# Toeplitz Operators on Abstract Hardy Spaces Built upon Banach Function Spaces

Authors: Alexei Yu. Karlovich
Journal: Journal of Function Spaces
 (2017)

---

## Abstract

LetX be a Banach function space over the unit circle T and let H[X] be the abstract Hardy space built upon X. If the Riesz projection P is bounded on X and a∈L∞, then the Toeplitz operator Taf=P(af) is bounded on H[X]. We extend well-known results by Brown and Halmos for X=L2 and show that, under certain assumptions on the space X, the Toeplitz operator Ta is bounded (resp., compact) if and only if a∈L∞ (resp., a=0). Moreover, aL∞≤TaB(H[X])≤PB(X)aL∞. These results are specified to the cases of abstract Hardy spaces built upon Lebesgue spaces with Muckenhoupt weights and Nakano spaces with radial oscillating weights.


---

## Body

## 1. Introduction

The Banach algebra of all bounded linear operators on a Banach spaceE will be denoted by B(E). Let T be the unit circle in the complex plane C. For n∈Z+≔{0,1,2,…}, a function of the form p(t)=∑k=-nnαktk, where αk∈C for all k∈{-n,…,n} and t∈T, is called a trigonometric polynomial of order n. The set of all trigonometric polynomials is denoted by P. The Riesz projection is the operator P which is defined on P by(1)P:∑k=-nnαktk⟼∑k=0nαktk.For 1≤p≤∞, let Lp≔Lp(T) be the Lebesgue space on the unit circle T in the complex plane. For f∈L1, let(2)f^n≔12π∫-ππfeiφe-inφdφ,n∈Z,be the sequence of the Fourier coefficients of f. The classical Hardy spaces Hp are given by(3)Hp≔f∈Lp:f^n=0∀n<0.It is well known that the Riesz projection extends to a bounded linear operator on Lp if and only if 1<p<∞. For a∈L∞, the Toeplitz operator Ta with symbol a on Hp, 1<p<∞, is given by(4)Taf=Paf.Toeplitz operators have attracted the mathematical community for the many decades since the classical paper by Toeplitz [1]. Brown and Halmos [2, Theorem  4] proved that a necessary and sufficient condition that an operator on H2 is a Toeplitz operator is that its matrix with respect to the standard basis of H2 is a Toeplitz matrix, that is, the matrix of the form (ak-j)j,k∈Z+. The norm of Ta on the Hardy space H2 coincides with the norm of its symbol in L∞ (actually, this result was already in a footnote of [1]). Brown and Halmos also observed, as a corollary, that the only compact Toeplitz operator on H2 is the zero operator. We here mention [3, Part B, Theorem  4.1.4] and [4, Theorem  1.8] for the proof of the Brown-Halmos theorem. An analogue of this result is true for Toeplitz operators acting on Hp, 1<p<∞ [5, Theorem  2.7].In this paper, we will consider the so-called Banach function spacesX in place of Lp. As usual, we equip the unit circle T with the normalized Lebesgue measure dm(τ)=dτ/(2π). Denote by L0 the set of all measurable complex-valued functions on T, and let L+0 be the subset of functions in L0 whose values lie in [0,∞]. The characteristic function of a measurable set E⊂T is denoted by IE. A mapping ρ:L+0→[0,∞] is called a function norm if, for all functions f, g, fn (n∈N) in L+0, for all constants c≥0, and for all measurable subsets E of T, the following properties hold:(a)
ρf=0⇔f=0 a.e., ρ(cf)=cρ(f), ρ(f+g)≤ρ(f)+ρ(g),(b)
0≤g≤f a.e. ⇒ρ(g)≤ρ(f) (the lattice property),(c)
0≤fn↑f a.e. ⇒ρ(fn)↑ρ(f) (the Fatou property),(d)
ρ(IE)<∞, ∫Ef(τ)dm(τ)≤CEρ(f),with CE∈(0,∞) depending on E and ρ but independent of f. When functions differing only on a set of measure zero are identified, the set X of all functions f∈L0 for which ρf<∞ is a Banach space under the norm fX≔ρf. Such a space X is called a Banach function space. If ρ is a function norm, its associate norm ρ′ is defined on L+0 by(5)ρ′g≔sup⁡∫Tfτgτdmτ:f∈L+0,ρf≤1,g∈L+0.The Banach function space X′ determined by the function norm ρ′ is called the associate space (or Köthe dual space) of X. The associate space X′ is a subspace of the dual space X∗. The simplest examples of Banach function spaces are the Lebesgue spaces Lp, 1≤p≤∞. The class of all Banach function spaces includes all Orlicz spaces, as well as all rearrangement-invariant Banach function spaces (see, e.g., [6, Chap. 3]). We are mainly interested in non-rearrangement-invariant Banach function spaces. Two typical examples of non-rearrangement-invariant Banach function spaces are weighted Lebesgue space and weighted Nakano spaces (weighted variable Lebesgue spaces) considered in the last section of the paper.Following [7, p. 877], we will consider abstract Hardy spaces H[X] built upon a Banach function space X over the unit circle T as follows:(6)HX≔f∈X:f^n=0∀n<0.This definition makes sense because X is continuously embedded in L1 in view of axiom (d). It can be shown that H[X] is a closed subspace of X. It is clear that if 1≤p≤∞, then H[Lp] is the classical Hardy space Hp.It follows from axiom (d) thatP⊂L∞⊂X. We will restrict ourselves to Banach function spaces X such that the Riesz projection defined initially on P by formula (1) extends to a bounded linear operator on the whole space X. The extension will again be denoted by P. If a∈L∞ and P∈B(X), then the Toeplitz operator defined by formula (4) is bounded on H[X] and(7)TaBHX≤PBXaL∞.The Brown-Halmos theorem [2, Theorem  4] was extended by the author [8, Theorem  4.5] to the case of reflexive rearrangement-invariant Banach function spaces with nontrivial Boyd indices. Note that the nontriviality of the Boyd indices implies the boundedness of the Riesz projection.The first aim of this paper is to show that the Brown-Halmos theorem remains true for abstract Hardy spacesH[X] built upon reflexive Banach function spaces X (not necessarily rearrangement-invariant) if P∈B(X). Further, we show that, under mild assumptions on a Banach function space X, a Toeplitz operator Ta is compact on the abstract Hardy space H[X] built upon X if and only if a=0. These results are specified to the case of Hardy spaces built upon Lebesgue spaces with Muckenhoupt weights and upon Nakano spaces with certain radial oscillating weights. Both classes of spaces in our examples are not rearrangement-invariant.Forf∈X and g∈X′, we will use the following pairing:(8)f,g≔∫Tfτgτ¯dmτ.For n∈Z and τ∈T, put χn(τ)=τn. Then the Fourier coefficients of a function f∈L1 can be expressed by f^(n)=f,χn for n∈Z.Theorem 1 (main result 1).
LetX be a reflexive Banach function space over the unit circle T such that the Riesz projection P is bounded on X. Suppose A∈B(H[X]) and there is a sequence {an}n∈Z of complex numbers such that(9)Aχj,χk=ak-j∀j,k∈Z+.Then there is a function a∈L∞ such that A=Ta and a^(n)=an for all n∈Z. Moreover(10)aL∞≤TaBHX≤PBXaL∞.We need the notion of a function with absolutely continuous norm to formulate the result on the noncompactness of nontrivial Toeplitz operators. Following [6, Chap. 1, Definition  3.1], a function f in a Banach function space X is said to have absolutely continuous norm in X if fIEnX→0 for every sequence {En}n∈N of measurable sets satisfying IEn→∅ almost everywhere as n→∞. The set of all functions in X of absolutely continuous norm is denoted by Xa. It is known that a Banach function space X is reflexive if and only if X and X′ have absolutely continuous norm (see [6, Chap. 1, Corollary  4.4]).Theorem 2 (main result 2).
LetX be a Banach function space over the unit circle T such that IE∈Xa for every measurable subset E⊂T. If the Riesz projection P is bounded on X and a∈L∞, then the Toeplitz operator Ta∈B(H[X]) is compact if and only if a=0.The paper is organized as follows. Section2 contains results on the density of the set of all trigonometric polynomials P (resp., the set of all analytic polynomials PA) in a Banach function space X (resp., in the abstract Hardy space H[X] built upon X). We also show that the norm of a function f in X can be calculated in terms of f,p, where p∈P, under the assumption that X′ is separable. Further, we prove that every bounded linear operator on a separable Banach function space, whose matrix is of the form (ak-j)j,k∈Z, is an operator of multiplication by a function a∈L∞ and the sequence of its Fourier coefficients is exactly {ak}k∈Z. Finally, we prove that if the characteristic functions of all measurable sets E⊂T have absolutely continuous norms in X, then the sequence {χk}k∈Z+ converges weakly to zero on the abstract Hardy space H[X]. In Section 3, we provide proofs of our main results, using auxiliary results from the previous section. In Section 4, we specify our main results to the case of Hardy spaces built upon weighted Lebesgue spaces Lp(w) with Muckenhoupt weights w and to the case of weighted Nakano spaces Lp(·)(w) with certain radial oscillating weights. In both cases, it is known that the Riesz projection is bounded.


