Research Article | Open Access
Alexei Yu. Karlovich, "Toeplitz Operators on Abstract Hardy Spaces Built upon Banach Function Spaces", Journal of Function Spaces, vol. 2017, Article ID 9768210, 8 pages, 2017. https://doi.org/10.1155/2017/9768210
Toeplitz Operators on Abstract Hardy Spaces Built upon Banach Function Spaces
Let be a Banach function space over the unit circle and let be the abstract Hardy space built upon . If the Riesz projection is bounded on and , then the Toeplitz operator is bounded on . We extend well-known results by Brown and Halmos for and show that, under certain assumptions on the space , the Toeplitz operator is bounded (resp., compact) if and only if (resp., ). Moreover, . 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.
The Banach algebra of all bounded linear operators on a Banach space will be denoted by . Let be the unit circle in the complex plane . For , a function of the form , where for all and , is called a trigonometric polynomial of order . The set of all trigonometric polynomials is denoted by . The Riesz projection is the operator which is defined on byFor , let be the Lebesgue space on the unit circle in the complex plane. For , letbe the sequence of the Fourier coefficients of . The classical Hardy spaces are given byIt is well known that the Riesz projection extends to a bounded linear operator on if and only if . For , the Toeplitz operator with symbol on , , is given byToeplitz operators have attracted the mathematical community for the many decades since the classical paper by Toeplitz . Brown and Halmos [2, Theorem 4] proved that a necessary and sufficient condition that an operator on is a Toeplitz operator is that its matrix with respect to the standard basis of is a Toeplitz matrix, that is, the matrix of the form . The norm of on the Hardy space coincides with the norm of its symbol in (actually, this result was already in a footnote of ). Brown and Halmos also observed, as a corollary, that the only compact Toeplitz operator on 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 , [5, Theorem 2.7].
In this paper, we will consider the so-called Banach function spaces in place of . As usual, we equip the unit circle with the normalized Lebesgue measure . Denote by the set of all measurable complex-valued functions on , and let be the subset of functions in whose values lie in . The characteristic function of a measurable set is denoted by . A mapping is called a function norm if, for all functions , , () in , for all constants , and for all measurable subsets of , the following properties hold:(a) a.e., , ,(b) a.e. (the lattice property),(c) a.e. (the Fatou property),(d), ,with depending on and but independent of . When functions differing only on a set of measure zero are identified, the set of all functions for which is a Banach space under the norm . Such a space is called a Banach function space. If is a function norm, its associate norm is defined on byThe Banach function space determined by the function norm is called the associate space (or Köthe dual space) of . The associate space is a subspace of the dual space . The simplest examples of Banach function spaces are the Lebesgue spaces , . 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 built upon a Banach function space over the unit circle as follows:This definition makes sense because is continuously embedded in in view of axiom (d). It can be shown that is a closed subspace of . It is clear that if , then is the classical Hardy space .
It follows from axiom (d) that . We will restrict ourselves to Banach function spaces such that the Riesz projection defined initially on by formula (1) extends to a bounded linear operator on the whole space . The extension will again be denoted by . If and , then the Toeplitz operator defined by formula (4) is bounded on and
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 spaces built upon reflexive Banach function spaces (not necessarily rearrangement-invariant) if . Further, we show that, under mild assumptions on a Banach function space , a Toeplitz operator is compact on the abstract Hardy space built upon if and only if . 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.
For and , we will use the following pairing:For and , put . Then the Fourier coefficients of a function can be expressed by for .
Theorem 1 (main result 1). Let be a reflexive Banach function space over the unit circle such that the Riesz projection is bounded on . Suppose and there is a sequence of complex numbers such thatThen there is a function such that and for all . Moreover
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 in a Banach function space is said to have absolutely continuous norm in if for every sequence of measurable sets satisfying almost everywhere as . The set of all functions in of absolutely continuous norm is denoted by . It is known that a Banach function space is reflexive if and only if and have absolutely continuous norm (see [6, Chap. 1, Corollary 4.4]).
Theorem 2 (main result 2). Let be a Banach function space over the unit circle such that for every measurable subset . If the Riesz projection is bounded on and , then the Toeplitz operator is compact if and only if .
