Properties of multiplication operators on the space of functions of bounded φ-variation

René E. Castillo; Julio C. Ramos-Fernández; Harold Vacca-González

doi:10.1515/math-2021-0050

Article Open Access

Properties of multiplication operators on the space of functions of bounded φ-variation

René E. Castillo , Julio C. Ramos-Fernández and Harold Vacca-González

Published/Copyright: June 14, 2021

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$Open Mathematics$

From the journal Open Mathematics Volume 19 Issue 1

Abstract

In this paper, the functions u ∈ B V φ [ 0 , 1 ] which define compact and Fredholm multiplication operators M u acting on the space of functions of bounded φ -variation are studied. All the functions u ∈ B V φ [ 0 , 1 ] which define multiplication operators M u : B V φ [ 0 , 1 ] → B V φ [ 0 , 1 ] with closed range are characterized.

Keywords: multiplication operator; bounded variation functions; compact operators; Fredholm operators

MSC 2010: 47B38; 26A45; 26B30; 46E40

1 Introduction

In 1881, Camille Jordan [1] introduced the notion of functions of bounded variation and established the relation between those functions and monotonic ones when he was studying convergence of Fourier series. Later on the concept of bounded variation was generalized in various directions by many mathematicians such as F. Riesz, N. Wiener, R. E. Love, H. Ursell, L. C. Young, W. Orlicz, J. Musielak, L. Tonelli, L. Cesari, R. Caccioppoli, E. de Giorgi, O. Olcinik, E. Conway, J. Smoller, A. Volpert, S. Hudjacv, L. Ambrosio, G. Dal Maso, among many others. It is noteworthy to mention that many of these generalizations were motivated by problems in such areas as calculus of variation, convergence of Fourier series, geometric measure theory, mathematical physics, etc. For many applications of functions of bounded variation see, e.g., the monograph [2].

On the other side, the space of functions of bounded p -variation, introduced by Wiener in 1924 (see [3]), led in 1937 to Young [4] to give a generalization of the concept of function of bounded variation by introducing the notion of φ -variation of a function; this concept, in turn, was generalized by Chistyakov, [5], for functions which take values in a linear normed vector space. We recommend the excellent book of Appell et al. [6] for the study of the properties of functions of bounded φ -variation.

The aim of this article is to study the properties of multiplication operator acting on B V φ [ 0 , 1 ] spaces, which we define in Section 2. Each function u ∈ B V φ [ 0 , 1 ] defines a linear and continuous operator M u : B V φ [ 0 , 1 ] → B V φ [ 0 , 1 ] given by M u ( f ) = u ⋅ f which is known as multiplication operator with symbol u . This operator has been widely studied in the context of spaces of measurable functions. We mention here the pioneering work by Singh and Kumar in [7,8] on properties of multiplication operators acting on spaces of measurable functions. These authors studied the compactness and closedness of the range of multiplication operators on L 2 ( μ ) . Additionally, we note that the work of Arora et al. in [9,10,11] examined properties of M u on Lorentz and Lorentz-Bochner spaces. Further significant results regarding multiplication operators were obtained by Castillo et al. in [12,13,14], in which these authors showed that the techniques used by previously mentioned authors can be modified to study multiplication operators on weak L p spaces, Orlicz-Lorentz spaces, and variable Lebesgue spaces. The most complete study of the properties of multiplication operators acting on measurable function spaces was carried out by Hudzik et al. [15] (see also [16] and [17]). However, in the context of spaces of functions with certain type of bounded variation, it is a recent topic, being the first work due to Astudillo-Villalba and Ramos-Fernández [18] (see also [19]) where the authors made an exhaustive study of the properties of multiplication operator acting on the space of functions of bounded variation.

Consequently, the main objective of this article is to extend the results achieved in [18] to more general setting of space of functions of bounded φ -variation. With the above end, in Section 2, we give some properties of functions of bounded φ -variation. Section 3 is dedicated to characterize all symbols u ∈ B V φ [ 0 , 1 ] which define multiplication operators having closed range; while in Section 4 we study the compactness of M u making special emphasis on the properties of the symbol u which define multiplication operators with finite dimensional range. Finally, in Section 5, we use the results obtained in Sections 4 and 5 to characterize the symbols u ∈ B V φ [ 0 , 1 ] which define Fredholm multiplication operators.

2 Some remarks on functions of bounded φ -variation

For the reader’s ease, in this section we collect some properties of φ -functions which will be of great utility in the proof of our results. A function φ : [ 0 , ∞ ) → [ 0 , ∞ ) is said to be a Young function (or φ -function) if it is convex, strictly increasing and such that φ ( 0 ) = 0 . In particular, according to this definition we also have that φ ( t ) → ∞ as t → ∞ . Thus, if φ is a fixed Young function, then we can define the φ -variation (also known as the Young variation) of a real-valued function f defined on [ 0 , 1 ] as

Var φ ( f ) = sup ∑ k = 1 n φ ( ∣ f ( t k ) − f ( t k − 1 ) ∣ ) ,

where the supremum is taken over all finite partitions P : 0 = t 0 < t 1 < ⋯ < t n = 1 of [ 0 , 1 ] . With this notation, we say that a real-valued function f defined on [ 0 , 1 ] belongs to the space of bounded φ -variation functions B V φ [ 0 , 1 ] (which was introduced by Young in 1937, see [4]) if there exists a λ > 0 such that

Var φ 1 λ f < ∞ .