## 2. Auxiliary Results

### 2.1. Density of Continuous Function and Trigonometric Polynomials in Banach Function Spaces

The following statement can be proved by analogy with [9, Lemma  1.3].Lemma 3.
LetX be a Banach function space over the unit circle T. The following statements are equivalent: (a)
the setP of all trigonometric polynomials is dense in the space X;(b)
the spaceC of all continuous functions on T is dense in the space X;(c)
the Banach function spaceX is separable.


### 2.2. Density of Analytic Polynomials in Abstract Hardy Spaces

Letm∈Z+. A function of the form q(t)=∑k=0mαktk, where αk∈C for all k∈{0,…,m} and t∈T, is said to be an analytic polynomial on T. The set of all analytic polynomials is denoted by PA.Lemma 4.
LetX be a separable Banach functions space over the unit circle T. If the Riesz projection P is bounded on X, then the set PA is dense in H[X].Proof.
Iff∈H[X]⊂X, then by Lemma 3, there exists a sequence pn∈P such that f-pnX→0 as n→∞. It is clear that f=Pf and Ppn∈PA. Since P∈B(X), we finally have(11)f-PpnX=Pf-PpnX≤PBXf-pnX⟶0as n→∞. Thus PA is dense in X.


### 2.3. A Formula for the Norm in a Banach Function Space

Lemma 5.
LetX be a Banach function space over the unit circle T. If the associate space X′ is separable, then for every f∈X(12)fX=sup⁡f,p:p∈P,pX′≤1.Proof.
By [6, Theorem  2.7 and Lemma  2.8], for every f∈X(13)fX=sup⁡f,g:g∈X′,gX′≤1.By the lattice property of the associate space X′, we have P⊂X′. Hence, equality (13) implies that for f∈X(14)fX≥sup⁡f,p:p∈P,pX′≤1.Fix g∈X′ such that 0<gX′≤1. Since X′ is separable, it follows from Lemma 3 that there exists a sequence qn∈P∖{0} such that qn-gX′→0 as n→∞. For n∈N, put pn≔(gX′/qnX′)qn∈P. Then for every n∈N(15)pnX′=gX′≤1,(16)g-pnX′≤g-qnX′+qnX′1-gX′qnX′.Hence(17)limn→∞⁡g-pnX′=0.It follows from Hölder’s inequality for Banach function spaces (see [6, Chap. 1, Theorem  2.4]) and (17) that(18)limn→∞⁡f,g-f,pn≤limn→∞⁡fXg-pnX′=0.Thus, taking into account (15) and (18), we deduce for every function g∈X′ satisfying 0<gX′≤1 that(19)f,g=limn→∞⁡f,pn≤supn∈N⁡f,pn≤sup⁡f,p:p∈P,pX′≤1.This inequality and equality (13) imply that(20)fX≤sup⁡f,p:p∈P,pX′≤1.Combining inequalities (14) and (20), we arrive at equality (12).


