On the Gevrey ultradifferentiability of weak solutions of an abstract evolution equation with a scalar type spectral operator on the open semi-axis

Marat V. Markin

doi:10.1515/math-2019-0083

Article Open Access

On the Gevrey ultradifferentiability of weak solutions of an abstract evolution equation with a scalar type spectral operator on the open semi-axis

Marat V. Markin

Published/Copyright: September 27, 2019

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$Open Mathematics$

From the journal Open Mathematics Volume 17 Issue 1

Abstract

Given the abstract evolution equation

y′(t)=Ay(t),t≥0,

with scalar type spectral operator A in a complex Banach space, found are conditions necessary and sufficient for all weak solutions of the equation, which a priori need not be strongly differentiable, to be strongly Gevrey ultradifferentiable of order β ≥ 1, in particular analytic or entire, on the open semi-axis (0, ∞). Also, revealed is a certain interesting inherent smoothness improvement effect.

Keywords: weak solution; scalar type spectral operator; Gevrey classes

MSC 2010: Primary 34G10; 47B40; 30D60; Secondary 47B15; 47D06; 47D60; 30D15

1 Introduction

We find conditions on a scalar type spectral operator A in a complex Banach space necessary and sufficient for all weak solutions of the evolution equation

y′(t)=Ay(t),t≥0, (1.1)

which a priori need not be strongly differentiable, to be strongly Gevrey ultradifferentiable of order β ≥ 1, in particular analytic, on the open semi-axis (0, ∞) and reveal a certain interesting inherent smoothness improvement effect.

The found results generalize the corresponding ones of paper [1], where similar consideration is given to equation (1.1) with a normal operator A in a complex Hilbert space, and the characterizations of the generation of Gevrey ultradifferentiable C₀-semigroups of Roumieu and Beurling types by scalar type spectral operators found in papers [2, 4] (see also [3]). They also develop the discourse of papers [5, 6], in which the strong differentiability of the weak solutions of equation (1.1) on [0, ∞) and (0, ∞) and their strong Gevrey ultradifferentiability of order β ≥ 1 on the closed semi-axis [0, ∞) are treated, respectively (cf. also [7]).

Definition 1.1

(Weak solution). Let A be a densely defined closed linear operator in a Banach space (X, ∥⋅∥). A strongly continuous vector function y : [0, ∞) → X is called a weak solution of equation (1.1) if, for any g^* ∈ D(A^*),

ddt〈y(t),g∗〉=〈y(t),A∗g∗〉,t≥0,

where D(⋅) is the domain of an operator, A^* is the operator adjoint to A, and 〈⋅,⋅〉 is the pairing between the space X and its dual X^* (cf. [8]).

Remarks 1.1

Due to the closedness of A, the weak solution of (1.1) can be equivalently defined to be a strongly continuous vector function y : [0, ∞) ↦ X such that, for all t ≥ 0,

∫0ty(s)ds∈D(A)andy(t)=y(0)+A∫0ty(s)ds

and is also called a mild solution (cf. [9, Ch. II, Definition 6.3], [10, Preliminaries]).
Such a notion of weak solution, which need not be differentiable in the strong sense, generalizes that of classical one, strongly differentiable on [0, ∞) and satisfying the equation in the traditional plug-in sense, the classical solutions being precisely the weak ones strongly differentiable on [0, ∞).
When a closed densely defined linear operator A in a complex Banach space X generates a C₀-semigroup {T(t)}_t≥0 of bounded linear operators (see, e.g., [9, 11]), i.e., the associated abstract Cauchy problem (ACP)

y′(t)=Ay(t),t≥0,y(0)=f (1.2)

is well-posed (cf. [9, Ch. II, Definition 6.8]), the weak solutions of equation (1.1) are the orbits

y(t)=T(t)f,t≥0, (1.3)

with f ∈ X[9, Ch. II, Proposition 6.4] (see also [8, Theorem]), whereas the classical ones are those with f ∈ D(A) (see, e.g., [9, Ch. II, Proposition 6.3]).
In our consideration, the associated ACP need not be well-posed, i.e., the scalar type spectral operator A need not generate a C₀-semigroup (cf. [12]).

2 Preliminaries

Here, for the reader’s convenience, we outline certain essential preliminaries.

2.1 Scalar type spectral operators

Henceforth, unless specified otherwise, A is supposed to be a scalar type spectral operator in a complex Banach space (X, ∥⋅∥) with strongly σ-additive spectral measure (the resolution of the identity) E_A(⋅) assigning to each Borel set δ of the complex plane ℂ a projection operator E_A(δ) on X and having the operator’s spectrum σ(A) as its support [13, 14, 15].

Observe that, on a complex finite-dimensional space, the scalar type spectral operators are all linear operators that furnish an eigenbasis for the space (see, e.g., [14, 15]) and, in a complex Hilbert space, the scalar type spectral operators are precisely all those that are similar to the normal ones [16].

Associated with a scalar type spectral operator in a complex Banach space is the Borel operational calculus analogous to that for a normal operator in a complex Hilbert space [17, 18], which assigns to any Borel measurable function F : σ(A) → ℂ a scalar type spectral operator

F(A):=∫σ(A)F(λ)dEA(λ)

(see [14, 15]).

In particular,

An=∫σ(A)λndEA(λ),n∈Z+,

(ℤ₊ := {0, 1, 2, …} is the set of nonnegative integers, A⁰ := I, I is the identity operator on X) and

ezA:=∫σ(A)ezλdEA(λ),z∈C. (2.4)

The properties of the spectral measure and operational calculus, exhaustively delineated in [14, 15], underlie the subsequent discourse. Here, we touch upon a few facts of particular importance.

Due to its strong countable additivity, the spectral measure E_A(⋅) is bounded [15, 19], i.e., there is such an M ≥ 1 that, for any Borel set δ⊆ ℂ,

∥EA(δ)∥≤M. (2.5)

Observe that the notation ∥⋅∥ is used here to designate the norm in the space L(X) of all bounded linear operators on X. We adhere to this rather conventional economy of symbols in what follows also adopting the same notation for the norm in the dual space X^*.

For any f ∈ X and g^* ∈ X^*, the total variation measure v(f, g^*, ⋅) of the complex-valued Borel measure 〈 E_A(⋅)f, g^* 〉 is a finite positive Borel measure with

v(f,g∗,C)=v(f,g∗,σ(A))≤4M∥f∥∥g∗∥ (2.6)

(see, e.g., [2, 20]).

Also (Ibid.), for a Borel measurable function F : ℂ → ℂ, f ∈ D(F(A)), g^* ∈ X^*, and a Borel set δ ⊆ ℂ,

∫δ|F(λ)|dv(f,g∗,λ)≤4M∥EA(δ)F(A)f∥∥g∗∥. (2.7)

In particular, for δ = σ(A),

∫σ(A)|F(λ)|dv(f,g∗,λ)≤4M∥F(A)f∥∥g∗∥. (2.8)

Observe that the constant M ≥ 1 in (2.6)–(2.8) is from (2.5).

Further, for a Borel measurable function F : ℂ → [0, ∞), a Borel set δ ⊆ ℂ, a sequence Δnn=1∞ of pairwise disjoint Borel sets in ℂ, and f ∈ X, g^* ∈ X^*,

∫δF(λ)dv(EA(∪n=1∞Δn)f,g∗,λ)=∑n=1∞∫δ∩ΔnF(λ)dv(EA(Δn)f,g∗,λ). (2.9)

Indeed, since, for any Borel sets δ, σ ⊆ ℂ,

EA(δ)EA(σ)=EA(δ∩σ)

[14, 15], for the total variation,

v(EA(δ)f,g∗,σ)=v(f,g∗,δ∩σ).

Whence, due to the nonnegativity of F(⋅) (see, e.g., [21, 22]),

∫δF(λ)dv(EA(∪n=1∞Δn)f,g∗,λ)=∫δ∩∪n=1∞ΔnF(λ)dv(f,g∗,λ)=∑n=1∞∫δ∩ΔnF(λ)dv(f,g∗,λ)

=∑n=1∞∫δ∩ΔnF(λ)dv(EA(Δn)f,g∗,λ).

The following statement, allowing to characterize the domains of Borel measurable functions of a scalar type spectral operator in terms of positive Borel measures, is fundamental for our consideration.

Proposition 2.1

([23, Proposition 3.1]). Let A be a scalar type spectral operator in a complex Banach space (X, ∥⋅∥) with spectral measure E_A(⋅) and F : σ(A) → ℂ be a Borel measurable function. Then f ∈ D(F(A)) iff

for each g∗∈X∗,∫σ(A)|F(λ)|dv(f,g∗,λ)<∞ and
sup{g∗∈X∗|∥g∗∥=1}∫{λ∈σ(A)||F(λ)|>n}|F(λ)|dv(f,g∗,λ)→0,n→∞,

where v(f, g^*, ⋅) is the total variation measure of 〈 E_A(⋅)f, g^* 〉.