For each f ∈ B V φ [ 0 , 1 ] , we can define the quantity

‖ f ‖ φ = inf λ > 0 : Var φ 1 λ f ≤ 1 ,

which is a seminorm for B V φ [ 0 , 1 ] (known as the Luxemburg seminorm) and B V φ [ 0 , 1 ] is a Banach space with the norm

‖ f ‖ B V φ = ‖ f ‖ ∞ + ‖ f ‖ φ ,

where

‖ f ‖ ∞ ≔ sup t ∈ [ 0 , 1 ] ∣ f ( t ) ∣ .

In fact, for all f , g ∈ B V φ [ 0 , 1 ] we have that f ⋅ g ∈ B V φ [ 0 , 1 ] and

(2.1) ‖ f ⋅ g ‖ B V φ ≤ ‖ f ‖ B V φ ‖ g ‖ B V φ ,

which means that B V φ [ 0 , 1 ] is a Banach algebra. When φ ( t ) = t we have the classical space of bounded variation; while when φ ( t ) = t p , the space B V φ [ 0 , 1 ] becomes into W B V p [ 0 , 1 ] , the space of functions of bounded p -variation, introduced by Wiener in 1924 (see [3]).

We fix a Young function φ and consider the space B V φ [ 0 , 1 ] defined by φ . Next, we have the following:

If f ∈ B V φ [ 0 , 1 ] , then there exists λ > 0 such that
Var φ 1 λ f < ∞ .
Let 0 < β ≤ 1 λ . Then 0 < β λ ≤ 1 . Since φ is convex and φ ( 0 ) = 0 ,
Var φ ( β f ) = Var φ λ β f λ + ( 1 − λ β ) 0 ≤ λ β Var φ 1 λ f < ∞ .
Hence,
0 ≤ lim β → 0 Var φ ( β f ) ≤ lim β → 0 λ β Var φ 1 λ f = 0
and so
lim β → 0 Var φ ( β f ) = 0 .
By (a), there exists β > 0 such that
Var φ ( β f ) ≤ 1 .
Let λ = 1 β . And so, there exists λ > 0 such that
Var φ 1 λ f ≤ 1 .
Hence, we have proved that
λ > 0 : Var φ 1 λ f ≤ 1 ≠ ∅ .
The quantity
(2.2) ‖ f ‖ φ = inf λ > 0 : Var φ 1 λ f ≤ 1
satisfies the following property:
(2.3) Var φ f ‖ f ‖ φ ≤ 1
whenever f ∈ B V φ [ 0 , 1 ] is not a constant function. In fact, f is a constant function on [ 0 , 1 ] if and only if Var φ ( f ) = 0 which occurs if and only if ‖ f ‖ φ = 0 . In particular, we have
λ > 0 : Var φ 1 λ f ≤ 1 = ( ‖ f ‖ φ , + ∞ ) .
Hence, ‖ f ‖ φ ≤ k if and only if Var φ 1 k f ≤ 1 . Furthermore, the relation (2.3) implies that for all f ∈ B V φ [ 0 , 1 ] and t ∈ [ 0 , 1 ] we have
∣ f ( t ) ∣ ≤ ∣ f ( 0 ) ∣ + φ − 1 ( 1 ) ‖ f ‖ φ
and B V φ [ 0 , 1 ] ⊊ B [ 0 , 1 ] , the space of all bounded functions defined on [ 0 , 1 ] , and hence we can write
‖ f ‖ ∞ ≤ ∣ f ( 0 ) ∣ + φ − 1 ( 1 ) ‖ f ‖ φ .
We also have that if f , g ∈ B V φ [ 0 , 1 ] , then f ⋅ g ∈ B V φ [ 0 , 1 ] and
(2.4) ‖ f ⋅ g ‖ φ ≤ ‖ f ‖ ∞ ‖ g ‖ φ + ‖ g ‖ ∞ ‖ f ‖ φ .
We include a proof of the above inequality to illustrate the use of the property (2.3) and the definition of Luxemburg seminorm (2.2). We can suppose that ‖ f ‖ φ > 0 and ‖ g ‖ φ > 0 . We set
M = ‖ f ‖ ∞ ‖ g ‖ φ + ‖ g ‖ ∞ ‖ f ‖ φ ,
then for any partition P : 0 = t 0 < t 1 < ⋯ < t n = 1 we can write
∑ k = 1 n φ 1 M ∣ f ( t k ) g ( t k ) − f ( t k − 1 ) g ( t k − 1 ) ∣ = ∑ k = 1 n φ 1 M ∣ ( f ( t k ) − f ( t k − 1 ) ) g ( t k ) + f ( t k − 1 ) ( g ( t k ) − g ( t k − 1 ) ) ∣ ≤ ∑ k = 1 n φ ‖ g ‖ ∞ M ∣ f ( t k ) − f ( t k − 1 ) ∣ + ‖ f ‖ ∞ M ∣ g ( t k ) − g ( t k − 1 ) ∣ = ∑ k = 1 n φ ‖ g ‖ ∞ ‖ f ‖ φ M ∣ f ( t k ) − f ( t k − 1 ) ∣ ‖ f ‖ φ + ‖ f ‖ ∞ ‖ g ‖ φ M ∣ g ( t k ) − g ( t k − 1 ) ∣ ‖ g ‖ φ ≤ 1 M ‖ g ‖ ∞ ‖ f ‖ φ Var φ f ‖ f ‖ φ + 1 M ‖ f ‖ ∞ ‖ g ‖ φ Var φ g ‖ g ‖ φ ≤ 1 M ‖ g ‖ ∞ ‖ f ‖ φ + 1 M ‖ f ‖ ∞ ‖ g ‖ φ = 1 ,
where we have used the convexity of φ in the second inequality and the property (2.3) in the last inequality. Thus, since the partition P was arbitrary, we obtain that
Var φ f ⋅ g M ≤ 1
which proves that ‖ f ⋅ g ‖ φ ≤ M as affirmed.