The paper is organized as follows. Section 2 contains results on the density of the set of all trigonometric polynomials (resp., the set of all analytic polynomials ) in a Banach function space (resp., in the abstract Hardy space built upon ). We also show that the norm of a function in can be calculated in terms of , where , under the assumption that is separable. Further, we prove that every bounded linear operator on a separable Banach function space, whose matrix is of the form , is an operator of multiplication by a function and the sequence of its Fourier coefficients is exactly . Finally, we prove that if the characteristic functions of all measurable sets have absolutely continuous norms in , then the sequence converges weakly to zero on the abstract Hardy space . 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 with Muckenhoupt weights and to the case of weighted Nakano spaces 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. Let be a Banach function space over the unit circle . The following statements are equivalent: (a)the set of all trigonometric polynomials is dense in the space ;(b)the space of all continuous functions on is dense in the space ;(c)the Banach function space is separable.
2.2. Density of Analytic Polynomials in Abstract Hardy Spaces
Let . A function of the form , where for all and , is said to be an analytic polynomial on . The set of all analytic polynomials is denoted by .
Lemma 4. Let be a separable Banach functions space over the unit circle . If the Riesz projection is bounded on , then the set is dense in .
Proof. If , then by Lemma 3, there exists a sequence such that as . It is clear that and . Since , we finally haveas . Thus is dense in .
2.3. A Formula for the Norm in a Banach Function Space
Lemma 5. Let be a Banach function space over the unit circle . If the associate space is separable, then for every
Proof. By [6, Theorem 2.7 and Lemma 2.8], for every By the lattice property of the associate space , we have . Hence, equality (13) implies that for Fix such that . Since is separable, it follows from Lemma 3 that there exists a sequence such that as . For , put . Then for every HenceIt follows from Hölder’s inequality for Banach function spaces (see [6, Chap. 1, Theorem 2.4]) and (17) thatThus, taking into account (15) and (18), we deduce for every function satisfying thatThis inequality and equality (13) imply thatCombining 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]). Let be a Banach function space over the unit circle . If , then the multiplication operatoris bounded on if and only if and .
It is easy see thatThe following lemma shows that every bounded operator with such a property is a multiplication operator.
Lemma 7. Let be a separable Banach functions space over the unit circle . Suppose and there exists a sequence of complex numbers such thatThen there exists a function such that and for all .
Proof. Put . Since , we infer from (23) thatIf , then and the th Fourier coefficient of is calculated byOn the other hand, from (23), we get for By (25) and (26), for all . Therefore, for all in view of the uniqueness theorem for Fourier series (see, e.g., [11, Chap. I, Theorem 2.7]). Since the space is separable, the set is dense in by Lemma 3. Therefore for . This means that . It remains to apply Lemma 6.
2.5. Weak Convergence of the Sequence to Zero on the Abstract Hardy Space
Recall that the annihilator of a subspace of a Banach space is the set of all linear functionals such that for all (see, e.g., [12, p. 110]).
Lemma 8. If is a Banach function space such that for every measurable subset , then converges weakly to zero on .
Proof. By [12, Theorem 7.1], is isometrically isomorphic to . Since for all , in view of the above fact, it is sufficient to prove that converges weakly to zero on the whole space instead of the subspace .
By [6, Chap. 1, Corollary 3.14], if for every measurable subset , then is isometrically isomorphic to . In view of [6, Chap. 1, Theorem 2.2], is a Banach function space, which is continuously embedded into due to axiom (d) of the definition of a Banach function norm. Thus, for every , there exists a function such that for all . In particular, if with , thenBy the Riemann-Lebesgue lemma (see, e.g., [11, Chap. I, Theorem 2.8]) and (27), as for every ; that is, converges weakly to zero on , which completes the proof.
3. Proof of the Main Results
3.1. Proof of Theorem 1
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 is nonzero. For , put . Then taking into account Lemma 6 and that , we get Consider the following subset of the associate space:It follows from Hölder’s inequality for Banach function spaces (see [6, Chap. 1, Theorem 2.4]) and (28) and (29) thatSince is reflexive, in view of [6, Chap. 1, Corollaries 4.3-4.4], we know that is canonically isometrically isomorphic to . Applying the Banach-Alaoglu theorem (see, e.g., [13, Theorem 3.17]) to , , and , we deduce that there exists a such that some subsequence of converges to in the weak topology on . In particularOn the other hand, the definition of and equality (9) imply thatIt follows from (31) and (32) thatNow define the mapping byAssume that and are trigonometric polynomials of orders and , respectively. ThenIt follows from (9) and (33) that for It is clear that for . Therefore, taking into account Lemma 6, we see that for By Hölder’s inequality for Banach function spaces (see [6, Chap. 1, Theorem 2.4]) from (36) and (37), we obtainSince a Banach function space is reflexive and the Lebesgue measure is separable, it follows from [6, Chap. 1, Corollaries 4.4 and 5.6] that the spaces and are separable. Then Lemma 5 and inequality (38) yieldfor all . In view of Lemma 3, is dense in . Then (39) implies that the linear mapping defined in (34) extends to an operator such thatWe deduce from (33) thatBy Lemma 7, there exists a function such that and for all . MoreoverIt follows from the definition of the Toeplitz operator thatCombining this fact with equality (9), we arrive atSince , by the uniqueness theorem for Fourier series (see, e.g., [11, Chap. I, Theorem 2.7]), it follows from (44) that for all . ThereforeIn view of Lemma 4, the set is dense in . This fact and equality (45) imply that andCombining 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 Theorem 2
It is clear that if , then is the zero operator, which is compact. Now assume that is compact. Then it maps weakly convergent sequences in into strongly convergent sequences in (see, e.g., [14, Section 7.5, Theorem 4]). Since converges to zero weakly on in view of Lemma 8, we haveBy [6, Chap. 1, Theorem 2.7 and Lemma 2.8], for ,Since , there exists a constant such thatFor all and all such that , we haveThen from (48)–(50) we obtain for all and all such that Passing in this inequality to the limit as and taking into account (47), we see that for all . By the uniqueness theorem for Fourier series (see, e.g., [11, Chap. I, Theorem 2.7]), this implies that a.e. on .