### 2.4. Multiplication Operators

We start this subsection with the following result by Maligranda and Persson on multiplication operators acting on Banach function spaces.Lemma 6 (see [10, Theorem  1]).
LetX be a Banach function space over the unit circle T. If a∈L0, then the multiplication operator(21)Ma:X⟶X,f⟼af,is bounded on X if and only if a∈L∞ and MaB(X)=aL∞.It is easy see that(22)Maχj,χk=a,χk-j=a^k-j∀j,k∈Z.The following lemma shows that every bounded operator with such a property is a multiplication operator.Lemma 7.
LetX be a separable Banach functions space over the unit circle T. Suppose A∈B(X) and there exists a sequence {an}n∈N of complex numbers such that(23)Aχj,χk=ak-j∀j,k∈Z.Then there exists a function a∈L∞ such that A=Ma and a^(n)=an for all n∈Z.Proof.
Puta≔Aχ0∈X. Since X⊂L1, we infer from (23) that(24)a^n=a,χn=Aχ0,χn=an,n∈Z.If f=∑k=-mmf^(k)χk∈P, then af∈X⊂L1 and the jth Fourier coefficient of af is calculated by(25)af^j=∑k∈Za^j-kf^k=∑k=-mmaj-kf^k.On the other hand, from (23), we get for j∈Z(26)Af^j=Af,χj=∑k=-mmf^kAχk,χj=∑k=-mmaj-kf^k.By (25) and (26), (af)^(j)=(Af)^(j) for all j∈Z. Therefore, Af=af for all f∈P in view of the uniqueness theorem for Fourier series (see, e.g., [11, Chap. I, Theorem  2.7]). Since the space X is separable, the set P is dense in X by Lemma 3. Therefore Af=af for f∈X. This means that A=Ma∈B(X). It remains to apply Lemma 6.


### 2.5. Weak Convergence of the Sequence{χk}k∈Z+ to Zero on the Abstract Hardy Space

Recall that the annihilator of a subspaceS of a Banach space E is the set S⊥ of all linear functionals Λ∈E∗ such that Λ(x)=0 for all x∈S (see, e.g., [12, p. 110]).Lemma 8.
IfX is a Banach function space such that IE∈Xa for every measurable subset E⊂T, then {χk}k∈Z+ converges weakly to zero on H[X].Proof.
By [12, Theorem  7.1], (H[X])∗ is isometrically isomorphic to X∗/(H[X])⊥. Since χk∈H[X] for all k≥0, in view of the above fact, it is sufficient to prove that {χk}k∈Z+ converges weakly to zero on the whole space X instead of the subspace H[X].
By [6, Chap. 1, Corollary  3.14], if IE∈Xa for every measurable subset E⊂T, then (Xa)∗ is isometrically isomorphic to X′. In view of [6, Chap. 1, Theorem  2.2], X′ is a Banach function space, which is continuously embedded into L1 due to axiom (d) of the definition of a Banach function norm. Thus, for every Λ∈X∗, there exists a function g∈X′⊂L1 such that Λ(f)=f,g for all f∈X. In particular, if f=χk with k≥0, then(27)Λχk=χk,g=g,χk¯=g^k¯.By the Riemann-Lebesgue lemma (see, e.g., [11, Chap. I, Theorem  2.8]) and (27), Λ(χk)→0 as k→∞ for every Λ∈X∗; that is, {χk}k∈Z+ converges weakly to zero on X, which completes the proof.


## 2.1. Density of Continuous Function and Trigonometric Polynomials in Banach Function Spaces

The following statement can be proved by analogy with [9, Lemma  1.3].Lemma 3.
LetX be a Banach function space over the unit circle T. The following statements are equivalent: (a)
the setP of all trigonometric polynomials is dense in the space X;(b)
the spaceC of all continuous functions on T is dense in the space X;(c)
the Banach function spaceX is separable.


## 2.2. Density of Analytic Polynomials in Abstract Hardy Spaces

Letm∈Z+. A function of the form q(t)=∑k=0mαktk, where αk∈C for all k∈{0,…,m} and t∈T, is said to be an analytic polynomial on T. The set of all analytic polynomials is denoted by PA.Lemma 4.
LetX be a separable Banach functions space over the unit circle T. If the Riesz projection P is bounded on X, then the set PA is dense in H[X].Proof.
Iff∈H[X]⊂X, then by Lemma 3, there exists a sequence pn∈P such that f-pnX→0 as n→∞. It is clear that f=Pf and Ppn∈PA. Since P∈B(X), we finally have(11)f-PpnX=Pf-PpnX≤PBXf-pnX⟶0as n→∞. Thus PA is dense in X.


## 2.3. A Formula for the Norm in a Banach Function Space

Lemma 5.
LetX be a Banach function space over the unit circle T. If the associate space X′ is separable, then for every f∈X(12)fX=sup⁡f,p:p∈P,pX′≤1.Proof.
By [6, Theorem  2.7 and Lemma  2.8], for every f∈X(13)fX=sup⁡f,g:g∈X′,gX′≤1.By the lattice property of the associate space X′, we have P⊂X′. Hence, equality (13) implies that for f∈X(14)fX≥sup⁡f,p:p∈P,pX′≤1.Fix g∈X′ such that 0<gX′≤1. Since X′ is separable, it follows from Lemma 3 that there exists a sequence qn∈P∖{0} such that qn-gX′→0 as n→∞. For n∈N, put pn≔(gX′/qnX′)qn∈P. Then for every n∈N(15)pnX′=gX′≤1,(16)g-pnX′≤g-qnX′+qnX′1-gX′qnX′.Hence(17)limn→∞⁡g-pnX′=0.It follows from Hölder’s inequality for Banach function spaces (see [6, Chap. 1, Theorem  2.4]) and (17) that(18)limn→∞⁡f,g-f,pn≤limn→∞⁡fXg-pnX′=0.Thus, taking into account (15) and (18), we deduce for every function g∈X′ satisfying 0<gX′≤1 that(19)f,g=limn→∞⁡f,pn≤supn∈N⁡f,pn≤sup⁡f,p:p∈P,pX′≤1.This inequality and equality (13) imply that(20)fX≤sup⁡f,p:p∈P,pX′≤1.Combining inequalities (14) and (20), we arrive at equality (12).


## 2.4. Multiplication Operators

We start this subsection with the following result by Maligranda and Persson on multiplication operators acting on Banach function spaces.Lemma 6 (see [10, Theorem  1]).
LetX be a Banach function space over the unit circle T. If a∈L0, then the multiplication operator(21)Ma:X⟶X,f⟼af,is bounded on X if and only if a∈L∞ and MaB(X)=aL∞.It is easy see that(22)Maχj,χk=a,χk-j=a^k-j∀j,k∈Z.The following lemma shows that every bounded operator with such a property is a multiplication operator.Lemma 7.
LetX be a separable Banach functions space over the unit circle T. Suppose A∈B(X) and there exists a sequence {an}n∈N of complex numbers such that(23)Aχj,χk=ak-j∀j,k∈Z.Then there exists a function a∈L∞ such that A=Ma and a^(n)=an for all n∈Z.Proof.
Puta≔Aχ0∈X. Since X⊂L1, we infer from (23) that(24)a^n=a,χn=Aχ0,χn=an,n∈Z.If f=∑k=-mmf^(k)χk∈P, then af∈X⊂L1 and the jth Fourier coefficient of af is calculated by(25)af^j=∑k∈Za^j-kf^k=∑k=-mmaj-kf^k.On the other hand, from (23), we get for j∈Z(26)Af^j=Af,χj=∑k=-mmf^kAχk,χj=∑k=-mmaj-kf^k.By (25) and (26), (af)^(j)=(Af)^(j) for all j∈Z. Therefore, Af=af for all f∈P in view of the uniqueness theorem for Fourier series (see, e.g., [11, Chap. I, Theorem  2.7]). Since the space X is separable, the set P is dense in X by Lemma 3. Therefore Af=af for f∈X. This means that A=Ma∈B(X). It remains to apply Lemma 6.