The succeeding key theorem provides a description of the weak solutions of equation (1.1) with a scalar type spectral operator A in a complex Banach space.

Theorem 2.1

([23, Theorem 4.2]). Let A be a scalar type spectral operator in a complex Banach space (X, ∥⋅∥). A vector function y : [0, ∞) → X is a weak solution of equation (1.1) iff there is an f∈⋂t≥0D(etA) such that

y(t)=etAf,t≥0, (2.10)

the operator exponentials understood in the sense of the Borel operational calculus (see (2.4)).

Remarks 2.1

Theorem 2.1 generalizing [24, Theorem 3.1]), its counterpart for a normal operator A in a complex Hilbert space, in particular, implies
1. that the subspace ⋂_t≥0D(e^tA) of all possible initial values of the weak solutions of equation (1.1) is the largest permissible for the exponential form given by (2.10), which highlights the naturalness of the notion of weak solution, and
2. that associated ACP (1.2), whenever solvable, is solvable uniquely.
Observe that the initial-value subspace ⋂_t≥0 D(e^tA) of equation (1.1) is dense in X since it contains the subspace

⋃α>0EA(Δα)X,whereΔα:=λ∈C||λ|≤α,α>0,

which is dense in X and coincides with the class ℰ^{0}(A) of entire vectors of A of exponential type [25, 26].
When a scalar type spectral operator A in a complex Banach space generates a C₀-semigroup {T(t)}_t≥0,

T(t)=etAandD(etA)=X,t≥0,

[12], and hence, Theorem 2.1 is consistent with the well-known description of the weak solutions for this setup (see (1.3)).

Subsequently, the frequent terms “spectral measure” and “operational calculus” are abbreviated to s.m. and o.c., respectively.

2.2 Gevrey classes of functions

Definition 2.1

(Gevrey classes of functions). Let (X, ∥⋅∥) be a (real or complex) Banach space, C^∞(I, X) be the space of all X-valued functions strongly infinite differentiable on an interval I ⊆ ℝ, and 0 ≤ β < ∞.

The following subspaces of C^∞(I, X)

E{β}(I,X):={g(⋅)∈C∞(I,X)|∀[a,b]⊆I∃α>0∃c>0:maxa≤t≤b∥g(n)(t)∥≤cαn[n!]β,n∈Z+},E(β)(I,X):={g(⋅)∈C∞(I,X)|∀[a,b]⊆I∀α>0∃c>0:maxa≤t≤b∥g(n)(t)∥≤cαn[n!]β,n∈Z+},

are called the βth-order Gevrey classes of strongly ultradifferentiable vector functions on I of Roumieu and Beurling type, respectively (see, e.g., [27, 28, 29, 30]).

Remarks 2.2

In view of Stirling’s formula, the sequence [n!]βn=0∞ can be replaced with nβnn=0∞ .
For 0 ≤ β < β′ < ∞, the inclusions

E(β)(I,X)⊆E{β}(I,X)⊆E(β′)(I,X)⊆E{β′}(I,X)⊆C∞(I,X)

hold.
For 1 < β < ∞, the Gevrey classes are non-quasianalytic (see, e.g., [29]).
For β = 1, ℰ^{1}(I, X) is the class of all analytic on I, i.e., analytically continuable into complex neighborhoods of I, vector functions and ℰ⁽¹⁾(I, X) is the class of all entire, i.e., allowing entire continuations, vector functions [31].
For 0 ≤ β < 1, the Gevrey class ℰ^{β}(I, X) (ℰ^(β)(I, X)) consists of all functions g(⋅) ∈ ℰ⁽¹⁾(I, X) such that, for some (any) γ > 0, there is an M > 0 for which

∥g(z)∥≤Meγ|z|1/(1−β),z∈C, (2.11)

[32] (see also [33]). In particular, for β = 0, ℰ^{0}(I, X) and ℰ⁽⁰⁾(I, X) are the classes of entire vector functions of exponential and minimal exponential type, respectively (see, e.g., [34]).

2.3 Gevrey classes of vectors

One can consider the Gevrey classes in a more general sense.

Definition 2.2

(Gevrey classes of vectors). Let (A, D(A)) be a densely defined closed linear operator in a (real or complex) Banach space (X, ∥⋅∥), 0 ≤ β < ∞, and

C∞(A):=⋂n=0∞D(An)

be the subspace of infinite differentiable vectors of A.

The following subspaces of C^∞(A)

E{β}(A):=x∈C∞(A)|∃α>0∃c>0:∥Anx∥≤cαn[n!]β,n∈Z+,E(β)(A):=x∈C∞(A)|∀α>0∃c>0:∥Anx∥≤cαn[n!]β,n∈Z+

are called the βth-order Gevrey classes of ultradifferentiable vectors of A of Roumieu and Beurling type, respectively (see, e.g., [ 35, 36, 37]).

Remarks 2.3

In view of Stirling’s formula, the sequence [n!]βn=0∞ can be replaced with nβnn=0∞ .
For 0 ≤ β < β′ < ∞, the inclusions E(β)(A)⊆E{β}(A)⊆E(β′)(A)⊆E{β′}(A)⊆C∞(A)

hold.
In particular, ℰ^{1}(A) and ℰ⁽¹⁾(A) are the classes of analytic and entire vectors of A, respectively [38, 39] and ℰ^{0}(A) and ℰ⁽⁰⁾(A) are the classes of entire vectors of A of exponential and minimal exponential type, respectively (see, e.g., [26, 36]).
In view of the closedness of A, it is easily seen that the class ℰ⁽¹⁾(A) forms the subspace of the initial values f ∈ X generating the (classical) solutions of (1.1), which are entire vector functions represented by the power series

∑n=0∞tnn!Anf,t≥0,

the classes ℰ^{β}(A) and ℰ^(β)(A) with 0 ≤ β < 1 being the subspaces of such initial values for which the solutions satisfy growth estimate (2.11) with some (any) γ > 0 and some M > 0, respectively (cf. [34]).

As is shown in [35] (see also [36, 37]), if 0 < β < ∞, for a normal operator A in a complex Hilbert space,

E{β}(A)=⋃t>0D(et|A|1/β)andE(β)(A)=⋂t>0D(et|A|1/β), (2.12)

the operator exponentials e^{t|A|^1/β}, t > 0, understood in the sense of the Borel operational calculus (see, e.g., [17, 18]).

In [20, 25], descriptions (2.12) are extended to scalar type spectral operators in a complex Banach space, in which form they are basic for our discourse. In [25], similar nature descriptions of the classes ℰ^{0}(A) and ℰ⁽⁰⁾(A) (β = 0), known for a normal operator A in a complex Hilbert space (see, e.g., [36]), are also generalized to scalar type spectral operators in a complex Banach space. In particular [25, Theorem 5.1],

E{0}(A)=⋃α>0EA(Δα)X,

where

Δα:=λ∈C||λ|≤α,α>0.

We also need the following characterization of a particular weak solution’s of equation (1.1) with a scalar type spectral operator A in a complex Banach space being strongly Gevrey ultradifferentiable on a subinterval I of [0, ∞).

Proposition 2.2

([6, Proposition 3.2]). Let A be a scalar type spectral operator in a complex Banach space (X, ∥⋅∥), 0 ≤ β < ∞, and I be a subinterval of [0, ∞). Then the restriction of a weak solution y(⋅) of equation (1.1) to I belongs to the Gevrey class ℰ^{β}(I, X) (ℰ^(β)(I, X)) iff, for each t ∈ I,

y(t)∈E{β}(A)(E(β)(A),respectively),

in which case, for every n ∈ ℕ,

y(n)(t)=Any(t),t∈I.

3 Gevrey ultradifferentiability of weak solutions

The case of the strong Gevrey ultradifferentiability of the weak solutions of equation (1.1) with a scalar type spectral operator in a complex Banach space on the open semi-axis (0, ∞), similarly to the analogous setup with a normal operator A in a complex Hilbert space [1], significantly differs from its counterpart over the closed semi-axis [0, ∞) studied in [6].

First, let us consider the Roumieu type strong Gevrey ultradifferentiability of order β ≥ 1.

