. Algorithm selection model based on fuzzy multi-criteria decision in big data information mining
  83. A new machine learning approach based on spatial fuzzy data correlation for recognizing sports activities
  84. Benchmarking the efficiency of distribution warehouses using a four-phase integrated PCA-DEA-improved fuzzy SWARA-CoCoSo model for sustainable distribution
  85. Special Issue on Application of Fractional Calculus: Mathematical Modeling and Control - Part II
  86. A study on a type of degenerate poly-Dedekind sums
  87. Efficient scheme for a category of variable-order optimal control problems based on the sixth-kind Chebyshev polynomials
  88. Special Issue on Mathematics for Artificial intelligence and Artificial intelligence for Mathematics
  89. Toward automated hail disaster weather recognition based on spatio-temporal sequence of radar images
  90. The shortest-path and bee colony optimization algorithms for traffic control at single intersection with NetworkX application
  91. Neural network quaternion-based controller for port-Hamiltonian system
  92. Matching ontologies with kernel principle component analysis and evolutionary algorithm
  93. Survey on machine vision-based intelligent water quality monitoring techniques in water treatment plant: Fish activity behavior recognition-based schemes and applications
  94. Artificial intelligence-driven tone recognition of Guzheng: A linear prediction approach
  95. Transformer learning-based neural network algorithms for identification and detection of electronic bullying in social media
  96. Squirrel search algorithm-support vector machine: Assessing civil engineering budgeting course using an SSA-optimized SVM model
  97. Special Issue on International E-Conference on Mathematical and Statistical Sciences - Part I
  98. Some fixed point results on ultrametric spaces endowed with a graph
  99. On the generalized Mellin integral operators
  100. On existence and multiplicity of solutions for a biharmonic problem with weights via Ricceri's theorem
  101. Approximation process of a positive linear operator of hypergeometric type
  102. On Kantorovich variant of Brass-Stancu operators
  103. A higher-dimensional categorical perspective on 2-crossed modules
  104. Special Issue on Some Integral Inequalities, Integral Equations, and Applications - Part I
  105. On parameterized inequalities for fractional multiplicative integrals
  106. On inverse source term for heat equation with memory term
  107. On Fejér-type inequalities for generalized trigonometrically and hyperbolic k-convex functions
  108. New extensions related to Fejér-type inequalities for GA-convex functions
  109. Derivation of Hermite-Hadamard-Jensen-Mercer conticrete inequalities for Atangana-Baleanu fractional integrals by means of majorization
  110. Some Hardy's inequalities on conformable fractional calculus
  111. The novel quadratic phase Fourier S-transform and associated uncertainty principles in the quaternion setting
  112. Special Issue on Recent Developments in Fixed-Point Theory and Applications - Part I
  113. A novel iterative process for numerical reckoning of fixed points via generalized nonlinear mappings with qualitative study
  114. Some new fixed point theorems of α-partially nonexpansive mappings
  115. Generalized Yosida inclusion problem involving multi-valued operator with XOR operation
  116. Periodic and fixed points for mappings in extended b-gauge spaces equipped with a graph
  117. Convergence of Peaceman-Rachford splitting method with Bregman distance for three-block nonconvex nonseparable optimization
  118. Topological structure of the solution sets to neutral evolution inclusions driven by measures
  119. (α, F)-Geraghty-type generalized F-contractions on non-Archimedean fuzzy metric-unlike spaces
  120. Solvability of infinite system of integral equations of Hammerstein type in three variables in tempering sequence spaces c 0 β and ℓ 1 β
  121. Special Issue on Nonlinear Evolution Equations and Their Applications - Part I
  122. Fuzzy fractional delay integro-differential equation with the generalized Atangana-Baleanu fractional derivative
  123. Klein-Gordon potential in characteristic coordinates
  124. Asymptotic analysis of Leray solution for the incompressible NSE with damping
  125. Special Issue on Blow-up Phenomena in Nonlinear Equations of Mathematical Physics - Part I
  126. Long time decay of incompressible convective Brinkman-Forchheimer in L2(ℝ3)
  127. Numerical solution of general order Emden-Fowler-type Pantograph delay differential equations
  128. Global smooth solution to the n-dimensional liquid crystal equations with fractional dissipation
  129. Spectral properties for a system of Dirac equations with nonlinear dependence on the spectral parameter
  130. A memory-type thermoelastic laminated beam with structural damping and microtemperature effects: Well-posedness and general decay
  131. The asymptotic behavior for the Navier-Stokes-Voigt-Brinkman-Forchheimer equations with memory and Tresca friction in a thin domain
  132. Absence of global solutions to wave equations with structural damping and nonlinear memory
  133. Special Issue on Differential Equations and Numerical Analysis - Part I
  134. Vanishing viscosity limit for a one-dimensional viscous conservation law in the presence of two noninteracting shocks
  135. Limiting dynamics for stochastic complex Ginzburg-Landau systems with time-varying delays on unbounded thin domains
  136. A comparison of two nonconforming finite element methods for linear three-field poroelasticity
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