Compact Hankel Operators Between Distinct Hardy Spaces and Commutators

The paper is devoted to the study of compactness of Hankel operators acting between distinct Hardy spaces generated by Banach function lattices. We prove an analogue of Hartman’s theorem characterizing compact Hankel operators in terms of properties of their symbols. As a byproduct we give an estimation of the essential norm of such operators. Furthermore, compactness of commutators and semicommutators of Toeplitz operators for unbounded symbols is discussed.


Introduction
Let H 2 be the space of analytic and square-integrable functions on the unit circle T = {e iθ : θ ∈ [0, 2π)}. For a measurable and bounded function a on T (called a symbol ), the Toeplitz operator T a is defined on H 2 as T a f = P (af ), f ∈ H 2 . Here P is the orthogonal projection from L 2 onto H 2 (i.e., the Riesz projection). Similarly, the Hankel operator H a is defined on H 2 by the formula H a f = P (aJf ), where J is the flip operator.
The theory of Toeplitz and Hankel operators has been constantly developed since the beginning of the twentieth century and plays a pivotal role in functional and harmonic analysis as well as in the operator theory (see [3,31]). Through intimate connection with Toeplitz and Hankel matrices, these operators are of great importance in many applications far beyond pure mathematics, for instance in signal and image processing, queueing theory or quantum mechanics.
It is worth mentioning that from the historical point of view, the main lines of research in the field of Toeplitz and Hankel operators are related to the case when a symbol of operators is bounded, that is the discussed operators act on (the same) Hardy space (mainly H 2 ). The case of Toeplitz and Hankel operators acting from one to another Hardy space (further we will refer to the generality of the topic being considered, we recall the main definitions and notations below.

Quasi-Banach Function Spaces
Recall that T is the unit circle and m is the normalized Lebesgue measure on T, that is dm(t) = (2π) −1 |dt|. Let L 0 := L 0 (T) be the space of (equivalence classes of) all m-measurable complex-valued almost everywhere finite functions on T.
A quasi-Banach space X ⊂ L 0 is called a quasi-Banach function space if it has the so-called ideal property (that is, g ∈ X and g X f X for all f ∈ X and g ∈ L 0 satisfying |g| |f | a.e. on T) and L ∞ ⊂ X. If in addition X is a Banach space, then we will call it a Banach function space (it is also common to refer to such a space a Köthe space or Banach function lattice).
A quasi-Banach function space X has the Fatou property when given a sequence (f n ) n∈N ⊂ X and f ∈ L 0 satisfying 0 ≤ f n ↑ f a.e. as n → ∞ and sup n∈N f n X < ∞ we have f ∈ X and f X = sup{ f n X : n ∈ N}. A quasi-Banach function space X has the semi-Fatou property if for each Recall that f ∈ X is said to be an order continuous element if for each (f n ) n∈N ⊂ X satisfying 0 ≤ f n ≤ |f | for all n ∈ N and f n → 0 a.e. as n → ∞, there holds f n X → 0 as n → ∞. The subspace of order continuous elements of X is denoted by X o . Evidently, X o has the semi-Fatou property. We say that X is order continuous, when X = X o . Notice that this is equivalent to the separability of X.
For a quasi-Banach function space X, its Köthe dual X is defined as the space of functions g ∈ L 0 satisfying fg ∈ L 1 for all f ∈ X and equipped with a quasi-norm g X = sup{ fg L 1 : f X 1}. Notice that X is nontrivial and possesses the Fatou property if X is a Banach function space. Note also that X may be trivial, that is X = {0}, when X is just a quasi-Banach function space. For example, (L p ) = {0} when p ∈ (0, 1). It is known that a Banach function space X has the Fatou property if and only if X ≡ X (see [23, p. 30]).
We will work within an important class of quasi-Banach function spaces, that is with rearrangement invariant spaces. Recall that for f ∈ L 0 it's distribution μ f is given by A quasi-Banach function space X is called rearrangement invariant (r.i. for short) if for every pair of equimeasurable functions f, g ∈ L 0 , f ∈ X implies that g ∈ X and f X = g X . We point out that many important examples of quasi-Banach function spaces are rearrangement-invariant, for example Lebesgue, Orlicz and Lorentz spaces. We refer the reader to [19,23] and [2] for more information on non-increasing rearrangements and r.i. spaces. Let X be a r.i. quasi-Banach function space. For each s ∈ R + the dilation operator D s is defined by It is well-known (see, for example, [19] for the case of Banach spaces, or [5,28] for the case of quasi-Banach spaces on R and R + , respectively) that D s is bounded on X for each s > 0 and the following limits exist The numbers α X and β X are called the lower and the upper Boyd index of X, respectively. For an arbitrary r.i. Banach function space X, its Boyd indices belong to [0, 1] and α X ≤ β X . We say that Boyd indices are nontrivial, if α X , β X ∈ (0, 1).