In this kind of spaces, the following functions will play an important role. For a subset B of [ 0 , 1 ] , the characteristic function of B is denoted by 1 B and it is defined by

1 B ( t ) = 0 , if t ∉ B , 1 , if t ∈ B .

This is known as an indicatrix function. In particular, we have the following very useful property:

Proposition 2.1

For each a ∈ [ 0 , 1 ] , the function 1 { a } belongs to B V φ [ 0 , 1 ] and

(2.5) 1 + 1 φ − 1 ( 1 ) ≤ ‖ 1 { a } ‖ B V φ ≤ 1 + 1 φ − 1 1 2 .

Proof

Indeed, clearly ‖ 1 { a } ‖ ∞ = 1 , hence, we shall estimate ‖ 1 { a } ‖ φ . To this end, let P : 0 = t 0 < t 1 < ⋯ < t n = 1 be any partition of [ 0 , 1 ] . If a ∉ P , then

∑ k = 1 n φ 1 λ ∣ 1 { a } ( t k ) − 1 { a } ( t k − 1 ) ∣ = 0

for all λ > 0 . If a ∈ P , then there exists k 0 ∈ { 0 , 1 , 2 , … , n } such that t k 0 = a . If k 0 = 0 or k 0 = n , then

∑ k = 1 n φ 1 λ ∣ 1 { a } ( t k ) − 1 { a } ( t k − 1 ) ∣ = φ 1 λ ∣ 1 { a } ( t k 0 ) ∣ = φ 1 λ ≤ 1

whenever λ ≥ 1 φ − 1 ( 1 ) ; while if k 0 ∈ { 1 , 2 , … , n − 1 } , then we have

∑ k = 1 n φ 1 λ ∣ 1 { a } ( t k ) − 1 { a } ( t k − 1 ) ∣ = 2 φ 1 λ ∣ 1 { a } ( t k 0 ) ∣ = 2 φ 1 λ ≤ 1

whenever λ ≥ 1 φ − 1 1 2 . Thus, since φ is an increasing function, we conclude that

‖ 1 { a } ‖ φ ≤ 1 φ − 1 1 2

and we have the upper bound in (2.5). Finally, if we consider the partition P 0 = { 0 , a , 1 } , then we have

Var φ 1 1 φ − 1 ( 1 ) 1 { a } = Var φ ( φ − 1 ( 1 ) 1 { a } ) ≥ φ ( φ − 1 ( 1 ) 1 { a } ( a ) ) = 1

and by definition of Luxemburg seminorm, we conclude that

‖ 1 { a } ‖ φ ≥ 1 φ − 1 ( 1 ) .

And so the proposition is proved.□

Remark

It is important to remark that from the proof of the above proposition, we can see that

∥ 1 { 0 } ∥ B V φ = ∥ 1 { 1 } ∥ B V φ = 1 + 1 φ − 1 ( 1 ) ,

while if a ∈ ( 0 , 1 ) , then

‖ 1 { a } ‖ B V φ = 1 + 1 φ − 1 1 2

which gives us a more precise information.

3 Multiplication operators with closed range on B V φ [ 0 , 1 ]

In this section, we characterize all functions u ∈ B V φ [ 0 , 1 ] which define multiplication operators M u : B V φ [ 0 , 1 ] → B V φ [ 0 , 1 ] with closed range. It is convenient, from now, to define the following set:

Z u = { t ∈ [ 0 , 1 ] : u ( t ) = 0 } ;

hence, Z u c = [ 0 , 1 ] ⧹ Z u = { t ∈ [ 0 , 1 ] : u ( t ) ≠ 0 } . Also, for ε > 0 fixed, we set

(3.1) E ε = { t ∈ [ 0 , 1 ] : ∣ u ( t ) ∣ ≥ ε } ,

with these notations, we have the following result.

Theorem 3.1

Suppose that u ∈ B V φ [ 0 , 1 ] . The operator M u : B V φ [ 0 , 1 ] → B V φ [ 0 , 1 ] has closed range if and only if there exists an ε > 0 such that Z u c = E ε .

Proof

The result is trivial if u is the null function. Let us suppose first that there exists an ε > 0 such that Z u c ⊆ E ε . Then ∣ u ( t ) ∣ ≥ ε for all t ∈ Z u c . We shall prove that Ran ( M u ) is a closed subset of B V φ [ 0 , 1 ] . To see this, suppose that h ∈ Ran ( M u ) ¯ . Then there exists a sequence { h n } ⊆ Ran ( M u ) such that

lim n → ∞ ‖ h n − h ‖ B V φ = 0 .

Then, for each n ∈ N we can find f n ∈ B V φ [ 0 , 1 ] such that h n = u ⋅ f n . In particular, h n ( t ) = 0 for all t ∈ Z u . This last fact implies that h ( t ) = 0 for all t ∈ Z u since

∣ h n ( t ) − h ( t ) ∣ ≤ ‖ h n − h ‖ ∞ ≤ ‖ h n − h ‖ B V φ

for all t ∈ [ 0 , 1 ] .

