Compact Hankel Operators Research Articles

Compact and finite rank operators satisfying a Hankel type equationT 2X=XT 1 *

In 1997, V. Ptak defined the notion of generalized Hankel operator as follows: Given two contractions $$\mathcal{B}(\mathcal{H}_1 )$$ and $$\mathcal{B}(\mathcal{H}_2 )$$ , an operatorX: $$X:\mathcal{H}_1 \to \mathcal{H}_2 $$ is said to be a generalized Hankel operator ifT 2 X=XT 1 * andX satisfies a boundedness condition that depends on the unitary parts of the minimal isometric dilations ofT 1 andT 2. The purpose behind this kind of generalization is to study which properties of classical Hankel operators depend on their characteristic intertwining relation rather than on the theory of analytic functions. Following this spirit, we give appropriate versions of a number of results about compact and finite rank Hankel operators that hold within Ptak's generalized framework. Namely, we extend Adamyan, Arov and Krein's estimates of the essential norm of a Hankel operator, Hartman's characterization of compact Hankel operators and Kronecker's characterization of finite rank Hankel operators.

The essential norm of Hankel operator on the Bergman spaces of strongly pseudoconvex domains

Let D be a bounded strongly pseudoconvex domain with smooth boundary in Cn and let f∈L2(D). For the Hankel operator Hf on the Bergman space A2(D), it is shown that the essential norm of Hf in L2(D) is comparable to the distance norm from Hf to compact Hankel operators. The result extends the previous corresponding version in the disc proved by Lin and Rochberg in Integ.Equat.Oper.Theory 361–372,17 (1993).

Compact Hankel operators with conjugate analytic symbols

In 1986 S. Axler [3] proved that forf∈L a 2 the Hankel operator $$H_{\bar f} :L_a^2 \to (L^2 )^ \bot $$ is compact if and only iff is in the little Bloch space {itB}{in0}. In this note we show that the same is true for $$H_{\bar f} :L_a^p \to L^p $$ , 1<p<∞. Moreover we prove that $$H_{\bar f} :L_a^1 \to L^1 $$ is ⋆-compact if and only if $$|f'(z)|(1 - |z|^2 )\log \tfrac{1}{{1 - |z|^2 }} \to 0$$ as |z|→1−.

Compact operators via the Berezin transform

AbstractIn this paper we prove that if S equals a finite sum of finite productsof Toeplitz operators on the Bergman space of the unit disk, thenS is compact if and only if the Berezin transform of S equals 0 on∂D. This result is new even when S equals a single Toeplitz operator.Our main result can be used to prove, via a unified approach, severalpreviously known results about compact Toeplitz operators, compactHankel operators, and appropriate products of these operators. 1 Introduction Let dA denote Lebesgue area measure on the unit disk D, normalizedso that the measure of D equals 1. The Bergman space L 2a is the Hilbertspace consisting of the analytic functions on D that are also in L 2 (D,dA).For z ∈ D, the Bergman reproducing kernel is the function K z ∈ L 2a suchthatf(z) = hf,K z ifor every f ∈ L 2a . The normalized Bergman reproducing kernel k z is thefunction K z /kK z k 2 . Here, as elsewhere in this paper, the norm k k 2 and theinner product h , i are taken in the space L 2 (D,dA).For S a bounded operator on L

Collectively Compact Hankel Operator Sequences on Hardy Spaces

In this paper, an equivalent condition for collectively compact Hankel operator sequences is given. Our result extends the well-known characterization of single compact Hankel operators due to Hartman.

Abstract, weighted, and multidimensional Adamjan-Arov-Krein theorems, and the singular numbers of Sarason commutants

The classical Adamjan-Arov-Krein (A-A-K) theorem relating the singular numbers of Hankel operators to best approximations of their symbols by rational functions is given an abstract version. This provides results for Hankel operators acting in weightedH2(T; μ), as well as inH2(Td), and an A-A-K type extension of Sarason's interpolation theorem. In particular, it is shown that all compact Hankel operators inH2(Td) are zero.