Pointwise Multipliers and Products
Let X and Y be Banach function spaces. The space of pointwise multipliers equipped with the operator norm The notion of multipliers is associated with a concept of products of spaces. Given two quasi-Banach spaces X and Y , the pointwise product X Y is defined as X Y = gh : g ∈ X, h ∈ Y . It should be noted that even if X, Y are Banach function spaces, then X Y need not to be a Banach space. Indeed, the product space is, in general, a quasi-Banach space when equipped with the quasi-norm For example, if p, q ∈ (0, ∞], then L p L q ≡ L r , where 1 r = 1 p + 1 q . For further reading we refer to the papers [17,34] and references included therein.

Hardy Spaces, Toeplitz and Hankel Operators
For n ∈ Z and t ∈ T, let χ n (t) := t n . The Fourier coefficients of a function f ∈ L 1 are given by Let X be a quasi-Banach function space such that X ⊂ L 1 (note that X ⊂ L 1 for each r.i. Banach function space). The Hardy space H[X] is defined by with the quasi-norm inherited from X (see for example [14,26,27,38]). If p ∈ [1, ∞], then H p := H[L p ] is the classical Hardy space (see for example [6,9,15]). We shall also use the following variants of Hardy spaces: The Riesz projection P is for f ∈ L 1 given by the formula where f is the conjugate function to f (see for example [15,Chapter III] for the definition of f ). We recall that if a quasi-Banach function space X has nontrivial Boyd indices, then from the Boyd interpolation theorem it follows that P : X → X is a bounded operator (see [5,23]). A little more can be said in the case of r.i. Banach function spaces. Indeed, in [7] it was proved that P : X → X is bounded if and only if X has nontrivial Boyd indices. In such the case, the following equivalent formula for P is meaningful Below we state definitions of Toeplitz and Hankel operators. Roughly speaking we follow [20], however, notice that we allow also quasi-Banach spaces, which will be used as a tool in the last section. Let X and Y be r.i. quasi-Banach function spaces such that X ⊂ Y ⊂ L 1 and assume that P is bounded on Y . For a ∈ M (X, Y ), the Toeplitz operator T a is defined by the formula Analogously, the Hankel operator is given by where J is the so-called flip operator, that is It is well known that a Toeplitz operator T a : k, j 0. The converse statement is known as the Brown-Halmos theorem and holds for a wide class of spaces (see for example [3], or [14,20] for nonalgebraic setting). An analogous role, but for Hankel operators is played by the Nehari theorem. We will need in the sequel the following general version of it (see [20,Theorem 5.2]).
Then a continuous linear operator A : where the constant c > 0 depends only on the spaces X, Y .

Compact Hankel Operators
Before we state the main result we need to prove an analogue of Lemma 5.4 from [31]. Recall that a Banach space Y has the approximation property, if for every Banach space X, the set of finite rank operators from X into Y is dense in the subspace K(X, Y ) of compact operators from X into Y (cf. [ In what follows S : X → X denotes the shift operator given by Since X is order continuous, then polynomials are dense in X . Thus there exist polynomials p k ∈ X such that Further observe that Notice that for a fixed k and arbitrary f ∈ H[X] we have whenever n > deg p k . As a consequence, for n > max{deg p k : k ∈ {1, . . . , m}} there holds KS n ε and the proof is finished. Recall that the essential norm T ess of an operator T ∈ L(X, Y ), where X, Y are Banach spaces, is the distance of T from the subspace of compact operators K(X, Y ), that is In 1958, Hartman [8] characterized the compactness of Hankel operators H a : H 2 → H 2 and showed that H a is compact if and only if there exists a continuous function g on T such that g(n) = a n for n ∈ N. Using the notion of the essential norm even more can be said. The following result attributed to Hartman, Adamyan, Arov and Krein, contains estimates on the essential norm of Hankel operators on H p spaces.
where c p is the norm of the Riesz projection P : L p → H p .
Here the space C + H ∞ is a closed subalgebra of L ∞ (this statement is usually referred to as Sarason's theorem) consisting of functions φ ∈ L ∞ such that φ admits a representation φ = g + f , where g ∈ C and f ∈ H ∞ (see [31, p. 25]).
Our description of compact Hankel operators will be of similar fashion, but C will be replaced by M (X, Y ) o , while M (X, Y ) and H[M (X, Y )] will IEOT play the role of L ∞ and H ∞ , respectively. At first, we need, however, one more auxiliary result, i.e., an analogue of Sarason's theorem.