## 2.5. Weak Convergence of the Sequence{χk}k∈Z+ to Zero on the Abstract Hardy Space

Recall that the annihilator of a subspaceS of a Banach space E is the set S⊥ of all linear functionals Λ∈E∗ such that Λ(x)=0 for all x∈S (see, e.g., [12, p. 110]).Lemma 8.
IfX is a Banach function space such that IE∈Xa for every measurable subset E⊂T, then {χk}k∈Z+ converges weakly to zero on H[X].Proof.
By [12, Theorem  7.1], (H[X])∗ is isometrically isomorphic to X∗/(H[X])⊥. Since χk∈H[X] for all k≥0, in view of the above fact, it is sufficient to prove that {χk}k∈Z+ converges weakly to zero on the whole space X instead of the subspace H[X].
By [6, Chap. 1, Corollary  3.14], if IE∈Xa for every measurable subset E⊂T, then (Xa)∗ is isometrically isomorphic to X′. In view of [6, Chap. 1, Theorem  2.2], X′ is a Banach function space, which is continuously embedded into L1 due to axiom (d) of the definition of a Banach function norm. Thus, for every Λ∈X∗, there exists a function g∈X′⊂L1 such that Λ(f)=f,g for all f∈X. In particular, if f=χk with k≥0, then(27)Λχk=χk,g=g,χk¯=g^k¯.By the Riemann-Lebesgue lemma (see, e.g., [11, Chap. I, Theorem  2.8]) and (27), Λ(χk)→0 as k→∞ for every Λ∈X∗; that is, {χk}k∈Z+ converges weakly to zero on X, which completes the proof.


## 3. Proof of the Main Results

### 3.1. Proof of Theorem1

We follow the scheme of the proof of [8, Theorem  4.5] (see also [5, Theorem  2.7]). Without loss of generality, we may assume that the operator A is nonzero. For n∈Z+, put bn≔χ-nAχn. Then taking into account Lemma 6 and that A∈B(H[X]), we get(28)bnX≤χ-nL∞AχnX=AχnHX≤ABHXχnHX=ABHX1X. Consider the following subset of the associate space:(29)V≔y∈X′:yX′<1ABHX1X.It follows from Hölder’s inequality for Banach function spaces (see [6, Chap. 1, Theorem  2.4]) and (28) and (29) that(30)bn,y≤bnXyX′<1∀y∈V,n∈Z+.Since X is reflexive, in view of [6, Chap. 1, Corollaries  4.3-4.4], we know that X′ is canonically isometrically isomorphic to X∗. Applying the Banach-Alaoglu theorem (see, e.g., [13, Theorem  3.17]) to V, X′, and {bn}n∈Z+⊂X=X∗∗=(X′)∗, we deduce that there exists a b∈X such that some subsequence {bnk}k∈Z+ of {bn}n∈Z+ converges to b in the weak topology on X. In particular(31)limk→+∞⁡bnk,χj=b,χj∀j∈Z.On the other hand, the definition of bn and equality (9) imply that(32)bnk,χj=Aχnk,χnk+j=ajwhenevernk+j∈Z+.It follows from (31) and (32) that(33)b,χj=aj∀j∈Z.Now define the mapping B by(34)B:P⟶X,f⟼bf.Assume that f and g are trigonometric polynomials of orders m and r, respectively. Then(35)f=∑k=-mmf^kχk,g=∑j=-rrg^jχj.It follows from (9) and (33) that for n≥max⁡{m,r}(36)Bf,g=∑k=-mm∑j=-rrf^kg^jbχk,χj=∑k=-mm∑j=-rrf^kg^jaj-k=∑k=-mm∑j=-rrf^kg^jAχk+n,χj+n=∑k=-mm∑j=-rrf^kg^jχ-nAχnχk,χj=χ-nAχnf,g.It is clear that χnf∈H[X] for n≥max⁡m,r. Therefore, taking into account Lemma 6, we see that for n≥max⁡{m,r}(37)Mχ-nAMχnfX≤χ-nL∞AχnfHX≤ABHXχnfHX≤ABHXχnL∞fX=ABHXfX.By Hölder’s inequality for Banach function spaces (see [6, Chap. 1, Theorem  2.4]) from (36) and (37), we obtain(38)Bf,g≤lim⁡supn→∞⁡Mχ-nAMχnf,g≤lim⁡supn→∞⁡Mχ-nAMχnfXgX′≤ABHXfXgX′.Since a Banach function space X is reflexive and the Lebesgue measure is separable, it follows from [6, Chap. 1, Corollaries  4.4 and 5.6] that the spaces X and X′ are separable. Then Lemma 5 and inequality (38) yield(39)BfX=supBf,g:g∈P,gX′≤1≤ABHXfX,for all f∈P. In view of Lemma 3, P is dense in X. Then (39) implies that the linear mapping B defined in (34) extends to an operator B∈B(X) such that(40)BBX≤ABHX.We deduce from (33) that(41)Bχj,χk=b,χk-j=ak-j∀j,k∈Z.By Lemma 7, there exists a function a∈L∞ such that B=Ma and an=a^(n) for all n∈Z. Moreover(42)BBX=MaBX=aL∞.It follows from the definition of the Toeplitz operator Ta that(43)Taχj,χk=a^k-j,j,k∈Z+.Combining this fact with equality (9), we arrive at(44)Taχj,χk=ak-j=Aχj,χk,j,k∈Z+.Since Taχj,Aχj∈H[X]⊂H1, by the uniqueness theorem for Fourier series (see, e.g., [11, Chap. I, Theorem  2.7]), it follows from (44) that Taχj=Aχj for all j∈Z+. Therefore(45)Tap=Ap∀p∈PA.In view of Lemma 4, the set PA is dense in H[X]. This fact and equality (45) imply that Ta=A and(46)TaBHX=ABHX.Combining inequality (40) with equalities (42) and (46), we arrive at the first inequality in (10). The second inequality in (10) is obvious.