Theorem 3.1

Let A be a scalar type spectral operator in a complex Banach space (X, ∥⋅∥) with spectral measure E_A(⋅) and 1 ≤ β < ∞. Every weak solution of equation (1.1) belongs to the βth-order Roumieu type Gevrey class ℰ^{β}((0, ∞), X) iff there exist b₊ > 0 and b_– > 0 such that the set σ(A) ∖ Pb−,b+β , where

Pb−,b+β:=λ∈C|Reλ≤−b−|Imλ|1/βor Reλ≥b+|Imλ|1/β,

is bounded (see Figure 1).

$Figure 1 Gevrey ultradifferentiability of order 1 ≤ β < ∞.$

Figure 1

Gevrey ultradifferentiability of order 1 ≤ β < ∞.

Proof

“If” Part. Suppose that there exist b₊ > 0 and b_– > 0 such that the set σ(A) ∖ Pb−,b+β is bounded and let y(⋅) be an arbitrary weak solution of equation (1.1).

By Theorem 2.1,

y(t)=etAf,t≥0,with somef∈⋂t≥0D(etA).

Our purpose is to show that y(⋅) ∈ ℰ^{β} ((0, ∞), X), which, by Proposition 2.2 and (2.12), is attained by showing that, for each t > 0,

y(t)∈E{β}A=⋃s>0D(es|A|1/β).

Let us proceed by proving that, for each t > 0, there exists an s > 0 such that

y(t)∈D(es|A|1/β)

via Proposition 2.1.

For an arbitrary t > 0, let us set

s:=t(1+b−−β)−1/β>0. (3.13)

Then, for any g^* ∈ X^*,

Indeed,

∫σ(A)∖Pb−,b+βes|λ|1/βetReλdv(f,g∗,λ)<∞

and

∫λ∈σ(A)∩Pb−,b+β|−1<Reλ<1es|λ|1/βetReλdv(f,g∗,λ)<∞

due to the boundedness of the sets

σ(A)∖Pb−,b+βandλ∈σ(A)∩Pb−,b+β|−1<Reλ<1,

the continuity of the integrated function on ℂ, and the finiteness of the measure v(f, g^*, ⋅).

Further, for an arbitrary t > 0, s > 0 chosen as in (3.13), and any g^* ∈ X^*,

Observe that, for the finiteness of the three preceding integrals, the special choice of s > 0 is superfluous. Finally, for an arbitrary t > 0, s > 0 chosen as in (3.13), and any g^* ∈ X^*,

Also, for an arbitrary t > 0, s > 0 chosen as in (3.13), and any n ∈ ℕ,

sup{g∗∈X∗|∥g∗∥=1}∫λ∈σ(A)|es|λ|1/βetReλ>nes|λ|1/βetReλdv(f,g∗,λ)≤sup{g∗∈X∗|∥g∗∥=1}∫λ∈σ(A)∖Pb−,b+β|es|λ|1/βetReλ>nes|λ|1/βetReλdv(f,g∗,λ)+sup{g∗∈X∗|∥g∗∥=1}∫λ∈σ(A)∩Pb−,b+β|−1<Reλ<1,es|λ|1/βetReλ>nes|λ|1/βetReλdv(f,g∗,λ)+sup{g∗∈X∗|∥g∗∥=1}∫λ∈σ(A)∩Pb−,b+β|Reλ≥1,es|λ|1/βetReλ>nes|λ|1/βetReλdv(f,g∗,λ)+sup{g∗∈X∗|∥g∗∥=1}∫λ∈σ(A)∩Pb−,b+β|Reλ≤−1,es|λ|1/βetReλ>nes|λ|1/βetReλdv(f,g∗,λ)→0,n→∞. (3.17)

Indeed, since, due to the boundedness of the sets

σ(A)∖Pb−,b+βandλ∈σ(A)∩Pb−,b+β|−1<Reλ<1

and the continuity of the integrated function on ℂ, the sets

λ∈σ(A)∖Pb−,b+β|es|λ|1/βetReλ>n

and

λ∈σ(A)∩Pb−,b+β|−1<Reλ<1,es|λ|1/βetReλ>n

are empty for all sufficiently large n ∈ ℕ, we immediately infer that, for any t > 0 and s > 0 chosen as in (3.13),

limn→∞sup{g∗∈X∗|∥g∗∥=1}∫λ∈σ(A)∖Pb−,b+β|es|λ|1/βetReλ>nes|λ|1/βetReλdv(f,g∗,λ)=0

and

limn→∞sup{g∗∈X∗|∥g∗∥=1}∫λ∈σ(A)∩Pb−,b+β|−1<Reλ<1,es|λ|1/βetReλ>nes|λ|1/βetReλdv(f,g∗,λ)=0.

Further, for an arbitrary t > 0, s > 0 chosen as in (3.13), and any n ∈ ℕ,

sup{g∗∈X∗|∥g∗∥=1}∫λ∈σ(A)∩Pb−,b+β|Reλ≥1,es|λ|1/βetReλ>nes|λ|1/βetReλdv(f,g∗,λ)as in (3.15);≤sup{g∗∈X∗|∥g∗∥=1}∫λ∈σ(A)∩Pb−,b+β|Reλ≥1,es|λ|1/βetReλ>nes1+b+−β1/β+tReλdv(f,g∗,λ)since f∈⋂t≥0D(etA), by (2.7);≤sup{g∗∈X∗|∥g∗∥=1}4MEAλ∈σ(A)∩Pb−,b+β|Reλ≥1,es|λ|1/βetReλ>nes1+b+−β1/β+tAf∥g∗∥≤4MEAλ∈σ(A)∩Pb−,b+β|Reλ≥1,es|λ|1/βetReλ>nes1+b+−β1/β+tAfby the strong continuity of thes.m.;→4MEA∅es1+b+−β1/β+tAf=0,n→∞.

Finally, for an arbitrary t > 0, s > 0 chosen as in (3.13), and any n ∈ ℕ,

sup{g∗∈X∗|∥g∗∥=1}∫λ∈σ(A)∩ Pb−,b+β|Reλ≤−1,es|λ|1/βetReλ>nes|λ|1/βetReλdv(f,g∗,λ)as in (3.16);≤sup{g∗∈X∗|∥g∗∥=1}∫λ∈σ(A)∩ Pb−,b+β|Reλ≤−1,es|λ|1/βetReλ>net−s1+b−−β1/βReλdv(f,g∗,λ)since s:=t(1+b−−β)−1/β (see (3.13));=sup{g∗∈X∗|∥g∗∥=1}∫λ∈σ(A)∩ Pb−,b+β|Reλ≤−1,es|λ|1/βetReλ>n1dv(f,g∗,λ)by (2.7);≤sup{g∗∈X∗|∥g∗∥=1}4MEAλ∈σ(A)∩Pb−,b+β|Reλ≤−1,es|λ |1 /βetReλ>nf∥g∗∥≤4MEAλ∈σ(A)∩Pb−,b+β|Reλ≤−1,es|λ |1 /βetReλ>nfby the strong continuity of thes.m.;→4MEA∅f=0,n→∞.

By Proposition 2.1 and the properties of the o.c. (see [15, Theorem XVIII.2.11 (f)]), (3.14) and (3.17) jointly imply that, for any t > 0,

f∈D(es|A|1/βetA)

with s:=t(1+b−−β)−1/β>0, and hence, in view of (2.12), for each t > 0,

y(t)=etAf∈⋃s>0D(es|A|1/β)=E{β}(A).

By Proposition 2.2, we infer that

y(⋅)∈E{β}((0,∞),X),

which completes the proof of the “if” part.

“Only if” part. Let us prove this part by contrapositive assuming that, for any b₊ > 0 and b_– > 0, the set σ(A) ∖ Pb−,b+β is unbounded. In particular, this means that, for any n ∈ ℕ, unbounded is the set

σ(A)∖Pn−1,n−2β=λ∈σ(A)|−n−1|Imλ|1/β<Reλ<n−2|Imλ|1/β.

Hence, we can choose a sequence λnn=1∞ of points in the complex plane as follows:

The latter implies, in particular, that the points λ_n, n ∈ ℕ, are distinct (λ_i ≠ λ_j, i ≠ j).

Since, for each n ∈ ℕ, the set

λ∈C|−n−1|Imλ|1/β<Reλ<n−2|Imλ|1/β,|λ|>max[n,|λn−1|]

is open in ℂ, along with the point λ_n, it contains an open disk

Δn:=λ∈C||λ−λn|<εn

centered at λ_n of some radius ε_n > 0, i.e., for each λ ∈ Δ_n,

−n−1|Imλ|1/β<Reλ<n−2|Imλ|1/βand|λ|>max[n,|λn−1|]. (3.18)

Furthermore, we can regard the radii of the disks to be small enough so that

0<εn<1n,n∈N,andΔi∩Δj=∅,i≠j(i.e., the disks arepairwise disjoint). (3.19)

Whence, by the properties of the s.m.,

EA(Δi)EA(Δj)=0,i≠j,

where 0 stands for the zero operator on X.