Next, we consider the function f u defined by

(3.2) f u ( t ) = 1 u ( t ) , if t ∈ Z u c = [ 0 , 1 ] ⧹ Z u , 0 , otherwise .

We shall prove that f u ∈ B V φ [ 0 , 1 ] . To this end, let P : 0 = t 0 < t 1 < t 2 < ⋯ < t m = 1 be any partition of [ 0 , 1 ] ; then we have the following cases:

Case I

If t k , t k − 1 ∈ Z u , then f u ( t k ) = f u ( t k − 1 ) = 0 and in this case we have

φ 1 λ ∣ f u ( t k ) − f u ( t k − 1 ) ∣ = 0

for all λ > 0 .

Case II

If t k , t k − 1 ∈ Z u c , then

φ 1 λ ∣ f u ( t k ) − f u ( t k − 1 ) ∣ = φ 1 λ ∣ u ( t k ) − u ( t k − 1 ) ∣ ∣ u ( t k ) ∣ ∣ u ( t k − 1 ) ∣ ≤ φ 1 λ ε 2 ∣ u ( t k ) − u ( t k − 1 ) ∣ ≤ φ 1 ‖ u ‖ B V φ ∣ u ( t k ) − u ( t k − 1 ) ∣

whenever λ > 1 ε 2 ‖ u ‖ B V φ , where we have used that φ is increasing function and Z u c = E ε .

Case III

If t k ∈ Z u c and t k − 1 ∈ Z u , then

φ 1 λ ∣ f u ( t k ) − f u ( t k − 1 ) ∣ = φ 1 λ 1 ∣ u ( t k ) ∣ ≤ φ 1 λ ε 2 ∣ u ( t k ) − u ( t k − 1 ) ∣ ≤ φ 1 ‖ u ‖ B V φ ∣ u ( t k ) − u ( t k − 1 ) ∣

whenever λ > 1 ε 2 ‖ u ‖ B V φ where we have used that φ is an increasing function. The above estimation is also valid for the case t k ∈ Z u and t k − 1 ∈ Z u c .

Next, taking into account the sum in each of the four cases, we obtain

∑ k = 1 m φ 1 λ ∣ f u ( t k ) − f u ( t k − 1 ) ∣ ≤ ∑ k = 1 m φ 1 ‖ u ‖ B V φ ∣ u ( t k ) − u ( t k − 1 ) ∣ ≤ Var φ 1 ‖ u ‖ B V φ u ≤ 1

whenever λ > 1 ε 2 ‖ u ‖ B V φ and since the partition P : 0 = t 0 < t 1 < t 2 < ⋯ < t m = 1 was arbitrary, we obtain that

Var φ 1 λ f u ≤ 1

and f u ∈ B V φ [ 0 , 1 ] .

Finally, if we set f = h ⋅ f u , we have f ∈ B V φ [ 0 , 1 ] since this space is an algebra and h = u ⋅ f which means that h ∈ Ran ( M u ) . This proves that Ran ( M u ) is a closed subspace of B V φ [ 0 , 1 ] .

Next, we suppose that M u : B V φ [ 0 , 1 ] → B V φ [ 0 , 1 ] has closed range and that the conclusion is false. Then, for each n ∈ N we can find t n ∈ Z u c such that

(3.3) 0 < ∣ u ( t n ) ∣ < 1 n 2 .

In particular, the sequence { t n } is an infinite set, it is contained in [ 0 , 1 ] and hence it has a convergent subsequence. Thus, we can suppose { t n } is a convergent sequence. Furthermore, by passing to another subsequence we may assume that { t n } is an ordered and convergent set (recall that from a convergent sequence we can build a monotone subsequence) such that t n ≠ t m for n ≠ m . Also, from (3.3) we can see that the series ∑ k = 1 ∞ u ( t k ) is absolutely convergent and the order of the summation does not affect the convergence.

Next, for each n ∈ N we consider the set A 2 n + 1 = { t 1 , t 3 , t 5 , … , t 2 n + 1 } and we define the function

1 A 2 n + 1 = ∑ k = 0 n 1 { t 2 k + 1 } .

Then clearly 1 A 2 n + 1 ∈ B V φ [ 0 , 1 ] because it is a finite sum of functions in this space and hence h n = u ⋅ 1 A 2 n + 1 ∈ Ran ( M u ) . Furthermore, if n , m ∈ N and n > m , then

∥ h n − h m ∥ B V φ = u ⋅ ∑ k = m + 1 n 1 { t 2 k + 1 } B V φ = ∑ k = m + 1 n u ( t 2 k + 1 ) 1 { t 2 k + 1 } B V φ ≤ ∑ k = m + 1 n ∣ u ( t 2 k + 1 ) ∣ ‖ 1 { t 2 k + 1 } ‖ B V φ ≤ 1 + 1 φ − 1 1 2 ∑ k = m + 1 n ∣ u ( t 2 k + 1 ) ∣ → 0 as m → ∞ .

