On commutativity of weighted Hankel operators and their spectra

2015 ◽  
Vol 18 (03) ◽  
pp. 1550017 ◽  
Author(s):  
Gopal Datt ◽  
Deepak Kumar Porwal
Keyword(s):  
Hankel Operator ◽  
Hankel Operators ◽  
Weyl’S Theorem

In this paper, we describe the conditions on which the nonzero weighted Hankel operators [Formula: see text] and [Formula: see text] on H2(β) induced by ϕ ∈ L∞(β) and ψ ∈ L∞(β) respectively commute, where β = {βn}n∈ℤ is a sequence of positive numbers with β0 = 1. Spectrum of the weighted Hankel operator [Formula: see text], when ϕ(z) = az-1 + bz-2, is computed and it is also shown that the Weyl's theorem holds for the compact weighted Hankel operators.

scholarly journals Hankel Operators on the WeightedLP-Bergman Spaces with Exponential Type Weights

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Hong Rae Cho ◽  
Jeong Wan Seo
Keyword(s):  
Exponential Type ◽  
Hankel Operator ◽  
Bergman Spaces ◽  
Hankel Operators

We characterize the boundedness and compactness of the Hankel operator with conjugate analytic symbols on the weightedLP-Bergman spaces with exponential type weights.

scholarly journals Finite-rank intermediate Hankel operators on the Bergman space

2001 ◽  
Vol 25 (1) ◽  
pp. 19-31 ◽  
Author(s):  
Takahiko Nakazi ◽  
Tomoko Osawa
Keyword(s):  
Bergman Space ◽  
Orthogonal Projection ◽  
Finite Rank ◽  
Lebesgue Space ◽  
Unit Disc ◽  
Hankel Operator ◽  
Hankel Operators ◽  
Weighted Bergman Space ◽  
Open Unit ◽  
Open Unit Disc

LetL2=L2(D,r dr dθ/π)be the Lebesgue space on the open unit disc and letLa2=L2∩ℋol(D)be the Bergman space. LetPbe the orthogonal projection ofL2ontoLa2and letQbe the orthogonal projection ontoL¯a,02={g∈L2;g¯∈La2,   g(0)=0}. ThenI−P≥Q. The big Hankel operator and the small Hankel operator onLa2are defined as: forϕinL∞,Hϕbig(f)=(I−P)(ϕf)andHϕsmall(f)=Q(ϕf)(f∈La2). In this paper, the finite-rank intermediate Hankel operators betweenHϕbigandHϕsmallare studied. We are working on the more general space, that is, the weighted Bergman space.

A proof of Hartman's theorem on compact Hankel operators

1975 ◽  
Vol 78 (3) ◽  
pp. 447-450 ◽  
Author(s):  
F. F. Bonsall ◽  
S. C. Power
Keyword(s):  
Hilbert Space ◽  
Compact Operator ◽  
Hankel Operator ◽  
Hankel Operators ◽  
Unilateral Shift ◽  
Space Model ◽  
Model Free ◽  
Multiplicity One ◽  
Image Position ◽  
Algebra Of 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.

scholarly journals Essential norm estimates for Hankel operators on convex domains in $\mathbb{C}^2$

2017 ◽  
Vol 120 (2) ◽  
pp. 305
Author(s):  
Željko Čučković ◽  
Sönmez Şahutoğlu
Keyword(s):  
Convex Domain ◽  
Hankel Operator ◽  
Smooth Boundary ◽  
Essential Norm ◽  
Hankel Operators ◽  
Convex Domains ◽  
Norm Estimates ◽  
Bounded Convex Domain ◽  
Derivatives Of ◽  
Bounded Convex

Let $\Omega \subset \mathbb{C}^2$ be a bounded convex domain with $C^1$-smooth boundary and $\varphi \in C^1(\overline{\Omega})$ such that $\varphi $ is harmonic on the non-trivial disks in the boundary. We estimate the essential norm of the Hankel operator $H_{\varphi }$ in terms of the $\overline{\partial}$ derivatives of $\varphi$ “along” the non-trivial disks in the boundary.