### 3.2. Proof of Theorem2

It is clear that ifa=0, then Ta is the zero operator, which is compact. Now assume that Ta is compact. Then it maps weakly convergent sequences in H[X] into strongly convergent sequences in H[X] (see, e.g., [14, Section  7.5, Theorem  4]). Since {χk}k∈Z+ converges to zero weakly on H[X] in view of Lemma 8, we have(47)limk→∞⁡TaχkHX=0.By [6, Chap. 1, Theorem  2.7 and Lemma  2.8], for k∈Z+,(48)TaχkHX=TaχkX=sup⁡Taχk,g:g∈X′,gX′≤1.Since L∞⊂X′, there exists a constant c∈0,∞ such that(49)c-1χmX′≤χmL∞=1,m∈Z.For all n∈Z and all k∈Z+ such that k+n∈Z+, we have(50)a^n=Taχk,χk+n.Then from (48)–(50) we obtain for all n∈Z and all k∈Z+ such that k+n∈Z+(51)TaχkHX≥Taχk,c-1χk+n=c-1a^n.Passing in this inequality to the limit as k→∞ and taking into account (47), we see that a^(n)=0 for all n∈Z. By the uniqueness theorem for Fourier series (see, e.g., [11, Chap. I, Theorem  2.7]), this implies that a=0 a.e. on T.


## 3.1. Proof of Theorem1

We follow the scheme of the proof of [8, Theorem  4.5] (see also [5, Theorem  2.7]). Without loss of generality, we may assume that the operator A is nonzero. For n∈Z+, put bn≔χ-nAχn. Then taking into account Lemma 6 and that A∈B(H[X]), we get(28)bnX≤χ-nL∞AχnX=AχnHX≤ABHXχnHX=ABHX1X. Consider the following subset of the associate space:(29)V≔y∈X′:yX′<1ABHX1X.It follows from Hölder’s inequality for Banach function spaces (see [6, Chap. 1, Theorem  2.4]) and (28) and (29) that(30)bn,y≤bnXyX′<1∀y∈V,n∈Z+.Since X is reflexive, in view of [6, Chap. 1, Corollaries  4.3-4.4], we know that X′ is canonically isometrically isomorphic to X∗. Applying the Banach-Alaoglu theorem (see, e.g., [13, Theorem  3.17]) to V, X′, and {bn}n∈Z+⊂X=X∗∗=(X′)∗, we deduce that there exists a b∈X such that some subsequence {bnk}k∈Z+ of {bn}n∈Z+ converges to b in the weak topology on X. In particular(31)limk→+∞⁡bnk,χj=b,χj∀j∈Z.On the other hand, the definition of bn and equality (9) imply that(32)bnk,χj=Aχnk,χnk+j=ajwhenevernk+j∈Z+.It follows from (31) and (32) that(33)b,χj=aj∀j∈Z.Now define the mapping B by(34)B:P⟶X,f⟼bf.Assume that f and g are trigonometric polynomials of orders m and r, respectively. Then(35)f=∑k=-mmf^kχk,g=∑j=-rrg^jχj.It follows from (9) and (33) that for n≥max⁡{m,r}(36)Bf,g=∑k=-mm∑j=-rrf^kg^jbχk,χj=∑k=-mm∑j=-rrf^kg^jaj-k=∑k=-mm∑j=-rrf^kg^jAχk+n,χj+n=∑k=-mm∑j=-rrf^kg^jχ-nAχnχk,χj=χ-nAχnf,g.It is clear that χnf∈H[X] for n≥max⁡m,r. Therefore, taking into account Lemma 6, we see that for n≥max⁡{m,r}(37)Mχ-nAMχnfX≤χ-nL∞AχnfHX≤ABHXχnfHX≤ABHXχnL∞fX=ABHXfX.By Hölder’s inequality for Banach function spaces (see [6, Chap. 1, Theorem  2.4]) from (36) and (37), we obtain(38)Bf,g≤lim⁡supn→∞⁡Mχ-nAMχnf,g≤lim⁡supn→∞⁡Mχ-nAMχnfXgX′≤ABHXfXgX′.Since a Banach function space X is reflexive and the Lebesgue measure is separable, it follows from [6, Chap. 1, Corollaries  4.4 and 5.6] that the spaces X and X′ are separable. Then Lemma 5 and inequality (38) yield(39)BfX=supBf,g:g∈P,gX′≤1≤ABHXfX,for all f∈P. In view of Lemma 3, P is dense in X. Then (39) implies that the linear mapping B defined in (34) extends to an operator B∈B(X) such that(40)BBX≤ABHX.We deduce from (33) that(41)Bχj,χk=b,χk-j=ak-j∀j,k∈Z.By Lemma 7, there exists a function a∈L∞ such that B=Ma and an=a^(n) for all n∈Z. Moreover(42)BBX=MaBX=aL∞.It follows from the definition of the Toeplitz operator Ta that(43)Taχj,χk=a^k-j,j,k∈Z+.Combining this fact with equality (9), we arrive at(44)Taχj,χk=ak-j=Aχj,χk,j,k∈Z+.Since Taχj,Aχj∈H[X]⊂H1, by the uniqueness theorem for Fourier series (see, e.g., [11, Chap. I, Theorem  2.7]), it follows from (44) that Taχj=Aχj for all j∈Z+. Therefore(45)Tap=Ap∀p∈PA.In view of Lemma 4, the set PA is dense in H[X]. This fact and equality (45) imply that Ta=A and(46)TaBHX=ABHX.Combining inequality (40) with equalities (42) and (46), we arrive at the first inequality in (10). The second inequality in (10) is obvious.