Thus, { h n } is a Cauchy sequence in the closed set Ran ( M u ) and because of this fact, there exists a function f ∈ B V φ [ 0 , 1 ] such that ‖ h n − u ⋅ f ‖ B V φ → 0 as n → ∞ . In particular, ‖ u ⋅ 1 A 2 n + 1 − u ⋅ f ‖ ∞ → 0 as n → ∞ . Thus, if we fix m ∈ N and we consider t ∈ A 2 m + 1 , then for all n ≥ m we have that t ∈ A 2 n + 1 and therefore

∣ u ( t ) 1 A 2 m + 1 ( t ) − u ( t ) f ( t ) ∣ ≤ ∥ u ⋅ 1 A 2 n + 1 − u ⋅ f ∥ ∞

for all n ≥ m and hence ∣ u ( t ) 1 A 2 m + 1 ( t ) − u ( t ) f ( t ) ∣ = 0 . This last fact implies that f ( t ) = 1 for all t ∈ A ∞ ≔ { t 1 , t 3 , t 5 , … } because u ( t ) ≠ 0 for all t ∈ A ∞ and f ( t 2 n ) = 0 for all n ∈ N .

We will see that a function f with this property cannot belong to B V φ [ 0 , 1 ] . Indeed, if f ∈ B V φ [ 0 , 1 ] , then we can find a constant λ f > 0 such that

Var φ 1 λ f f < ∞ .

Thus for each n ∈ N , we consider the partition P = { 0 , t 1 , t 2 , … , t n − 1 , t n , 1 } (remember that { t n } is an ordered set) and we have

Var φ 1 λ f f ≥ ∑ k = 2 n φ 1 λ f ∣ f ( t k ) − f ( t k − 1 ) ∣ = ( n − 1 ) φ 1 λ f → ∞ ,

as n → ∞ . Where, in the last equality, we have used the fact that f ( t 2 k − 1 ) = 1 and f ( t 2 k ) = 0 . This implies that f does not belong to B V φ [ 0 , 1 ] and we have a contradiction. Therefore, there exists a δ > 0 such that ∣ u ( t ) ∣ ≥ δ for all t ∈ Z u c .□

Remark

As an important application of Theorem 3.1 we can see that when φ ( t ) = t , we have Theorem 8 in [18], while if φ ( t ) = t p with p > 1 we have Theorem 7 in [19].

4 On the compactness

In this section, we analyze when a function u ∈ B V φ [ 0 , 1 ] defines compact multiplication operators M u acting on B V φ [ 0 , 1 ] . We recall that an operator T : X → X is said to be compact if { T ( x n ) } has a convergent subsequence in X for all bounded sequences { x n } ⊂ X . It is known that the limit of compact operators is also a compact operator and that the identity operator I : X → X defined by I f = f is compact if and only if dim ( X ) < ∞ . We recall that for ε > 0 we set

(4.1) E ε = { t ∈ [ 0 , 1 ] : ∣ u ( t ) ∣ ≥ ε } ,

then we have the following result.

Theorem 4.1

Suppose that u ∈ B V φ [ 0 , 1 ] . If the operator M u is compact on B V φ [ 0 , 1 ] , then for each ε > 0 the set E ε is finite.

Proof

Suppose that M u : B V φ [ 0 , 1 ] → B V φ [ 0 , 1 ] is a compact operator and that there exists an ε > 0 such that E ε is an infinite set. Then E ε has a sequence { t n } such that t n ≠ t m for n ≠ m and the sequence of functions { f n } defined by

(4.2) f n ( t ) = 1 t n ( t ) = 1 , if t = t n , 0 , if t ≠ t n ,

are linearly independent and belong to the space

X ε = { f ∈ B V φ [ 0 , 1 ] : f ( t ) = 0 ∀ t ∈ [ 0 , 1 ] ⧹ E ε } ,

hence we can conclude that dim ( X ε ) = ∞ . Furthermore, if { g n } is a sequence in X ε such that ‖ g n − g ‖ B V φ goes to zero as n → ∞ for some g ∈ B V φ [ 0 , 1 ] , then in particular ‖ g n − g ‖ ∞ → 0 as n → ∞ and h ( t ) = 0 for all t ∈ [ 0 , 1 ] ⧹ E ε which means that g ∈ X ε and X ε is a closed subspace of B V φ [ 0 , 1 ] .

The inclusion operator i X ε : X ε → B V φ [ 0 , 1 ] defined by i X ε f = f is continuous, and therefore, the composition M u ∘ i X ε : X ε → B V φ [ 0 , 1 ] is a compact operator. We shall show that Ran ( M u ∘ i X ε ) = X E ε . It is clear that Ran ( M u ∘ i X ε ) ⊆ X E ε and if f ∈ X ε , then we consider the function h : [ 0 , 1 ] → R defined by

h ( t ) = f ( t ) u ( t ) , if t ∈ E ε , 0 , otherwise .

We have that f ( t ) = u ( t ) ⋅ h ( t ) for all t ∈ [ 0 , 1 ] , hence we need to show that h ∈ B V φ [ 0 , 1 ] . To see that, we consider, as in the proof of Theorem 3.1, four cases for the function

h u ( t ) = 1 u ( t ) , if t ∈ E ε , 0 , otherwise ,

depending on t k , t k − 1 ∈ E ε c , t k , t k − 1 ∈ E ε , t k ∈ E ε c and t k − 1 ∈ E ε or t k ∈ E ε and t k − 1 ∈ E ε c and we conclude that h u ∈ B V φ [ 0 , 1 ] , which implies that h = f ⋅ h u ∈ B V φ [ 0 , 1 ] and f ∈ Ran ( M u ∘ i X ε ) . This proves what we claimed.