Hankel operators with PC symbols and the space H∞ + PC

1981 ◽  
Vol 89 (1-2) ◽  
pp. 17-24 ◽  
Author(s):  
F. F. Bonsall ◽  
T. A. Gillespie
Keyword(s):  
Unit Circle ◽  
Explicit Formula ◽  
Hardy Spaces ◽  
Hankel Operator ◽  
Essential Norm ◽  
Hankel Operators ◽  
Continuous Functions ◽  
Piecewise Continuous ◽  
The Way

SynopsisWe obtain an explicit formula for the essential norm of a Hankel operator with its symbol in the space PC, which is the closure in L∞ of the space of piecewise continuous functions on the unit circle . It follows from this formula that functions in PC can be approximated as closely by functions in C, the continuous functions on the circle, as by functions in the much larger space H∞ + C. This is an example of the way in which properties of the Hardy spaces can be derived from properties of Hankel operators.

scholarly journals The Essential Norm of the Generalized Hankel Operators on the Bergman Space of the Unit Ball inCn

2010 ◽  
Vol 2010 ◽  
pp. 1-13 ◽  
Author(s):  
Luo Luo ◽  
Yang Xuemei
Keyword(s):  
Unit Ball ◽  
Bergman Space ◽  
Hankel Operator ◽  
Equivalent Form ◽  
Essential Norm ◽  
Hankel Operators

In 1993, Peloso introduced a kind of operators on the Bergman spaceA2(B)of the unit ball that generalizes the classical Hankel operator. In this paper, we estimate the essential norm of the generalized Hankel operators on the Bergman spaceAp(B)  (p>1)of the unit ball and give an equivalent form of the essential norm.

scholarly journals Symbols for trace class Hankel operators with good estimates for norms

1986 ◽  
Vol 28 (1) ◽  
pp. 47-54 ◽  
Author(s):  
F. F. Bonsall ◽  
D. Walsh
Keyword(s):  
Hardy Space ◽  
Bergman Space ◽  
Unit Disc ◽  
Hankel Operator ◽  
Trace Class ◽  
Hankel Operators ◽  
Open Unit ◽  
Second Derivative ◽  
Open Unit Disc ◽  
Symbol Function

Peller [4, 5] has proved that a Hankel operator S on the Hardy space H2 is in the trace class if and only if with h analytic on the open unit disc Dand with its second derivative belonging to the Bergman space L1a. This theorem does not include an estimate for the trace class norm ∥S∥1, of the operator in terms of the symbol function. In fact it is clear that cannot give an estimate for since the first two terms in the coefficient sequence of the Hankel operator have been removed by differentiation.

On an Extension Problem for Contractive Block Hankel Operator Matrices

Analysis ◽  
2005 ◽  
Vol 25 (1) ◽  
Author(s):  
Bernd Fritzsche ◽  
Bernd Hirstein ◽  
Jürgen Lorenz
Keyword(s):  
Hankel Operator ◽  
Hankel Operators ◽  
Linear Operators ◽  
Extension Problem ◽  
Operator Matrices ◽  
Bounded Linear ◽  
Nondegenerate Case ◽  
Operator Version ◽  
Operator Ball ◽  
Operator Extension

AbstractThe paper deals with an operator extension problem for contractive block Hankel operators which arose in the context of the operator version of the classical Nehari interpolation problem. V.M. Adamjan, D.Z. Arov, and M.G. Krein [5] obtained that the solution set of this operator extension problem is an operator ball. Hereby, they constructed the parameters of this operator ball via a regularization procedure using the corresponding expressions of the first studied nondegenerate case. The main aim of this paper is to derive more explicit formulas for the parameters of this operator ball. Hereby we use Moore-Penrose inverses of bounded linear operators in Hilbert space.

scholarly journals Finite rank intermediate Hankel operators and the big Hankel operator

2006 ◽  
Vol 2006 ◽  
pp. 1-3 ◽  
Author(s):  
Tomoko Osawa
Keyword(s):  
Bergman Space ◽  
Finite Rank ◽  
Hankel Operator ◽  
Closed Subspace ◽  
Hankel Operators ◽  
Coordinate Function ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

LetLa2be a Bergman space. We are interested in an intermediate Hankel operatorHφMfromLa2to a closed subspaceMofL2which is invariant under the multiplication by the coordinate functionz. It is well known that there do not exist any nonzero finite rank big Hankel operators, but we are studying same types in caseHφMis close to big Hankel operator. As a result, we give a necessary and sufficient condition aboutMthat there does not exist a finite rankHφMexceptHφM=0.

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