Compact Hankel operators on multiply connected domains

Compact Hankel operators on multiply connected domains

On the convergence rate ofs-numbers of compact Hankel operators

It is well known that the sequence ofs-numbers {sn},n=0, 1,..., of a compact operator, and particularly a compact Hankel operator Γ=[h j+k −1], converges monotonically to zero. Since the (n + 1)sts-number sn measures the error ofL ∞(¦z¦=1) approximation, modulo an additive H∞ function, by nth degree proper rational functions whose poles are restricted to ¦z¦ < 1, it is very important to study how fast {s n } converges to zero. It is not difficult to see that ifh n =O(n −α), for someα > 1, thens n =O(n −α). In this paper we construct, for any given sequenceɛ n ↓ 0, a compact Hankel operator Γ such thats n ≥ɛ n for alln.

Rate of convergence of schmidt pairs and rational functions corresponding to best approximants of truncated hankel operators

The problem of approximating Hankel operators of infinite rank by finite-rank Hankel operators is considered. For efficiency, truncated infinite Hankel matrices Γn of Γ are utilized. In this paper for any compact Hankel operator Γ of the Wiener class, we derive the rate of l2-convergence of the Schmidt pairs of Γn to the corresponding Schmidt pairs of Γ. For a certain subclass of Hankel operators of the Wiener class, we also obtain the rate of l1-convergence. In addition, an upper bound for the rate of uniform convergence of the rational symbols of best rank-k Hankel approximants of Γn to the corresponding rational symbol of the best rank-k Hankel approximant to Γ asn → ∞ is derived.

WEIGHTED BERGMAN SPACES, BLOCH SPACE AND DING SPACE

In this paper we give some alternate descriptions of the Bloch space and a relations between the Bloch space and a special class of Ba space which generalize some of the results by Axlex, He and Ouyang respectively, and we give a proof of the theorem about the bounded and compact Hankel operator on weighted Bergman space.

The Compact Hankel Operators Form an M-Ideal in the Space of Hankel Operators

The Compact Hankel Operators Form an M-Ideal in the Space of Hankel Operators

The compact Hankel operators form an 𝑀-ideal in the space of Hankel operators

The theorem in the title is proved and used to give a new proof that elements of L ∞ {L^\infty } have best approximations in H ∞ + C {H^\infty } + C .

A proof of Hartman's theorem on compact Hankel operators

Let U be a unilateral shift of arbitrary (perhaps uncountable) multiplicity on a Hilbert space. Following Rosenblum (5), an operator A is said to be a Hankel operator relative to U ifHartman (2) has characterized the compact Hankel operators relative to the unilateral shift of multiplicity one as the Hankel operators with symbol in H∞ + C(T). Using the usual function space model for representing the unilateral shift, Page ((4), theorem 10) has extended Hartman's theorem to unilateral shifts of countable multiplicity. We give a model-free proof of Hartman's theorem which applies to shifts of arbitrary multiplicity. The proof turns on the observation that a compact operator acts compactly on a certain algebra of operators.

Compact Hankel operators and the F. and M. Riesz theorem

Compact Hankel operators and the F. and M. Riesz theorem

Bounded and Compact Vectorial Hankel Operators

Operators $H$ satisfying ${S^ \ast }H = HS$ where $S$ is a unilateral shift on Hilbert space are called Hankel operators. For a fixed shift $S$ of arbitrary multiplicity the Banach spaces of bounded Hankel operators and of compact Hankel operators are described, and it is shown that the former is always the second dual of the latter. Representations for bounded and for compact Hankel operators are given in a standard function space model.

Bounded and compact vectorial Hankel operators

Operators H H satisfying S ∗ H = H S {S^ \ast }H = HS where S S is a unilateral shift on Hilbert space are called Hankel operators. For a fixed shift S S of arbitrary multiplicity the Banach spaces of bounded Hankel operators and of compact Hankel operators are described, and it is shown that the former is always the second dual of the latter. Representations for bounded and for compact Hankel operators are given in a standard function space model.