Next, we have that the operator M u ∘ i X ε : X ε → X ε is a surjective and compact operator and since M u ∘ i X ε : X ε → X ε is injective, we conclude that M u ∘ i X ε : X ε → X ε is a bijective and compact operator. Hence, we have arrived to the following conclusion: X ε is a finite dimensional space, which leads a contradiction with the conclusion of the first part of the proof.□

For the converse, we need to impose a reasonable condition to the Young function φ . The Young function φ : [ 0 , ∞ ) → [ 0 , ∞ ) satisfies the global Δ 2 condition if there exists a constant K Δ > 0 such that

φ ( 2 t ) ≤ K Δ φ ( t )

for all t ≥ 0 . For instance, the Young function φ 1 ( t ) = t log ( 1 + t ) satisfies the global Δ 2 condition but the Young function φ 2 ( t ) = e t − 1 does not satisfy the global Δ 2 condition. We also recall that a continuous operator S : X → X has finite range if dim ( Ran ( S ) ) < ∞ . It is well known that each operator having finite range is a compact operator. With this notation and definition, we have the following result:

Theorem 4.2

Suppose that the Young function φ : [ 0 , ∞ ) → [ 0 , ∞ ) satisfies the global Δ 2 condition and that u ∈ B V φ [ 0 , 1 ] . If for each ε > 0 the set E ε is finite, then the operator M u is compact on B V φ [ 0 , 1 ] .

Proof

Suppose that for each ε > 0 the set E ε is finite. Then, since

Z u c = ⋃ n = 1 ∞ E 1 n = ⋃ n = 1 ∞ t ∈ [ 0 , 1 ] : ∣ u ( t ) ∣ ≥ 1 n ,

we conclude that Z u c is a finite set or a numerable set.

If Z u c is a finite set, then we can write Z u c = { t 1 , … , t m } for some m ∈ N . Thus, the functions { h 1 , h 2 , … , h m } defined by h k = u ⋅ f k , being f k as in (4.2) are linearly independent and belong to Ran ( M u ) . Furthermore, if h ∈ Ran ( M u ) , then h = u ⋅ f for some f ∈ B V φ [ 0 , 1 ] . Thus, if we consider the scalars α k = f ( t k ) , with k = 1 , 2 , … , m , we have

h = ∑ k = 1 m α k h k

and dim ( Ran ( M u ) ) = m < ∞ , that is, M u has finite range. This last fact implies that the operator M u is compact on B V φ [ 0 , 1 ] .

If Z u c is an infinite numerable set, then we can write

Z u c = { t 1 , t 2 , … , t n , … } ⊂ [ 0 , 1 ]

with t n ≠ t m for n ≠ m . We claim that the series

(4.3) ∑ k = 1 ∞ φ ( ∣ u ( t k ) ∣ )

is convergent. Indeed, since u ∈ B V φ [ 0 , 1 ] , then there exists λ u > 0 such that

Var φ u λ u < ∞ .

We fix m ∈ N , then we can suppose that the set { t 1 , t 2 , … , t m } is ordered and t k < t k + 1 for all k = 1 , 2 , … , m − 1 . Thus, for each k ∈ { 1 , 2 , … , m } we can find x k ∈ Z u such that t k < x k < t k + 1 and t m < x m < 1 . Hence, we can build the partition

P m : 0 < t 1 < x 1 < t 2 < x 2 < ⋯ < t m < x m < 1 .

Then we have

∑ k = 1 m φ ∣ u ( t k ) ∣ λ u = ∑ k = 1 m φ ∣ u ( x k ) − u ( t k ) ∣ λ u ≤ Var φ u λ u < ∞

and the series in (4.3) converges absolutely since φ satisfies a Δ 2 condition and the claim is proved. In particular,

lim k → ∞ ∣ u ( t k ) ∣ = 0

because φ ( t ) = 0 if and only if t = 0 .

Next, for each n ∈ N , we set the symbol u n by

u n = ∑ k = 1 n u ( t k ) 1 { t k } .

Then u n ∈ B V φ [ 0 , 1 ] since it is a finite sum of functions in B V φ [ 0 , 1 ] , and Z u n c = { t 1 , t 2 , … , t n } is a finite set. Thus by the first part of the proof we can conclude that the operator M u n is compact on B V φ [ 0 , 1 ] for each n ∈ N .

Also, for any t ∈ [ 0 , 1 ] we have

u ( t ) − u n ( t ) = ∑ k = n + 1 ∞ u ( t k ) 1 { t k } ( t )

and hence ‖ u − u n ‖ ∞ = sup k ≥ n + 1 ∣ u ( t k ) ∣ . This last fact implies that

(4.4) lim n → ∞ ‖ u − u n ‖ ∞ = lim n → ∞ sup k ≥ n + 1 ∣ u ( t k ) ∣ = lim sup k → ∞ ∣ u ( t k ) ∣ = 0 .

Next, we shall prove that

lim n → ∞ ‖ u − u n ‖ φ = inf λ > 0 : Var φ u − u n λ ≤ 1 = 0 .

With this end, we fix ε > 0 . There exists H ∈ N , depending on ε , such that

2 ε ≤ 2 H .