Theorem 4.3. Let X be r.i. Banach function spaces with the Fatou property. Then X o + H[X] is a closed subspace of X. In particular,
Proof. First notice that if X = X o or X has nontrivial Boyd indices, then there is nothing to prove. Suppose that X = X o . We will follow ideas of [31, Theorem 5.1]. At first we will show that for each (4.1) We need to prove only the inequality dist Denote for f ∈ L 1 and r ∈ (0, 1) where P r is the Poisson kernel for the unit disc D (see [6,9,15]). Let now f ∈ X o and g ∈ H [X]. Then where the last inequality follows from the Calderón-Mitjagin theorem (cf. [2]) and the fact that P r H 1 = 1 for each r ∈ (0, 1). As a consequence, However, f, f r ∈ X o and so f − f r Xo = f − f r X → 0 as r → ∞, because (P r ) is the approximation kernel and X o is a homogeneous Banach space in the sense of [15,Theorem 2.11]. Hence, (ii) If in addition spaces X and Y satisfy assumptions of Theorem 3.1, Y has the Fatou property and X, X are order continuous, then also for some constant c > 0 depending only on spaces X and Y .
Proof. (i). At first assume that a ∈ M (X, Y ) o . Thanks to the assumption L ∞ M (X, Y ), the set of all polynomials is included in M (X, Y ) o and is dense therein (see [20, Lemma 3.1 (a) and Lemma 3.4]). Note also that any Hankel operator induced by a polynomial has a finite rank (cf. Kronecker's theorem [31]). Let (p n ) be a sequence of polynomials such that a − p n M (X,Y ) → 0 as n → ∞. By the definition of the Hankel operator we have Hence H a is compact. Assume now that a ∈ M (X, Y ). Then evidently where the inequality comes from the definition and properties of Hankel operator (cf. [20,Theorem 5.2]). Besides, we get Eventually, the calculations above lead to the conclusion that (ii). Let a ∈ M (X, Y ) and assume that K : Before we proceed with further results let us comment on assumptions of the above theorem and discuss some special cases excluded from it. First of all, we have assumed L ∞ M (X, Y ). Nevertheless, this assumption is essentially not necessary to prove the point (i). The reason to assume it is that in case L ∞ = M (X, Y ) we can not use the notion of M (X, Y ) o = {0}. However, in such circumstances the role of M (X, Y ) o is played by the space of continuous functions C = C(T). Thus, when X ⊂ Y , but M (X, Y ) = L ∞ , point (i) holds (with exactly the same proof, under the assumption that Y has nontrivial Boyd indices) and takes the form For the point (ii) much more is needed. Namely, we have two possibilities. If the assumption of Theorem 3.1(ii) is satisfied, it implies that L ∞ M (X, Y ). Notice however that the assumption from Theorem 3.1(ii) is much stronger than L ∞ M (X, Y ). On the other hand, the assumption of Theorem 3.1(i) is satisfied when X = Y and X has the Fatou property. In this case Theorem 4.4 has the following form. Finally, notice that it may happen that L ∞ = M (X, Y ) even for X Y (for example, M (L p,q , L p,r ) = L ∞ when p ∈ [1, ∞) and 1 q < r ∞, see the definition of the Lorentz space L p,q below). In such a case, however, neither assumption (i), nor (ii) of Theorem 3.1 is satisfied, and thus estimation from Theorem 4.4(ii) is problematic-with current knowledge we can not decide whether it holds.

Commutators and Semicommutators of Toeplitz and Hankel Operators
For two operators T, S : X → X their commutator is defined as The commutator inspects to what extent operators T and S fail to be commutative, which constitutes an important question in the field of operator algebras. Seemingly, the definition of commutator requires that both operators belong to the same operator algebra and in general can not be directly extended to nonalgebraic settings. Nonetheless, it is quite interesting that we may define commutators and semicommutators for Toeplitz and Hankel operators acting between distinct Hardy spaces.
The following example serves as an inspiration for considerations in this part of the paper. Therefore, finalizing the previous discussion, we see that in the case (ii) one can select a and b such that Then the natural question is whether for some a, b as above [T a , T b ], (T b , T a ]: H p1,q1 → H p3,q3 may still be compact?