Observe also that the subspaces E_A(Δ_n)X, n ∈ ℕ, are nontrivial since

Δn∩σ(A)≠∅,n∈N,

with Δ_n being an open set in ℂ.

By choosing a unit vector e_n ∈ E_A(Δ_n)X for each n ∈ ℕ, we obtain a sequence enn=1∞ in X such that

∥en∥=1,n∈N,andEA(Δi)ej=δijej,i,j∈N, (3.20)

where δ_ij is the Kronecker delta.

As is easily seen, (3.20) implies that the vectors e_n, n ∈ ℕ, are linearly independent.

Furthermore, there exists an ε > 0 such that

dn:=disten,spanei|i∈N,i≠n≥ε,n∈N. (3.21)

Indeed, the opposite implies the existence of a subsequence dn(k)k=1∞ such that

dn(k)→0,k→∞.

Then, by selecting a vector

fn(k)∈spanei|i∈N,i≠n(k),k∈N,

such that

∥en(k)−fn(k)∥<dn(k)+1/k,k∈N,

we arrive at

1=∥en(k)∥since, by (3.20),EA(Δn(k))fn(k)=0;=∥EA(Δn(k))(en(k)−fn(k))∥≤∥EA(Δn(k))∥∥en(k)−fn(k)∥by (2.5));≤M∥en(k)−fn(k)∥≤Mdn(k)+1/k→0,k→∞,

which is a contradiction proving (3.21).

As follows from the Hahn-Banach Theorem, for any n ∈ ℕ, there is an en∗ ∈ X^* such that

∥en∗∥=1,n∈N,and〈ei,ej∗〉=δijdi,i,j∈N. (3.22)

Let us consider separately the two possibilities concerning the sequence of the real parts {Reλn}n=1∞ : its being bounded or unbounded.

First, suppose that the sequence {Reλn}n=1∞ is bounded, i.e., there exists an ω > 0 such that

|Reλn|≤ω,n∈N, (3.23)

and consider the element

f:=∑k=1∞k−2ek∈X,

which is well defined since k−2k=1∞ ∈ l₁ (l₁ is the space of absolutely summable sequences) and ∥e_k∥ = 1, k ∈ ℕ (see (3.20)).

In view of (3.20), by the properties of the s.m.,

EA(∪k=1∞Δk)f=fandEA(Δk)f=k−2ek,k∈N. (3.24)

For an arbitrary t ≥ 0 and any g^* ∈ X^*,

∫σ(A)etReλdv(f,g∗,λ) by (3.24);=∫σ(A)etReλdv(EA(∪k=1∞Δk)f,g∗,λ) by (2.9);=∑k=1∞∫σ(A)∩ΔketReλdv(EA(Δk)f,g∗,λ) by (3.24);=∑k=1∞k−2∫σ(A)∩ΔketReλdv(ek,g∗,λ) since, for λ∈Δk; by (3.23) and (3.19),Reλ=Reλk+(Reλ−Reλk)≤Reλk+|λ−λk|≤ω+εk≤ω+1;≤et(ω+1)∑k=1∞k−2∫σ(A)∩Δk1dv(ek,g∗,λ)=et(ω+1)∑k=1∞k−2v(ek,g∗,Δk) by (2.6);≤et(ω+1)∑k=1∞k−24M∥ek∥∥g∗∥=4Met(ω+1)∥g∗∥∑k=1∞k−2<∞. (3.25)

Similarly, for an arbitrary t ≥ 0 and any n ∈ ℕ,

sup{g∗∈X∗|∥g∗∥=1}∫λ∈σ(A)|etReλ>netReλdv(f,g∗,λ)≤sup{g∗∈X∗|∥g∗∥=1}et(ω+1)∑k=1∞k−2∫λ∈σ(A)|etReλ>n∩Δk1dv(ek,g∗,λ)by (3.24);=et(ω+1)sup{g∗∈X∗|∥g∗∥=1}∑k=1∞∫λ∈σ(A)|etReλ>n∩Δk1dv(EA(Δk)f,g∗,λ by (2.9);=et(ω+1)sup{g∗∈X∗|∥g∗∥=1}∫{λ∈σ(A)|etReλ>n}1dv(EA(∪k=1∞Δk)f,g∗,λ)by (3.24);=et(ω+1)sup{g∗∈X∗|∥g∗∥=1}∫{λ∈σ(A)|etReλ>n}1dv(f,g∗,λ)by (2.7);≤et(ω+1)sup{g∗∈X∗|∥g∗∥=1}4MEAλ∈σ(A)|etReλ>nf∥g∗∥≤4Met(ω+1)EAλ∈σ(A)|etReλ>nf by the strong continuity of thes.m.;→4Met(ω+1)EA∅f=0,n→∞. (3.26)

By Proposition 2.1, (3.25) and (3.26) jointly imply that

f∈⋂t≥0D(etA),

and hence, by Theorem 2.1,

y(t):=etAf,t≥0,

is a weak solution of equation (1.1).

Let

h∗:=∑k=1∞k−2ek∗∈X∗, (3.27)

the functional being well defined since {k−2}k=1∞ ∈ l₁ and ∥ ek∗ ∥ = 1, k ∈ ℕ (see (3.22)).

In view of (3.22) and (3.21), we have:

〈ek,h∗〉=〈ek,k−2ek∗〉=dkk−2≥εk−2,k∈N. (3.28)

For any s > 0,

By Proposition 2.1 and the properties of the o.c. (see [15, Theorem XVIII.2.11 (f)]), (3.29) implies that, for any s > 0,

f∉D(es|A|1/βeA),

and hence, in view of (2.12),

y(1)=eAf∉⋃s>0D(es|A|1/β)=E{β}(A).

By Proposition 2.2, we infer that the weak solution y(t) = e^tAf, t ≥ 0, of equation (1.1) does not belong to the Roumieu type Gevrey class ℰ^{β} ((0, ∞), X), which completes our consideration of the case of the sequence’s {Reλn}n=1∞ being bounded.

Now, suppose that the sequence {Reλn}n=1∞ is unbounded.

Therefore, there is a subsequence {Reλn}n=1∞ such that

Reλn(k)→∞orReλn(k)→−∞,k→∞.

Let us consider separately each of the two cases.

First, suppose that

Reλn(k)→∞,k→∞.

Then, without loss of generality, we can regard that

Reλn(k)≥k,k∈N. (3.30)

Consider the elements

f:=∑k=1∞e−n(k)Reλn(k)en(k)∈Xandh:=∑k=1∞e−n(k)2Reλn(k)en(k)∈X,

well defined since, by (3.30),

e−n(k)Reλn(k)k=1∞,e−n(k)2Reλn(k)k=1∞∈l1

and ∥e_n(k)∥ = 1, k ∈ ℕ (see (3.20)).

By (3.20),

EA(∪k=1∞Δn(k))f=fandEA(Δn(k))f=e−n(k)Reλn(k)en(k),k∈N, (3.31)

and

EA(∪k=1∞Δn(k))h=handEA(Δn(k))h=e−n(k)2Reλn(k)en(k),k∈N. (3.32)

For an arbitrary t ≥ 0 and any g^* ∈ X^*,

∫σ(A)etReλdv(f,g∗,λ) by (2.9) as in (3.25);=∑k=1∞e−n(k)Reλn(k)∫σ(A)∩Δn(k)etReλdv(en(k),g∗,λ)since, for λ∈Δn(k), by (3.19),Reλ=Reλn(k)+(Reλ−Reλn(k))≤Reλn(k)+|λ−λn(k)|≤Reλn(k)+1;≤∑k=1∞e−n(k)Reλn(k)et(Reλn(k)+1)∫σ(A)∩Δn(k)1dv(en(k),g∗,λ)=et∑k=1∞e−[n(k)−t]Reλn(k)v(en(k),g∗,Δn(k)) by (2.6);≤et∑k=1∞e−[n(k)−t]Reλn(k)4M∥en(k)∥∥g∗∥=4Met∥g∗∥∑k=1∞e−[n(k)−t]Reλn(k)<∞. (3.33)

Indeed, for all k ∈ ℕ sufficiently large so that

n(k)≥t+1,

in view of (3.30),

e−[n(k)−t]Reλn(k)≤e−k.