## 3.2. Proof of Theorem2

It is clear that ifa=0, then Ta is the zero operator, which is compact. Now assume that Ta is compact. Then it maps weakly convergent sequences in H[X] into strongly convergent sequences in H[X] (see, e.g., [14, Section  7.5, Theorem  4]). Since {χk}k∈Z+ converges to zero weakly on H[X] in view of Lemma 8, we have(47)limk→∞⁡TaχkHX=0.By [6, Chap. 1, Theorem  2.7 and Lemma  2.8], for k∈Z+,(48)TaχkHX=TaχkX=sup⁡Taχk,g:g∈X′,gX′≤1.Since L∞⊂X′, there exists a constant c∈0,∞ such that(49)c-1χmX′≤χmL∞=1,m∈Z.For all n∈Z and all k∈Z+ such that k+n∈Z+, we have(50)a^n=Taχk,χk+n.Then from (48)–(50) we obtain for all n∈Z and all k∈Z+ such that k+n∈Z+(51)TaχkHX≥Taχk,c-1χk+n=c-1a^n.Passing in this inequality to the limit as k→∞ and taking into account (47), we see that a^(n)=0 for all n∈Z. By the uniqueness theorem for Fourier series (see, e.g., [11, Chap. I, Theorem  2.7]), this implies that a=0 a.e. on T.


## 4. Toeplitz Operators on Hardy Spaces Built upon Weighted Lebesgue and Nakano Spaces

### 4.1. The Case of Hardy Spaces Built upon Lebesgue Spaces with Muckenhoupt Weights

A measurable functionw:T→[0,∞] is referred to as a weight if 0<w(τ)<∞ almost everywhere on T. If X is a Banach function space over the unit circle and w is a weight, then(52)Xw≔f∈L0:fw∈Xis a normed space equipped with the norm fX(w)≔fwX. Moreover, if w∈X and 1/w∈X′, then X(w) is a Banach function space (see [15, Lemma  2.5]).Let1<p<∞ and w be a weight. It is well known that the Riesz projection P is bounded on the weighted Lebesgue space Lp(w) if and only if the weight w satisfies the Muckenhoupt Ap-condition; that is,(53)supI⊂T⁡1mI∫Iwpτdmτ1/p1mI∫Iw-p′τdmτ1/p′<∞,where the supremum is taken over all subarcs I of the unit circle T and 1/p+1/p′=1 (see [16] and also [5, Section  1.46], [3, Section  5.7.3(h)]). In the latter case, we will write w∈Ap(T). It is clear that if w∈Ap(T), then w∈Lp and 1/w∈Lp′. Hence Lp(w) is a Banach function space whenever w∈Ap(T). It is well known that if 1<p<∞, then Lp(w) is reflexive. We denote the corresponding Hardy space by Hp(w)≔H[Lp(w)].Surprisingly enough, we were not able to find the following results explicitly stated in the literature.Corollary 9.
Let1<p<∞ and w∈Ap(T). If A∈B(Hp(w)) and there exists a sequence {an}n∈Z of complex numbers satisfying (9), then there exists a function a∈L∞ such that A=Ta and a^(n)=an for all n∈Z. Moreover(54)aL∞≤TaBHpw≤PBLpwaL∞.This is an immediate consequence of Theorem1. For the weight w=1, it is proved in [5, Theorem  2.7].Corollary 10.
Let1<p<∞ and w∈Ap(T). If a∈L∞, then the Toeplitz operator Ta∈B(Hp(w)) is compact if and only if a=0.This corollary follows from Theorem2.


### 4.2. The Case of Hardy Spaces Built upon Nakano Spaces with Radial Oscillating Weights

We denote byPC(T) the set of all continuous functions p:T→(1,∞). For p∈PC(T), let Lp(·) be the set of all functions f∈L0 such that(55)∫Tfτλpτdmτ<∞,for some λ=λ(f)>0. This set becomes a Banach function space when equipped with the Luxemburg-Nakano norm(56)fLp·≔inf⁡λ>0:∫Tfτλpτdmτ≤1,(see, e.g., [17, p. 73] or [18, p. 77]). If p is constant, then Lp(·) is nothing but the Lebesgue space Lp. The spaces Lp(·) are referred to as Nakano spaces. We refer to Maligranda’s paper [19] for the role of Hidegoro Nakano in the study of these spaces.SinceT is compact, we have(57)1<mint∈T⁡pt,maxt∈T⁡pt<∞.In this case, the space Lp(·) is reflexive and its associate space is isomorphic to the space Lp′(·), where 1/p(τ)+1/p′τ=1 for all τ∈T (see, e.g., [17, Section  2.8] and [18, Section  3.2]).LetSf be the Cauchy singular integral of a function f∈L1(T) defined by(58)Sft≔limε→01πi∫T∖Tt,εfττ-tdτ,t∈T,where T(t,ε)≔{τ∈T:τ-t<ε}. For a weight w:T→[0,∞], consider the weighted Nakano space(59)Lp·w≔f∈L0:fw∈Lp·.It follows from [15, Theorem  6.1] that if the operator S is bounded on Lp(·)(w), then(60)supI⊂T⁡1mIwχILp·w-1χILp′·<∞,where the supremum is taken over all subarcs I⊂T. In particular, in this case, w∈Lp(·) and 1/w∈Lp′(·), whence Lp(·)(w) is a Banach function space by [15, Lemma  2.5(b)].We say that an exponentp∈PC(T) is locally log-Hölder continuous (cf. [17, Definition  2.2]) if there exists a constant Cp(·)∈(0,∞) such that(61)pt-pτ≤Cp·-log⁡t-τ∀t,τ∈Tsatisfyingt-τ<12.The class of all locally log-Hölder continuous exponents will be denoted by LH(T). Notice that some authors also denote this class by Plog(T) (see, e.g., [20, Section  1.1.4]).Following [21, Section  2.3], denote by W the class of all continuous functions ϱ:[0,2π]→0,∞ such that ϱ(0)=0, ϱ(x)>0, if 0<x≤2π, and ϱ is almost increasing; that is, there is a universal constant C>0 such that ϱ(x)≤Cϱ(y) whenever x≤y. Further, let W be the set of all functions ϱ:[0,2π]→[0,∞] such that xαϱ(x)∈W and xβ/ϱ(x)∈W for some α,β∈R. Clearly, the functions ϱ(x)=xγ belong to W for all γ∈R. For ϱ∈W, put(62)Φϱ0x≔lim⁡supy→0⁡ϱxyϱy,x∈0,∞.Since ϱ∈W, one can show that the limits(63)mϱ≔limx→0⁡log⁡Φϱ0xlog⁡x,Mϱ≔limx→∞⁡log⁡Φϱ0xlog⁡xexist and -∞<m(ϱ)≤Mϱ<+∞. These numbers were defined under some extra assumptions on ϱ by Matuszewska and Orlicz [22, 23] (see also [24] and [25, Chapter  11]). We refer to m(ϱ) (resp., M(ϱ)) as the lower (resp., upper) Matuszewska-Orlicz index of ϱ. For ϱ(x)=xγ, one has m(ϱ)=M(ϱ)=γ. Examples of functions ϱ∈W with m(ϱ)<M(ϱ) can be found, for instance, in [25, p. 93]. Fix pairwise distinct points t1,…,tn∈Γ and functions w1,…,wn∈W. Consider the following weight:(64)wt≔∏k=1nwkt-tk,t∈Γ.Each function wkt-tk is a radial oscillating weight. This is a natural generalization of the so-called Khvedelidze weights w(t)=∏k=1nt-tkλk, where λk∈R (see, e.g., [5, Section  5.8]).Theorem 11.
Letp∈LH(T). Suppose w1,…,wn∈W and the weight w is given by (64). The Cauchy singular integral operator S is bounded on Lp(·)(w) if and only if for all k∈{1,…,n}(65)0<1ptk+mwk,1ptk+Mwk<1.The sufficiency portion of Theorem11 was obtained by Kokilashvili et al. [21, Theorem  4.3] (see also [20, Corollary  2.109]) for more general finite Carleson curves in place of T. The necessity portion was proved by the author [26, Corollary  4.3] for Jordan Carleson curves.Lemma 12.
Letp∈LH(T). Suppose w1,…,wn∈W and the weight w is given by (64). Then the weighted Nakano space Lp(·)(w) is a reflexive Banach function space and the Riesz projection P is bounded on Lp(·)(w).Proof.
In view of Theorem11, the operator S is bounded on the space Lp(·)(w). As was observed above, the boundedness of the operator S on the space Lp(·)(w) implies that w∈Lp(·) and 1/w∈Lp′(·)(w) by [15, Theorem  6.1]. Hence Lp(·)(w) is a reflexive Banach function space thanks to [15, Lemma  2.5 and Corollary  2.8]. By [15, Lemma  6.4], the operator P=(I+S)/2 is bounded on Lp(·)(w).Consider the Hardy spaceHp(·)(w)≔H[Lp(·)(w)] built upon the weighted Nakano space Lp(·)(w), where p∈LH(T) and w is a weight as in Theorem 11.Theorem1 and Lemma 12 yield the following.Corollary 13.
Letp∈LH(T). Suppose w1,…,wn∈W and the weight w is given by (64). If A∈B(Hp(·)(w)) and there exists a sequence {an}n∈Z of complex numbers satisfying (9), then there exists a function a∈L∞ such that A=Ta and a^(n)=an for all n∈Z. Moreover(66)aL∞≤TaBHp·w≤PBLp·waL∞.Similarly, Theorem2 and Lemma 12 imply the following.Corollary 14.
Letp∈LH(T). Suppose w1,…,wn∈W and the weight w is given by (64). If a∈L∞, then the Toeplitz operator Ta∈B(Hp(·)(w)) is compact if and only if a=0.