Thus, since φ is an increasing function and it satisfies a global Δ 2 condition, we can find a constant K Δ ( ε ) > 0 , depending only on φ and ε , such that

(4.5) φ 2 ε t ≤ K Δ ( ε ) φ ( t )

for all t ≥ 0 . Next, since the series ∑ k = 1 ∞ φ ( ∣ u ( t k ) ∣ ) converges, then we can find N ∈ N such that

2 K Δ ( ε ) ∑ k = n + 1 ∞ φ ( ∣ u ( t k ) ∣ ) ≤ 1

for all n ≥ N . For such n ≥ N fixed, we consider the set A n = { t n + 1 , t n + 2 , t n + 3 , … } and we shall estimate

∑ k = 1 m φ 1 ε ∣ h n ( x k ) − h n ( x k − 1 ) ∣ ,

where h n = u − u n and P : 0 = x 0 < x 1 < x 2 < ⋯ < x m = 1 is any partition of [ 0 , 1 ] . As in the proof of Theorem 3.1, we consider four cases but now we will use the Δ 2 -condition:

Case I

If x k , x k − 1 ∉ A n , then h n ( x k ) = h n ( x k − 1 ) = 0 and in this case we have

φ 1 ε ∣ h n ( x k ) − h n ( x k − 1 ) ∣ = 0 .

Case II

If x k , x k − 1 ∈ A n , then there are i , j ≥ n + 1 such that x k = t j and x k − 1 = t i . In this case h n ( x k ) = u ( t j ) , h n ( x k − 1 ) = u ( t i ) and we can write

φ 1 ε ∣ h n ( x k ) − h n ( x k − 1 ) ∣ ≤ φ 1 ε ∣ u ( t j ) ∣ + 1 ε ∣ u ( t i ) ∣ ≤ 1 2 φ 2 ε ∣ u ( t j ) ∣ + 1 2 φ 2 ε ∣ u ( t i ) ∣ ≤ 1 2 K Δ ( ε ) φ ( ∣ u ( t j ) ∣ ) + 1 2 K Δ ( ε ) φ ( ∣ u ( t i ) ∣ ) ,

where we have used that φ is increasing, convex and satisfies the condition (4.5).

Case III

If x k ∈ A n and x k − 1 ∉ A n , then there exists j ≥ n + 1 such that x k = t j . In this case, h n ( x k − 1 ) = 0 , h n ( x k ) = u ( t j ) and we have

φ 1 ε ∣ h n ( x k ) − h n ( x k − 1 ) ∣ = φ 1 ε ∣ u ( t j ) ∣ ≤ 1 2 φ 2 ε ∣ u ( t j ) ∣ ≤ 1 2 K Δ ( ε ) φ ( ∣ u ( t j ) ∣ ) ,

where we have used that φ is a convex function with φ ( 0 ) = 0 . The above estimation is also valid for the case x k ∉ A n and x k − 1 ∈ A n .

Next, taking into account the sum in each of the four cases, we obtain

∑ k = 1 m φ 1 ε ∣ h n ( x k ) − h n ( x k − 1 ) ∣ ≤ 2 K Δ ( ε ) ∑ i = n + 1 ∞ φ ( ∣ u ( t i ) ∣ ) < 1

and since the partition P : 0 = x 0 < x 1 < x 2 < ⋯ < x m = 1 was arbitrary, we obtain that

Var φ 1 ε h n ≤ 1

whenever n ≥ N , which means that ‖ u − u n ‖ φ < ε whenever n ≥ N , that is,

(4.6) lim n → ∞ ‖ u − u n ‖ φ = 0 .

Finally, from (4.4) and (4.6) we can conclude that

lim n → ∞ ∥ M u n − M u ∥ = lim n → ∞ ‖ M u n − u ‖ = lim n → ∞ ‖ u n − u ‖ B V φ = lim n → ∞ ( ‖ u − u n ‖ ∞ + ‖ u − u n ‖ φ ) = 0 .

This proves that M u : B V φ [ 0 , 1 ] → B V φ [ 0 , 1 ] is the uniform limit of compact operators and therefore it is a compact operator.□

Remark

As an application of Theorems 4.1 and 4.2 we can see that when φ ( t ) = t , we have Theorem 10 in [18], while if φ ( t ) = t p with p > 1 we have Theorem 5 in [19].

Remark

We finish this section remarking that in the first part of the proof of the above theorem we have proven that if the set Z u c is finite, then dim ( Ran ( M u ) ) < ∞ . We shall prove that the converse is also true.

Proposition 4.3

Suppose that u ∈ B V φ [ 0 , 1 ] . Then dim ( Ran ( M u ) ) < ∞ if and only if Z u c is a finite set.

Proof

Indeed, if Z u c is an infinite set, then we can find a sequence { t n } ⊆ Z u c such that t n ≠ t m whenever n ≠ m and the functions h n ≔ u ⋅ 1 t n are linearly independent in Ran ( M u ) . This means that dim ( Ran ( M u ) ) = ∞ .□

5 Fredholm multiplication operators on B V φ [ 0 , 1 ]

In this section, we shall use the technical and the results of the above sections to characterize all functions u ∈ B V φ [ 0 , 1 ] which define Fredholm multiplication operators on B V φ [ 0 , 1 ] . First, we can observe that if t 1 , t 2 ∈ Z u and t 1 ≠ t 2 , then { 1 { t 1 } , 1 { t 2 } } is a linearly independent set in Ker ( M u ) . This simple fact allows us to write the following proposition:

Proposition 5.1

Suppose that u ∈ B V φ [ 0 , 1 ] . Then dim ( Ker ( M u ) ) < ∞ if and only if Z u is a finite set.