Similarly, for an arbitrary t ≥ 0 and any n ∈ ℕ,

sup{g∗∈X∗|∥g∗∥=1}∫λ∈σ(A)|etReλ>netReλdv(f,g∗,λ)≤sup{g∗∈X∗|∥g∗∥=1}et∑k=1∞e−[n(k)−t]Reλn(k)∫λ∈σ(A)|etReλ>n∩Δn(k)1dv(en(k),g∗,λ)=etsup{g∗∈X∗|∥g∗∥=1}∑k=1∞e−n(k)2−tReλn(k)e−n(k)2Reλ(k)∫λ∈σ(A)|etReλ>n∩Δn(k)1dv(en(k),g∗,λ)since, by (3.30), there is an L>0 such that e−n(k)2−tReλn(k)≤L,k∈N;≤Letsup{g∗∈X∗|∥g∗∥=1}∑k=1∞e−n(k)2Reλn(k)∫λ∈σ(A)|etReλ>n∩Δn(k)1dv(en(k),g∗,λ)by (3.32);=Letsup{g∗∈X∗|∥g∗∥=1}∑k=1∞∫λ∈σ(A)|etReλ>n∩Δn(k)1dv(EA(Δn(k))h,g∗,λ) by (2.9);=Letsup{g∗∈X∗|∥g∗∥=1}∫λ∈σ(A)|etReλ>n1dv(EA(∪k=1∞Δn(k))h,g∗,λ)by (3.32);=Letsup{g∗∈X∗|∥g∗∥=1}∫{λ∈σ(A)|etReλ>n}1dv(h,g∗,λ)by (2.7);≤Letsup{g∗∈X∗|∥g∗∥=1}4MEAλ∈σ(A)|etReλ>nh∥g∗∥≤4LMet∥EA({λ∈σ(A)|etReλ>n})h∥by the strong continuity of thes.m.;→4LMetEA∅h=0,n→∞. (3.34)

By Proposition 2.1, (3.33) and (3.34) jointly imply that

f∈⋂t≥0D(etA),

and hence, by Theorem 2.1,

y(t):=etAf,t≥0,

is a weak solution of equation (1.1).

Since, for any λ ∈ Δ_n(k), k ∈ ℕ, by (3.19), (3.30),

Reλ=Reλn(k)−(Reλn(k)−Reλ)≥Reλn(k)−|Reλn(k)−Reλ|≥Reλn(k)−εn(k)≥Reλn(k)−1/n(k)≥k−1≥0

and, by (3.18),

Reλ<n(k)−2|Imλ|1/β,

we infer that, for any λ ∈ Δ_n(k), k ∈ ℕ,

|λ|≥|Imλ|≥n(k)2Reλβ≥n(k)2(Reλn(k)−1/n(k))β.

Using this estimate, for an arbitrary s > 0 and the functional h^* ∈ X^* defined by (3.27), we have:

Indeed, for all k ∈ ℕ sufficiently large so that

sn(k)≥s+2,

in view of (3.30),

e(sn(k)−1)n(k)Reλn(k)−sn(k)n(k)−2≥e(s+1)n(k)−sn(k)n(k)−2=en(k)n(k)−2→∞,k→∞.

By Proposition 2.1 and the properties of the o.c. (see [15, Theorem XVIII.2.11 (f)]), (3.35) implies that, for any s > 0,

f∉D(es|A|1/βeA),

which, in view of (2.12), further implies that

y(1)=eAf∉⋃s>0D(es|A|1/β)=E{β}(A).

Whence, by Proposition 2.2, we infer that the weak solution y(t) = e^tAf, t ≥ 0, of equation (1.1) does not belong to the Roumieu type Gevrey class ℰ^{β}((0, ∞), X).

Now, suppose that

Reλn(k)→−∞,k→∞

Then, without loss of generality, we can regard that

Reλn(k)≤−k,k∈N. (3.36)

Consider the element

f:=∑k=1∞k−2en(k)∈X,

which is well defined since {k−2}k=1∞ ∈ l₁ and ∥e_n(k)∥ = 1, k ∈ ℕ (see (3.20)).

By (3.20),

EA(∪k=1∞Δn(k))f=fandEA(Δn(k))f=k−2en(k),k∈N. (3.37)

For an arbitrary t ≥ 0 and any g^* ∈ X^*,

∫σ(A)etReλdv(f,g∗,λ) by (2.9) as in (3.25);=∑k=1∞k−2∫σ(A)∩Δn(k)etReλdv(en(k),g∗,λ) since, for λ∈Δn(k), by (3.36) and (3.19), Reλ=Reλn(k)+(Reλ−Reλn(k))≤Reλn(k)+|λ−λn(k)|≤−k+1≤0;≤∑k=1∞k−2∫σ(A)∩Δn(k)1dv(en(k),g∗,λ)=∑k=1∞k−2v(en(k),g∗,Δn(k))by (2.6);≤∑k=1∞k−24M∥en(k)∥∥g∗∥=4M∥g∗∥∑k=1∞k−2<∞. (3.38)

Similarly, for an arbitrary t ≥ 0 and any g^* ∈ X^*,

sup{g∗∈X∗|∥g∗∥=1}∫λ∈σ(A)|etReλ>netReλdv(f,g∗,λ)≤sup{g∗∈X∗|∥g∗∥=1}∑k=1∞k−2∫λ∈σ(A)|etReλ>n∩Δn(k)1dv(en(k),g∗,λ) by (3.37);≤sup{g∗∈X∗|∥g∗∥=1}∑k=1∞∫λ∈σ(A)|etReλ>n∩Δn(k)1dv(EA(Δn(k))f,g∗,λ) by (2.9);=sup{g∗∈X∗|∥g∗∥=1}∫{λ∈σ(A)|etReλ>n}1dv(EA(∪k=1∞Δn(k))f,g∗,λ) by (3.37);=sup{g∗∈X∗|∥g∗∥=1}∫{λ∈σ(A)|etReλ>n}1dv(f,g∗,λ) by (2.7);≤sup{g∗∈X∗|∥g∗∥=1}4MEAλ∈σ(A)|etReλ>nf∥g∗∥≤4M∥EA({λ∈σ(A)|etReλ>n})f∥by the strong continuity of the s.m.;→4MEA∅f=0,n→∞. (3.39)

By Proposition 2.1, (3.38) and (3.39) jointly imply that

f∈⋂t≥0D(etA),

and hence, by Theorem 2.1,

y(t):=etAf,t≥0,

is a weak solution of equation (1.1).

Let

h∗:=∑k=1∞k−2en(k)∗∈X∗, (3.40)

the functional being well defined since {k−2}k=1∞ ∈ l₁ and ∥ en(k)∗ ∥ = 1, k ∈ ℕ (see (3.22)).

In view of (3.22) and (3.21), we have:

〈en(k),h∗〉=〈en(k),k−2en(k)∗〉=dn(k)k−2≥εk−2,k∈N. (3.41)

Since, for any λ ∈ Δ_n(k), k ∈ ℕ, by (3.36) and (3.19),

Reλ=Reλn(k)+(Reλ−Reλn(k))≤Reλn(k)+|Reλ−Reλn(k)|≤Reλn(k)+εn(k)≤−k+1≤0 (3.42)

and, by (3.18),

−n(k)−1|Imλ|1/β<Reλ,

we infer that, for any λ ∈ Δ_n(k), k ∈ ℕ,

|λ|≥|Imλ|≥n(k)(−Reλ)β.

Using this estimate, for an arbitrary s > 0 and the functional h^* ∈ X^* defined by (3.59), we have:

∫σ(A)es|λ|1/βeReλdv(f,h∗,λ)by (2.9) as in (3.25);=∑k=1∞k−2∫σ(A)∩Δn(k)es|λ|1/βeReλdv(en(k),h∗,λ)≥∑k=1∞k−2∫σ(A)∩Δn(k)e[sn(k)−1](−Reλ)dv(en(k),h∗,λ)=∞. (3.43)

Indeed, for all k ∈ ℕ sufficiently large so that

sn(k)≥2,

we have:

k−2∫σ(A)∩Δn(k)e[sn(k)−1](−Reλ)dv(en(k),h∗,λ)≥k−2∫σ(A)∩Δn(k)e−Reλdv(en(k),h∗,λ)by (3.42);≥k−2ek−1∫σ(A)∩Δn(k)1dv(en(k),h∗,λ)=k−2ek−1v(en(k),h∗,Δn(k))≥k−2ek−1|〈EA(Δn(k))en(k),h∗〉|by (3.20) and(3.41);≥εk−4ek−1→∞,k→∞.

By Proposition 2.1 and the properties of the o.c. (see [15, Theorem XVIII.2.11 (f)]), (3.43) implies that, for any s > 0,

f∉D(es|A|1/βeA),

which, in view of (2.12), further implies that

y(1)=eAf∉⋃s>0D(es|A|1/β)=E{β}(A).