### 4.3. Concluding Remarks

After this paper was submitted for publication, Leśnik posted the paper [27] in arXiv, where among other results he proved analogues of Theorems 1 and 2 for Toeplitz operators acting between abstract Hardy spaces H[X] and H[Y] built upon distinct rearrangement-invariant Banach function spaces X and Y. The set of allowed symbols in [27] coincides with the set M(X,Y) of pointwise multipliers from X to Y, which may contain unbounded functions. Thus, his results complement ours in a nontrivial way but are not more general than ours, because Leśnik restricts himself to rearrangement-invariant spaces X and Y only. On the other hand, the main aim of this paper is to consider the questions of the boundedness and compactness of Toeplitz operators on an abstract Hardy space H[X] in the case when X is an arbitrary, not necessarily rearrangement-invariant, Banach function space.


## 4.1. The Case of Hardy Spaces Built upon Lebesgue Spaces with Muckenhoupt Weights

A measurable functionw:T→[0,∞] is referred to as a weight if 0<w(τ)<∞ almost everywhere on T. If X is a Banach function space over the unit circle and w is a weight, then(52)Xw≔f∈L0:fw∈Xis a normed space equipped with the norm fX(w)≔fwX. Moreover, if w∈X and 1/w∈X′, then X(w) is a Banach function space (see [15, Lemma  2.5]).Let1<p<∞ and w be a weight. It is well known that the Riesz projection P is bounded on the weighted Lebesgue space Lp(w) if and only if the weight w satisfies the Muckenhoupt Ap-condition; that is,(53)supI⊂T⁡1mI∫Iwpτdmτ1/p1mI∫Iw-p′τdmτ1/p′<∞,where the supremum is taken over all subarcs I of the unit circle T and 1/p+1/p′=1 (see [16] and also [5, Section  1.46], [3, Section  5.7.3(h)]). In the latter case, we will write w∈Ap(T). It is clear that if w∈Ap(T), then w∈Lp and 1/w∈Lp′. Hence Lp(w) is a Banach function space whenever w∈Ap(T). It is well known that if 1<p<∞, then Lp(w) is reflexive. We denote the corresponding Hardy space by Hp(w)≔H[Lp(w)].Surprisingly enough, we were not able to find the following results explicitly stated in the literature.Corollary 9.
Let1<p<∞ and w∈Ap(T). If A∈B(Hp(w)) and there exists a sequence {an}n∈Z of complex numbers satisfying (9), then there exists a function a∈L∞ such that A=Ta and a^(n)=an for all n∈Z. Moreover(54)aL∞≤TaBHpw≤PBLpwaL∞.This is an immediate consequence of Theorem1. For the weight w=1, it is proved in [5, Theorem  2.7].Corollary 10.
Let1<p<∞ and w∈Ap(T). If a∈L∞, then the Toeplitz operator Ta∈B(Hp(w)) is compact if and only if a=0.This corollary follows from Theorem2.