Proof

Indeed, if Z u is an infinite set, then we can find a sequence { t n } ⊆ Z u c such that t n ≠ t m whenever n ≠ m and the functions { 1 { t n } } are linearly independent in Ker ( M u ) . This means that dim ( Ker ( M u ) ) = ∞ .

Conversely, if Z u = { t 1 , t 2 , … , t m } and f ∈ Ker ( M u ) , then u ( t ) ⋅ f ( t ) = 0 for all t ∈ [ 0 , 1 ] and hence { t ∈ [ 0 , 1 ] : f ( t ) ≠ 0 } ⊆ Z u . This last fact implies that

f = ∑ k = 1 m α k 1 { t k } ,

which proves that Ker ( M u ) = span { 1 { t 1 } , 1 { t 2 } , … , 1 { t m } } and dim ( Ker ( M u ) ) = m < ∞ .□

We recall that for a Banach space X , a bounded operator T : X → X is called upper semi-Fredholm if dim ( Ker ( T ) ) < ∞ and Ran ( T ) is a closed set of X , while T is lower semi-Fredholm if codim ( Ran ( T ) ) < ∞ . It is well known that the condition codim ( Ran ( T ) ) < ∞ implies that Ran ( M u ) is a closed set of X . The operator T : X → X is Fredholm if and only if T : X → X is lower and upper semi-Fredholm. The following is the main result of this section:

Theorem 5.2

Suppose that u ∈ B V φ [ 0 , 1 ] . The following statements are equivalent:

M u : B V φ [ 0 , 1 ] → B V φ [ 0 , 1 ] is upper semi-Fredholm;
M u : B V φ [ 0 , 1 ] → B V φ [ 0 , 1 ] is lower semi-Fredholm;
M u : B V φ [ 0 , 1 ] → B V φ [ 0 , 1 ] is Fredholm;
Z u is a finite set and there exists a δ > 0 such that E δ ⊆ Z u c .

Proof

It is enough to show that if Ran ( M u ) is a closed subset of B V φ [ 0 , 1 ] , then

(5.1) codim ( Ran ( M u ) ) < ∞ ⇔ Z u is a finite set .

Indeed, as before, if Z u = { t 1 , t 2 , … , t m } and h ˆ ∈ B V φ [ 0 , 1 ] / Ran ( M u ) is arbitrary, then by definition of quotient space we can write

h ˆ = h + Ran ( M u ) = { h + g : g ∈ Ran ( M u ) }

for some h ∈ B V φ [ 0 , 1 ] . Thus, since Ran ( M u ) is a closed subset of B V φ [ 0 , 1 ] , Theorem 3.1 and the three cases in the proof of Theorem 3.1 imply that the function

f u ( t ) = 1 u ( t ) , if t ∈ Z u c , 0 , otherwise ,

belongs to B V φ [ 0 , 1 ] and hence the function g = h ⋅ f u also belongs to B V φ [ 0 , 1 ] . Thus, if for k = 1 , 2 , … , m we set α k = h ( t k ) , then for t ∈ Z u c , we have

h ( t ) − ∑ k = 1 m α k 1 { t k } ( t ) = h ( t ) = u ( t ) ⋅ g ( t ) ;

while if t ∈ Z u , then

h ( t ) − ∑ k = 1 m α k 1 { t k } ( t ) = 0 = u ( t ) ⋅ g ( t ) .

This means that h − ∑ k = 1 m α k 1 { t k } ∈ Ran ( M u ) and

h ˆ = ∑ k = 1 m α k 1 { t k } + Ran ( M u ) .

Therefore, B V φ [ 0 , 1 ] / Ran ( M u ) ⊆ span { 1 { t k } + Ran ( M u ) : k = 1 , 2 , … , m } and

codim ( Ran ( M u ) ) = dim ( B V φ [ 0 , 1 ] / Ran ( M u ) ) ≤ m < ∞ .

If Z u is an infinite set, then we can find a sequence { t 1 , t 2 , … } ⊆ Z u such that t n ≠ t m whenever n ≠ m and the set { 1 { t k } + Ran ( M u ) : k ∈ N } is linearly independent in B V φ [ 0 , 1 ] / Ran ( M u ) which means that codim ( Ran ( M u ) ) = ∞ .

Finally, by definition, Proposition 5.1 and Theorem 3.1 we have that statements (1) and (4) are equivalent. While that if (2) holds, then Ran ( M u ) is a closed set of B V φ [ 0 , 1 ] and by the equivalence (5.1) we have that Z u is a finite set. Therefore, dim ( Ker ( M u ) ) < ∞ and M u : B V φ [ 0 , 1 ] → B V φ [ 0 , 1 ] is upper semi-Fredholm. That is, (2) and (4) also are equivalent. This proves the result.□

Remark

As an application of Theorem 5.2 we can see that when φ ( t ) = t , we have Theorem 11 in [18], while if φ ( t ) = t p with p > 1 we have Theorem 8 in [19].

Acknowledgment

The authors wish to express their sincere gratitude to the anonymous referees for a thorough review and insightful suggestions.

Funding information: The research of the second author has been financed by CIDC-grant No. 1-73-634-20, Universidad Distrital Francisco José de Caldas.
Conflict of interest: Authors state no conflict of interest.

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Received: 2020-12-20

Revised: 2021-03-22

Accepted: 2021-05-06

Published Online: 2021-06-14

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

Articles in the same Issue

https://doi.org/10.1515/math-2021-0050

Keywords for this article

multiplication operator; bounded variation functions; compact operators; Fredholm operators

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