Whence, by Proposition 2.2, we infer that the weak solution y(t) = e^tAf, t ≥ 0, of equation (1.1) does not belong to the Roumieu type Gevrey class ℰ^{β}((0, ∞), X), which completes our consideration of the case of the sequence’s {Reλn}n=1∞ being unbounded.

With every possibility concerning {Reλn}n=1∞ considered, the proof by contrapositive of the “only if” part is complete and so is the proof of the entire statement.□

For β = 1, we obtain the following important particular case.

Corollary 3.1

(Characterization of the analyticity of weak solutions on (0, ∞)). Let A be a scalar type spectral operator in a complex Banach space (X, ∥⋅∥). Every weak solution of the equation (1.1) is analytic on (0, ∞) iff there exist b₊ > 0 and b_– > 0 such that the set σ(A) ∖ Pb−,b+1 , where

Pb−,b+1:=λ∈C|Reλ≤−b−|Imλ|orReλ≥b+|Imλ|,

is bounded (see Figure 2).

$Figure 2 The case of β = 1.$

Figure 2

The case of β = 1.

Remark 3.1

Thus, we have obtained a generalization of [1, Theorem 4.2], the counterpart for a normal operator A in a complex Hilbert space, and of [2, Theorem 5.1] (cf. [3]), a characterization of the generation of a Roumieu type Gevrey ultradifferentiable C₀-semigroup by a scalar type spectral operator A.

Now, let us treat the Beurling type strong Gevrey ultradifferentiability of order β > 1. Observe that the case of entireness (β = 1) is included in [6, Theorem 4.1] (see also [6, Corollary 4.1]).

Theorem 3.2

Let A be a scalar type spectral operator in a complex Banach space (X, ∥⋅∥) with spectral measure E_A(⋅) and 1 < β < ∞. Every weak solution of equation (1.1) belongs to the βth-order Beurling type Gevrey class ℰ^(β)((0, ∞), X) iff there exists a b₊ > 0 such that, for any b_– > 0, the set σ(A) ∖ Pb−,b+β , where

Pb−,b+β:=λ∈C|Reλ≤−b−|Imλ|1/βorReλ≥b+|Imλ|1/β,

is bounded (see Figure 1).

Proof

“If” Part. Suppose that there exists a b₊ > 0 such that, for any b_– > 0, the set σ(A) ∖ Pb−,b+β is bounded and let y(⋅) be an arbitrary weak solution of equation (1.1).

By Theorem 2.1,

y(t)=etAf,t≥0,with somef∈⋂t≥0D(etA).

Our purpose is to show that y(⋅) ∈ ℰ^(β)((0, ∞), X), which, by Proposition 2.2 and (2.12), is attained by showing that, for each t > 0,

y(t)∈E(β)A=⋂s>0D(es|A|1/β).

Let us proceed by proving that, for any t > 0 and s > 0,

y(t)∈D(es|A|1/β)

via Proposition 2.1.

Since β > 1, for any b_– > 0, there exists a c(b_–) > 0 such that

x≤b−−βxβ,x≥c(b−). (3.44)

Fixing arbitrary t > 0 and s > 0, since b_– > 0 is random, we can set

b−:=21/βst−1>0, (3.45)

such a peculiar choice explaining itself in the process.

For arbitrary t > 0 and s > 0, b_– > 0 chosen as in (3.45), and any g^* ∈ X^*,

Indeed,

∫σ(A)∖Pb−,b+βes|λ|1/βetReλdv(f,g∗,λ)<∞

and

∫λ∈σ(A)∩Pb−,b+β|−c(b−)<Reλ<1es|λ|1/βetReλdv(f,g∗,λ)<∞

due to the boundedness of the sets

σ(A)∖Pb−,b+βandλ∈σ(A)∩Pb−,b+β|−c(b−)<Reλ<1,

the continuity of the integrated function on ℂ, and the finiteness of the measure v(f, g^*, ⋅).

Further, for arbitrary t > 0, s > 0, b_– > 0 chosen as in (3.45), and any g^* ∈ X^*,

Observe that, for the finiteness of the three preceding integrals, the choice of b_– > 0 is superfluous. Finally, for arbitrary t > 0 and s > 0, b_– > 0 chosen as in (3.45), and any g^* ∈ X^*,

Also, for arbitrary t > 0 and s > 0, b_– > 0 chosen as in (3.45), and any g^* ∈ X^*,

sup{g∗∈X∗|∥g∗∥=1}∫λ∈σ(A)|es|λ|1/βetReλ>nes|λ|1/βetReλdv(f,g∗,λ)≤sup{g∗∈X∗|∥g∗∥=1}∫λ∈σ(A)∖Pb−,b+β|es|λ|1/βetReλ>nes|λ|1/βetReλdv(f,g∗,λ)+sup{g∗∈X∗|∥g∗∥=1}∫λ∈σ(A)∩Pb−,b+β|−c(b−)<Reλ<1,es|λ|1/βetReλ>nes|λ|1/βetReλdv(f,g∗,λ)+sup{g∗∈X∗|∥g∗∥=1}∫λ∈σ(A)∩Pb−,b+β|Reλ≥1,es|λ|1/βetReλ>nes|λ|1/βetReλdv(f,g∗,λ)+sup{g∗∈X∗|∥g∗∥=1}∫λ∈σ(A)∩Pb−,b+β|Reλ≤−c(b−),es|λ|1/βetReλ>nes|λ|1/βetReλdv(f,g∗,λ)→0,n→∞. (3.49)

Indeed, since, due to the boundedness of the sets

σ(A)∖Pb−,b+βandλ∈σ(A)∩Pb−,b+β|−c(b−)<Reλ<1

and the continuity of the integrated function on ℂ, the sets

λ∈σ(A)∖Pb−,b+β|es|λ|1/βetReλ>n

and

λ∈σ(A)∩Pb−,b+β|−c(b−)<Reλ<1,es|λ|1/βetReλ>n

are empty for all sufficiently large n ∈ ℕ, we immediately infer that, for any t > 0, s > 0 and b_– > 0 chosen as in (3.45),

limn→∞sup{g∗∈X∗|∥g∗∥=1}∫λ∈σ(A)∖Pb−,b+β|es|λ|1/βetReλ>nes|λ|1/βetReλdv(f,g∗,λ)=0

and

limn→∞sup{g∗∈X∗|∥g∗∥=1}∫λ∈σ(A)∩ Pb−,b+β|−c(b−)<Reλ<1,es|λ|1 /βetReλ>nes|λ|1/βetReλdv(f,g∗,λ)=0.

Further, for arbitrary t > 0, s > 0, b_– > 0 chosen as in (3.45), and any g^* ∈ X^*,

sup{g∗∈X∗|∥g∗∥=1}∫λ∈σ(A)∩ Pb−,b+β|Reλ≥1,es|λ|1/βetReλ>nes|λ|1/βetReλdv(f,g∗,λ) as in (3.47);≤sup{g∗∈X∗|∥g∗∥=1}∫λ∈σ(A)∩ Pb−,b+β|Reλ≥1,es|λ|1/βetReλ>nes1+b+−β1/β+tReλdv(f,g∗,λ) since f∈⋂t≥0D(etA), by (2.7);≤sup{g∗∈X∗|∥g∗∥=1}4MEAλ∈σ(A)∩Pb−,b+β|Reλ≥1,es|λ |1 /βetReλ>nes1+b+−β1 /β+tAf∥g∗∥≤4MEAλ∈σ(A)∩Pb−,b+β|Reλ≥1,es|λ |1 /βetReλ>nes1+b+−β1 /β+tAfby the strong continuity of the s.m.;→4MEA∅es1+b+−β1 /β+tAf=0,n→∞.

Finally, for arbitrary t > 0 and s > 0, b_– > 0 chosen as in (3.45), and any g^* ∈ X^*,

sup{g∗∈X∗|∥g∗∥=1}∫λ∈σ(A)∩Pb−,b+β|Reλ≤−c(b−),es|λ |1 /βetReλ>nes|λ|1/βetReλdv(f,g∗,λ) as in (3.48);≤sup{g∗∈X∗|∥g∗∥=1}∫λ∈σ(A)∩Pb−,b+β|Reλ≤−c(b−),es|λ |1 /βetReλ>net−s21 /βb−−1Reλdv(f,g∗,λ) by the choice of b−>0 (see (3.45));=sup{g∗∈X∗|∥g∗∥=1}∫λ∈σ(A)∩Pb−,b+β|Reλ≤−c(b−),es|λ |1 /βetReλ>n1dv(f,g∗,λ) by (2.7);≤sup{g∗∈X∗|∥g∗∥=1}4MEAλ∈σ(A)∩Pb−,b+β|Reλ≤−c(b−),es |λ|1/βetReλ>nf∥g∗∥≤4MEAλ∈σ(A)∩Pb−,b+β|Reλ≤−c(b−),es |λ|1/βetReλ>nfby the strong continuity of the s.m.;→4MEA∅f=0,n→∞.