## 4.2. The Case of Hardy Spaces Built upon Nakano Spaces with Radial Oscillating Weights

We denote byPC(T) the set of all continuous functions p:T→(1,∞). For p∈PC(T), let Lp(·) be the set of all functions f∈L0 such that(55)∫Tfτλpτdmτ<∞,for some λ=λ(f)>0. This set becomes a Banach function space when equipped with the Luxemburg-Nakano norm(56)fLp·≔inf⁡λ>0:∫Tfτλpτdmτ≤1,(see, e.g., [17, p. 73] or [18, p. 77]). If p is constant, then Lp(·) is nothing but the Lebesgue space Lp. The spaces Lp(·) are referred to as Nakano spaces. We refer to Maligranda’s paper [19] for the role of Hidegoro Nakano in the study of these spaces.SinceT is compact, we have(57)1<mint∈T⁡pt,maxt∈T⁡pt<∞.In this case, the space Lp(·) is reflexive and its associate space is isomorphic to the space Lp′(·), where 1/p(τ)+1/p′τ=1 for all τ∈T (see, e.g., [17, Section  2.8] and [18, Section  3.2]).LetSf be the Cauchy singular integral of a function f∈L1(T) defined by(58)Sft≔limε→01πi∫T∖Tt,εfττ-tdτ,t∈T,where T(t,ε)≔{τ∈T:τ-t<ε}. For a weight w:T→[0,∞], consider the weighted Nakano space(59)Lp·w≔f∈L0:fw∈Lp·.It follows from [15, Theorem  6.1] that if the operator S is bounded on Lp(·)(w), then(60)supI⊂T⁡1mIwχILp·w-1χILp′·<∞,where the supremum is taken over all subarcs I⊂T. In particular, in this case, w∈Lp(·) and 1/w∈Lp′(·), whence Lp(·)(w) is a Banach function space by [15, Lemma  2.5(b)].We say that an exponentp∈PC(T) is locally log-Hölder continuous (cf. [17, Definition  2.2]) if there exists a constant Cp(·)∈(0,∞) such that(61)pt-pτ≤Cp·-log⁡t-τ∀t,τ∈Tsatisfyingt-τ<12.The class of all locally log-Hölder continuous exponents will be denoted by LH(T). Notice that some authors also denote this class by Plog(T) (see, e.g., [20, Section  1.1.4]).Following [21, Section  2.3], denote by W the class of all continuous functions ϱ:[0,2π]→0,∞ such that ϱ(0)=0, ϱ(x)>0, if 0<x≤2π, and ϱ is almost increasing; that is, there is a universal constant C>0 such that ϱ(x)≤Cϱ(y) whenever x≤y. Further, let W be the set of all functions ϱ:[0,2π]→[0,∞] such that xαϱ(x)∈W and xβ/ϱ(x)∈W for some α,β∈R. Clearly, the functions ϱ(x)=xγ belong to W for all γ∈R. For ϱ∈W, put(62)Φϱ0x≔lim⁡supy→0⁡ϱxyϱy,x∈0,∞.Since ϱ∈W, one can show that the limits(63)mϱ≔limx→0⁡log⁡Φϱ0xlog⁡x,Mϱ≔limx→∞⁡log⁡Φϱ0xlog⁡xexist and -∞<m(ϱ)≤Mϱ<+∞. These numbers were defined under some extra assumptions on ϱ by Matuszewska and Orlicz [22, 23] (see also [24] and [25, Chapter  11]). We refer to m(ϱ) (resp., M(ϱ)) as the lower (resp., upper) Matuszewska-Orlicz index of ϱ. For ϱ(x)=xγ, one has m(ϱ)=M(ϱ)=γ. Examples of functions ϱ∈W with m(ϱ)<M(ϱ) can be found, for instance, in [25, p. 93]. Fix pairwise distinct points t1,…,tn∈Γ and functions w1,…,wn∈W. Consider the following weight:(64)wt≔∏k=1nwkt-tk,t∈Γ.Each function wkt-tk is a radial oscillating weight. This is a natural generalization of the so-called Khvedelidze weights w(t)=∏k=1nt-tkλk, where λk∈R (see, e.g., [5, Section  5.8]).Theorem 11.
Letp∈LH(T). Suppose w1,…,wn∈W and the weight w is given by (64). The Cauchy singular integral operator S is bounded on Lp(·)(w) if and only if for all k∈{1,…,n}(65)0<1ptk+mwk,1ptk+Mwk<1.The sufficiency portion of Theorem11 was obtained by Kokilashvili et al. [21, Theorem  4.3] (see also [20, Corollary  2.109]) for more general finite Carleson curves in place of T. The necessity portion was proved by the author [26, Corollary  4.3] for Jordan Carleson curves.Lemma 12.
Letp∈LH(T). Suppose w1,…,wn∈W and the weight w is given by (64). Then the weighted Nakano space Lp(·)(w) is a reflexive Banach function space and the Riesz projection P is bounded on Lp(·)(w).Proof.
In view of Theorem11, the operator S is bounded on the space Lp(·)(w). As was observed above, the boundedness of the operator S on the space Lp(·)(w) implies that w∈Lp(·) and 1/w∈Lp′(·)(w) by [15, Theorem  6.1]. Hence Lp(·)(w) is a reflexive Banach function space thanks to [15, Lemma  2.5 and Corollary  2.8]. By [15, Lemma  6.4], the operator P=(I+S)/2 is bounded on Lp(·)(w).Consider the Hardy spaceHp(·)(w)≔H[Lp(·)(w)] built upon the weighted Nakano space Lp(·)(w), where p∈LH(T) and w is a weight as in Theorem 11.Theorem1 and Lemma 12 yield the following.Corollary 13.
Letp∈LH(T). Suppose w1,…,wn∈W and the weight w is given by (64). If A∈B(Hp(·)(w)) and there exists a sequence {an}n∈Z of complex numbers satisfying (9), then there exists a function a∈L∞ such that A=Ta and a^(n)=an for all n∈Z. Moreover(66)aL∞≤TaBHp·w≤PBLp·waL∞.Similarly, Theorem2 and Lemma 12 imply the following.Corollary 14.
Letp∈LH(T). Suppose w1,…,wn∈W and the weight w is given by (64). If a∈L∞, then the Toeplitz operator Ta∈B(Hp(·)(w)) is compact if and only if a=0.


## 4.3. Concluding Remarks

After this paper was submitted for publication, Leśnik posted the paper [27] in arXiv, where among other results he proved analogues of Theorems 1 and 2 for Toeplitz operators acting between abstract Hardy spaces H[X] and H[Y] built upon distinct rearrangement-invariant Banach function spaces X and Y. The set of allowed symbols in [27] coincides with the set M(X,Y) of pointwise multipliers from X to Y, which may contain unbounded functions. Thus, his results complement ours in a nontrivial way but are not more general than ours, because Leśnik restricts himself to rearrangement-invariant spaces X and Y only. On the other hand, the main aim of this paper is to consider the questions of the boundedness and compactness of Toeplitz operators on an abstract Hardy space H[X] in the case when X is an arbitrary, not necessarily rearrangement-invariant, Banach function space.



---

*Source: 9768210-2017-09-28.xml*