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 function is referred to as a weight if almost everywhere on . If is a Banach function space over the unit circle and is a weight, thenis a normed space equipped with the norm . Moreover, if and , then is a Banach function space (see [15, Lemma 2.5]).
Let and be a weight. It is well known that the Riesz projection is bounded on the weighted Lebesgue space if and only if the weight satisfies the Muckenhoupt -condition; that is,where the supremum is taken over all subarcs of the unit circle and (see  and also [5, Section 1.46], [3, Section 5.7.3(h)]). In the latter case, we will write . It is clear that if , then and . Hence is a Banach function space whenever . It is well known that if , then is reflexive. We denote the corresponding Hardy space by .
Surprisingly enough, we were not able to find the following results explicitly stated in the literature.
Corollary 9. Let and . If and there exists a sequence of complex numbers satisfying (9), then there exists a function such that and for all . Moreover
Corollary 10. Let and . If , then the Toeplitz operator is compact if and only if .
This corollary follows from Theorem 2.
4.2. The Case of Hardy Spaces Built upon Nakano Spaces with Radial Oscillating Weights
We denote by the set of all continuous functions . For , let be the set of all functions such thatfor some . This set becomes a Banach function space when equipped with the Luxemburg-Nakano norm(see, e.g., [17, p. 73] or [18, p. 77]). If is constant, then is nothing but the Lebesgue space . The spaces are referred to as Nakano spaces. We refer to Maligranda’s paper  for the role of Hidegoro Nakano in the study of these spaces.
Let be the Cauchy singular integral of a function defined bywhere . For a weight , consider the weighted Nakano spaceIt follows from [15, Theorem 6.1] that if the operator is bounded on , thenwhere the supremum is taken over all subarcs . In particular, in this case, and , whence is a Banach function space by [15, Lemma 2.5(b)].
We say that an exponent is locally log-Hölder continuous (cf. [17, Definition 2.2]) if there exists a constant such thatThe class of all locally log-Hölder continuous exponents will be denoted by . Notice that some authors also denote this class by (see, e.g., [20, Section 1.1.4]).
Following [21, Section 2.3], denote by the class of all continuous functions such that , , if , and is almost increasing; that is, there is a universal constant such that whenever . Further, let be the set of all functions such that and for some . Clearly, the functions belong to for all . For , putSince , one can show that the limitsexist and . These numbers were defined under some extra assumptions on by Matuszewska and Orlicz [22, 23] (see also  and [25, Chapter 11]). We refer to (resp., ) as the lower (resp., upper) Matuszewska-Orlicz index of . For , one has . Examples of functions with can be found, for instance, in [25, p. 93]. Fix pairwise distinct points and functions . Consider the following weight:Each function is a radial oscillating weight. This is a natural generalization of the so-called Khvedelidze weights , where (see, e.g., [5, Section 5.8]).
Theorem 11. Let . Suppose and the weight is given by (64). The Cauchy singular integral operator is bounded on if and only if for all
The sufficiency portion of Theorem 11 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 . The necessity portion was proved by the author [26, Corollary 4.3] for Jordan Carleson curves.
Lemma 12. Let . Suppose and the weight is given by (64). Then the weighted Nakano space is a reflexive Banach function space and the Riesz projection is bounded on .
Proof. In view of Theorem 11, the operator is bounded on the space . As was observed above, the boundedness of the operator on the space