By Proposition 2.1 and the properties of the o.c. (see [15, Theorem XVIII.2.11 (f)]), (3.46) and (3.49) jointly imply that, for any t > 0 and s > 0,

f∈D(es|A|1/βetA),

which, in view of (2.12), further implies that, for each t > 0,

y(t)=etAf∈⋂s>0D(es|A|1/β)=E(β)(A).

Whence, by Proposition 2.2, we infer that

y(⋅)∈E(β)((0,∞),X),

which completes the proof of the “if” part.

“Only if” part. Let us prove this part by contrapositive assuming that, for any b₊ > 0, there exists a b_– > 0 such that the set σ(A) ∖ Pb−,b+β is unbounded.

Let us show that, under the circumstances, we can equivalently set the following seemingly stronger hypothesis: there exists a b_– > 0 such that, for any b₊ > 0, the set σ(A) ∖ Pb−,b+β is unbounded.

Indeed, under the premise, there are two possibilities:

For some b_– > 0, the set

λ∈σ(A)|−b−|Imλ|1/β<Reλ≤0

is unbounded.
For any b_– > 0, the set

λ∈σ(A)|−b−|Imλ|1/β<Reλ≤0

is bounded.

In the first case, as is easily seen, the set σ(A) ∖ Pb−,b+β is also unbounded for some b_– > 0 and any b₊ > 0.

In the second case, by the premise, we infer that, for any b₊ > 0, unbounded is the set

λ∈σ(A)|0<Reλ<b+|Imλ|1/β,

which makes the set σ(A) ∖ Pb−,b+β unbounded for any b_– > 0 and b₊ > 0.

The foregoing equivalent version of the premise implies, in particular, that, for some b_– > 0 and any n ∈ ℕ, unbounded is the set

σ(A)∖Pb−,n−2β=λ∈σ(A)|−b−|Imλ|1/β<Reλ<n−2|Imλ|1/β.

Hence, we can choose a sequence λnn=1∞ of points in the complex plane as follows:

The latter implies, in particular, that the points λ_n, n ∈ ℕ, are distinct (λ_i ≠ λ_j, i ≠ j).

Since, for each n ∈ ℕ, the set

λ∈C|−b−|Imλ|1/β<Reλ<n−2|Imλ|1/β,|λ|>max[n,|λn−1|]

is open in ℂ, along with the point λ_n, it contains an open disk

Δn:=λ∈C||λ−λn|<εn

centered at λ_n of some radius ε_n > 0, i.e., for each λ ∈ Δ_n,

−b−|Imλ|1/β<Reλ<n−2|Imλ|1/βand|λ|>max[n,|λn−1|]. (3.50)

Furthermore, under the circumstances, we can regard the radii of the disks to be small enough so that

0<εn<1n,n∈N,andΔi∩Δj=∅,i≠j(i.e., the disks are pairwise disjoint). (3.51)

Whence, by the properties of the s.m.,

EA(Δi)EA(Δj)=0,i≠j,

where 0 stands for the zero operator on X.

Observe also that the subspaces E_A(Δ_n)X, n ∈ ℕ, are nontrivial since

Δn∩σ(A)≠∅,n∈N,

with Δ_n being an open set in ℂ.

By choosing a unit vector e_n ∈ E_A(Δ_n)X for each n ∈ ℕ, we obtain a sequence enn=1∞ such that

∥en∥=1,n∈N, and EA(Δi)ej=δijej, i,j∈N, (3.52)

where δ_ij is the Kronecker delta.

As is easily seen, (3.52) implies that the vectors e_n, n ∈ ℕ, are linearly independent.

Furthermore, there exists an ε > 0 such that

dn:=disten,spanei|i∈N,i≠n≥ε,n∈N. (3.53)

Indeed, the opposite implies the existence of a subsequence dn(k)k=1∞ such that

dn(k)→0,k→∞.

Then, by selecting a vector

fn(k)∈spanei|i∈N,i≠n(k),k∈N,

such that

∥en(k)−fn(k)∥<dn(k)+1/k,k∈N,

we arrive at

1=∥en(k)∥since, by (3.20),EA(Δn(k))fn(k)=0;=∥EA(Δn(k))(en(k)−fn(k))∥≤∥EA(Δn(k))∥∥en(k)−fn(k)∥by (2.5));≤M∥en(k)−fn(k)∥≤Mdn(k)+1/k→0,k→∞,

which is a contradiction proving (3.53).

As follows from the Hahn-Banach Theorem, for any n ∈ ℕ, there is an en∗ ∈ X^* such that

∥en∗∥=1,n∈N,and〈ei,ej∗〉=δijdi,i,j∈N. (3.54)

Let us consider separately the two possibilities concerning the sequence of the real parts {Reλn}n=1∞ : its being bounded or unbounded.

The case of the sequence’s {Reλn}n=1∞ being bounded is considered in absolutely the same manner as the corresponding case in the proof of the “only if” part of Theorem 3.1 and furnishes a weak solution y(⋅) of equation (1.1) such that

y(1)∉E{β}(A).

Hence, by Proposition 2.2, y(⋅) does not belong to the Roumieu type Gevrey class ℰ^{β}((0, ∞), X), and the more so, the narrower Beurling type Gevrey class ℰ^(β)((0, ∞), X).

Now, suppose that the sequence {Reλn}n=1∞ is unbounded.

Therefore, there is a subsequence {Reλn(k)}k=1∞ such that

Reλn(k)→∞orReλn(k)→−∞,k→∞.

Let us consider separately each of the two cases.

The case of

Reλn(k)→∞,k→∞

is also considered in the same manner as the corresponding case in the proof of the “only if” part of Theorem 3.1, and again furnishes a weak solution y(⋅) of equation (1.1) such that

y(1)∉E{β}(A).

Hence, by Proposition 2.2, y(⋅) does not belong to the Roumieu type Gevrey class ℰ^{β}((0, ∞), X), let alone, the narrower Beurling type Gevrey class ℰ^(β)((0, ∞), X).

Suppose that

Reλn(k)→−∞,k→∞.

Then, without loss of generality, we can regard that

Reλn(k)≤−k,k∈N. (3.55)

Consider the element

f:=∑k=1∞k−2en(k)∈X,

which is well defined since {k−2}k=1∞ ∈ l₁ and ∥e_n(k)∥ = 1, k ∈ ℕ (see (3.52)).

By (3.52),

EA(∪k=1∞Δn(k))f=fandEA(Δn(k))f=k−2en(k),k∈N. (3.56)

For arbitrary t ≥ 0 and any g^* ∈ X^*,

∫σ(A)etReλdv(f,g∗,λ) by (2.9) as in (3.25);=∑k=1∞k−2∫σ(A)∩Δn(k)etReλdv(en(k),g∗,λ) since, forλ∈Δn(k),by (3.55) and (3.51),Reλ=Reλn(k)+(Reλ−Reλn(k))≤Reλn(k)+|λ−λn(k)|≤−k+1≤0;≤∑k=1∞k−2∫σ(A)∩Δn(k)1dv(en(k),g∗,λ)=∑k=1∞k−2v(en(k),g∗,Δn(k))by (2.6);≤∑k=1∞k−24M∥en(k)∥∥g∗∥=4M∥g∗∥∑k=1∞k−2<∞. (3.57)

Similarly, for arbitrary t ≥ 0 and any g^* ∈ X^*,

sup{g∗∈X∗|∥g∗∥=1}∫λ∈σ(A)|etReλ>netReλdv(f,g∗,λ) as in (3.57);≤sup{g∗∈X∗|∥g∗∥=1}∑k=1∞k−2∫λ∈σ(A)|etReλ>n∩Δn(k)1dv(en(k),g∗,λ) by (3.56);=sup{g∗∈X∗|∥g∗∥=1}∑k=1∞∫λ∈σ(A)|etReλ>n∩Δn(k)1dv(EA(Δn(k))f,g∗,λ) by (2.9);=sup{g∗∈X∗|∥g∗∥=1}∫{λ∈σ(A)|etReλ>n}1dv(EA(∪k=1∞Δn(k))f,g∗,λ) by(3.56);=sup{g∗∈X∗|∥g∗∥=1}∫{λ∈σ(A)|etReλ>n}1dv(f,g∗,λ) by(2.7);≤sup{g∗∈X∗|∥g∗∥=1}4MEAλ∈σ(A)|etReλ>nf∥g∗∥≤4M∥EA({λ∈σ(A)|etReλ>n})f∥by the strong continuity of the s.m.;→4MEA∅f=0,n→∞. (3.58)

By Proposition 2.1, (3.57) and (3.58) jointly imply that

f∈⋂t≥0D(etA),

and hence, by Theorem 2.1,

y(t):=etAf,t≥0,

is a weak solution of equation (1.1).

