Schatten class membership of Hankel operators on the unit sphere

Let H²(S) be the Hardy space on the unit sphere S in Cn, n?2. Consider the Hankel operator Hf=(1−P)Mf|H²(S), where the symbol function f is allowed to be arbitrary in L²(S,dσ). We show that for p>2n, Hf is in the Schatten class Cp if and only if f−Pf belongs to the Besov space Bp. To be more precise, the “if” part of this statement is easy. The main result of the paper is the “only if” part. We also show that the membership Hf∈C2n implies f−Pf=0, i.e., Hf=0.     

Separation for kernels of Hankel operators

We prove that for two Hankel operators and on the Hardy space of the unit disk either the kernel of equals the kernel of or the kernel of equals the kernel of . In fact we prove a version of the above result for products of an arbitrary finite number of Hankel operators. Some immediate corollaries are generalizations of the result of Brown and Halmos on zero products of two Hankel operators and the result of Axler, Chang and Sarason on finite rank products of two Hankel operators. Simple examples show our results are sharp.

     

Generalized Hankel operators on the Fock space

In this paper we study generalized Hankel operators ofthe form : ?²(|z |²) → L²(|z |²). Here, (f):= (Id–P_l )( ^kf) and Pl is the projection onto A_l ²(?, |z |²):= cl(span{ ^m zⁿ | m, n ∈ N, m ≤ l }). The investigations in this article extend the ones in [11] and [6], where the special cases l = 0 and l = 1 are considered, respectively. The main result is that the operators are not bounded for l < k – 1. The proof relies on a combinatoric argument and a generalization to general conjugate holomorphic L² symbols, generalizing arguments from [6], seems possible and is planned for future work (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Hankel operators on the Dirichlet space

We study (small) Hankel operators on the Dirichlet space D with symbols in a class of function space, and show that such (small) Hankel operators are closely related to the corresponding Hankel operators on the Bergman space and the Hardy space H².     

Hankel operators and the Stieltjes moment problem

Let s be a non-vanishing Stieltjes moment sequence and let μ be a representing measure of it. We denote by μn the image measure in Cn of μ⊗σn under the map , where σn is the rotation invariant probability measure on the unit sphere. We show that the closure of holomorphic polynomials in L²(μn) is a reproducing kernel Hilbert space of analytic functions and describe various spectral properties of the corresponding Hankel operators with anti-holomorphic symbols. In particular, if n=1, we prove that there are nontrivial Hilbert-Schmidt Hankel operators with anti-holomorphic symbols if and only if s is exponentially bounded. In this case, the space of symbols of such operators is shown to be the classical Dirichlet space. We mention that the classical weighted Bergman spaces, the Hardy space and Fock type spaces fall in this setting.     

Boundedness and compactness of Volterra type integral operators

We introduce some nested classes of Volterra type integral operators. For the operators of these classes we establish criteria for boundedness and compactness in Lebesgue spaces.     

Strict convexity of some subsets of Hankel operators

We find some extreme points in the unit ball of the set of Hankel operators and show that the unit ball of the set of compact Hankel operators is strictly convex. We use this result to show that the collection of lower triangular Toeplitz contractions is strictly convex. We also find some extreme points in certain reduced Cowen sets and discuss cases in which they are or are not strictly convex.

     

Hankel operators with antiholomorphic symbols on the Fock space

We consider Hankel operators of the form . Here . We show that in the case of one complex dimension the Hankel operators are compact but not Hilbert-Schmidt if 2k$">.

     

The Dixmier trace of Hankel operators on the Bergman space

We give a formula for the Dixmier trace of (big) Hankel operators on the Bergman space of the disk or of finitely connected domains. For harmonic symbols we find the regularity required of the symbol for the formula to hold.     

Hankel operators and best Hankel approximation on the half-plane

The objective of this paper is to give an interrelation between Hankel oeprators on the unit disc and Hankel operators on the half-plane. As an application, the AAK result on the half-plane is established and the rate of best Hankel approximation on the halfplane is derived. Research partially supported by the University Research Grants and Fellowship Committee at UNLV.     

On the compactness of the product of Hankel operators on the sphere

Consider Hankel operators and on the unit sphere in . If , then a necessary condition for to be compact is . We show that when , this condition is no longer necessary for to be compact.

     

多重调和Bergman空间上的Toeplitz算子和Hankel算子

完全刻画多重调和Bergman空间上Toeplitz算子和Hankel算子的紧性.运用紧Toeplitz算子这个结果,建立了Toeplitz代数和小Hankel代数的短正合列,推广了单位圆盘上相应的结果.     

Truncations of multilinear Hankel operators

We extend to multilinear Hankel operators the fact that some truncations of bounded Hankel operators are still bounded. We prove and use a continuity property of bilinear Hilbert transforms on products of Lipschitz spaces and Hardy spaces.

     

Hankel and Toeplitz type operators on the unit disk

Let D be the unit disk in the complex plane with the weighted measure dμβ(z) = (β+1)--π(1-|z|2)βdm(z)(β＞-1). Then L2(D,dμβ(z)) = ∞k=0(Aβk Aβk) is the orthogonal direct sum decomposition.In this paper, we define the Hankel and Toeplitz type operators, and study the boundedness, compactness and Sp-criteria for them.     

Hankel operators, the Segal-Bargmann space, and symmetrically-normed ideals

Starting with the Segal-Bargmann space, we investigate the Hankel operators with symbol functions in a certain linear space. Given an appropriate symbol function, we consider the associated Hankel operator together with the Hankel operator associated with that symbol function's complex conjugate. We give a necessary and sufficient condition for the simultaneous membership of these two operators in the symmetrically-normed ideal associated with any given symmetric norming function.     

Algebraic properties of Toeplitz and small Hankel operators on the harmonic Bergman space

In this paper,we discuss some algebraic properties of Toeplitz operators and small Hankel operators with radial and quasihomogeneous symbols on the harmonic Bergman space of the unit disk in the complex plane C.We solve the product problem of quasihomogeneous Toeplitz operator and quasihomogeneous small Hankel operator.Meanwhile,we characterize the commutativity of quasihomogeneous Toeplitz operator and quasihomogeneous small Hankel operator.     

Boundedness,compactness, and Schattenp-classes of Hankel operators between weighted Dirichlet spaces

Part of the research for this paper was done at Institute Mittag-Leffler. The author is very grateful to the institute for providing excellent working environment and financial support.     

On the topology of Hankel multipliers and of Hankel convolution operators

The topologies of simple convergence and of bounded convergence are shown to coincide on the spaces of Hankel multipliers and of Hankel convolution operators. The properties of these spaces being bornological, nuclear, Montel, and reflexive are established.     

Lp-Norm inequality of the product of pseudo-differential operators involving Hankel transformation

Lp-norm type inequalities for pseudo-differential operators L(x,D) and H(x,D) are derived for 1 ≤ p < ∞. Product of pseudo-differential operators associated with Bessel operator Sμ are defined and its Lp-norm type inequalities are obtained.