Let

h∗:=∑k=1∞k−2en(k)∗∈X∗, (3.59)

the functional being well defined since {k−2}k=1∞ ∈ l₁ and ∥ en(k)∗ ∥ = 1, k ∈ ℕ (see (3.54)).

In view of (3.54) and (3.53), we have:

〈en(k),h∗〉=〈en(k),k−2en(k)∗〉=dn(k)k−2≥εk−2,k∈N. (3.60)

Since, for any λ ∈ Δ_n(k), k ∈ ℕ, by (3.55) and (3.51),

Reλ=Reλn(k)+(Reλ−Reλn(k))≤Reλn(k)+|Reλ−Reλn(k)|≤Reλn(k)+εn(k)≤−k+1≤0 (3.61)

and, by (3.50),

−b−|Imλ|1/β<Reλ,

we infer that, for any λ ∈ Δ_n(k), k ∈ ℕ,

|λ|≥|Imλ|≥b−−1(−Reλ)β.

Using this estimate, for

s:=2b−>0 (3.62)

and the functional h^* ∈ X^* defined by (3.59), we have:

∫σ(A)es|λ|1/βeReλdv(f,h∗,λ)by (2.9) as in (3.25);=∑k=1∞k−2∫σ(A)∩Δn(k)es|λ|1/βeReλdv(en(k),h∗,λ)≥∑k=1∞k−2∫Δn(k)e[sb−−1−1](−Reλ)dv(en(k),h∗,λ)since s:=2b−>0 (see (3.62));=∑k=1∞k−2∫σ(A)∩Δn(k)e−Reλdv(en(k),h∗,λ) by (3.61);≥∑k=1∞k−2ek−1∫σ(A)∩Δn(k)1dv(en(k),h∗,λ)=∑k=1∞k−2ek−1v(en(k),h∗,Δn(k))≥∑k=1∞k−2ek−1|〈EA(Δn(k))en(k),h∗〉|by (3.52) and (3.60);≥∑k=1∞εk−4ek−1=∞. (3.63)

By Proposition 2.1 and the properties of the o.c. (see [15, Theorem XVIII.2.11 (f)]), (3.63) implies that

f∉D(es|A|1/βeA)

with s = 2b_– > 0, which, in view of (2.12), further implies that

y(1)=eAf∉⋂s>0D(es|A|1/β)=E(β)(A).

Whence, by Proposition 2.2, we infer that the weak solution y(t) = e^tAf, t ≥ 0, of equation (1.1) does not belong to the Beurling type Gevrey class ℰ^(β)((0, ∞), X), which completes our consideration of the case of the sequence’s {Reλn}n=1∞ being unbounded.

With every possibility concerning {Reλn}n=1∞ considered, the proof by contrapositive of the “only if” part is complete and so is the proof of the entire statement.□

Remark 3.2

Thus, we have obtained a generalization of [1, Theorem 4.3], the counterpart for a normal operator A in a complex Hilbert space, and of [4, Corollary 4.1], a characterization of the generation of a Berling type Gevrey ultradifferentiable C₀-semigroup by a scalar type spectral operator A.

4 Inherent Smoothness Improvement Effect

Now, let us see that there is more to be said about the important particular case of analyticity (β = 1) in Theorem 3.1 (see Corollary 3.1).

Proposition 4.1

Let A be a scalar type spectral operator in a complex Banach space (X, ∥⋅∥). If every weak solution of equation (1.1) is analytically continuable into a complex neighborhood of (0, ∞) (each one into its own), then all of them are analytically continuable into the open sector

Σθ:=λ∈C||arg⁡λ|<θ∖{0}

with

θ:=sup0<φ<π/2|λ∈σ(A)|Reλ<0,|arg⁡λ|≤π/2+φis bounded,

where –π < argλ ≤ π is the principal value of the argument of λ (arg 0: = 0).

Proof

By Corollary 3.1, the analyticity of all weak solutions of equation (1.1) on (0, ∞) is equivalent to the existence of b₊ > 0 and b_– > 0 such that the set

σ(A)∖λ∈C|Re≤−b−|Imλ|orRe≥b+|Imλ|

is bounded (see Figure 2).

As is easily seen, this implies, in particular, that the set

Φ:=0<φ<π/2|λ∈σ(A)|Reλ<0,|arg⁡λ|≤π/2+φis bounded≠∅.

For any φ ∈ Φ,

A=Aφ−+Aφ+,

where the scalar type spectral operators Aφ− and Aφ+ are defined as follows:

Aφ−:=AEAλ∈σ(A)||arg⁡λ|≥π/2+φ,Aφ+:=AEAλ∈σ(A)||arg⁡λ|<π/2+φ

(see [15, Theorem XVIII.2.11 (f)]).

By the properties of the o.c. (see [15, Theorem XVIII.2.11 (h), (c)]), for any φ ∈ Φ,

σ(Aφ−)⊆λ∈σ(A)||arg⁡λ|≥π/2+φ∪{0},σ(Aφ+)⊆λ∈σ(A)||arg⁡λ|≤π/2+φ.

Hence, by [12, Proposition 4.1] (cf. also [2]), for any φ ∈ Φ, the operator Aφ− generates the C₀-semigroup etAφ−t≥0 of the operator exponentials (see Preliminaries) analytic in the open sector

Σφ:=λ∈C|arg⁡λ<φ∖{0}

(see also [9]).

As follows from the premise, for any φ ∈ Φ, the set

σ(Aφ+)∖λ∈C|Reλ≥b+|Imλ|,

is bounded, which, by [6, Corollary 4.1], implies that all weak solutions of the equation

y′(t)=Aφ+y(t),t≥0,

i.e., by Theorem 2.1, all vector functions of the form

y(t)=etAφ+f,t≥0,f∈⋂t≥0D(etAφ+)

are entire.

By the properties of the o.c. (see [15, Theorem XVIII.2.11]),

etA=etAφ−+etAφ+−I,t≥0.

In view of the fact that

D(etAφ−)=X,t≥0,

for each

f∈⋂t≥0D(etA)=⋂t≥0D(etAφ+),

the vector function

y+(t):=etAφ+−If,t≥0,

is entire, whereas the vector function

y−(t):=etAφ−f,t≥0,

is analytically continuable into the open sector Σ_φ, which makes the vector function

y(t):=etAf=y−(t)+y+(t),t≥0,

to be analytically continuable into the open sector Σ_φ.

Considering that

φ∈Φandf∈⋂t≥0D(etA)

are arbitrary, by Theorem 2.1, we infer that every weak solution of equation (1.1) is analytically continuable into the open sector

Σθ:=λ∈C||arg⁡λ|<θ∖{0}

with θ := sup Φ.□

Remarks 4.1

Thus, we have obtained a generalization of [1, Proposition 5.2], the counterpart for a normal operator A in a complex Hilbert space.
It is noteworthy that Corollary 3.1 (i.e., Theorem 3.1 with β = 1) and Proposition 4.1 with θ = π/2 apply to equation (1.1) with a self-adjoint operator in a complex Hilbert space, which implies that, for such an equation, all weak solutions are analytically continuable into the open right half-plane

λ∈C|Reλ>0

(see [1, Corollary 5.1] and, for symmetric operators, [1, Theorem 6.1]).

5 Concluding remark

Due to the scalar type spectrality of the operator A, Theorems 3.1 and 3.2 are stated exclusively in terms of the location of its spectrum in the complex plane, similarly to the celebrated Lyapunov stability theorem [40] (cf. [9, Ch. I, Theorem 2.10]), and thus, are intrinsically qualitative statements (cf. [5, 6, 41]).

Acknowledgments

The author extends sincere gratitude to his colleague, Dr. Maria Nogin of the Department of Mathematics, California State University, Fresno, for her kind assistance with the graphics.

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Received: 2019-05-28

Accepted: 2019-08-07

Published Online: 2019-09-27

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

Articles in the same Issue

https://doi.org/10.1515/math-2019-0083

Keywords for this article

weak solution; scalar type spectral operator; Gevrey classes

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