On Ptak's Generalization of Hankel Operators

In 1997 Ptak defined generalized Hankel operators as follows: Given two contractions and , an operator is said to be a generalized Hankel operator if and X satisfies a boundedness condition that depends on the unitary parts of the minimal isometric dilations of T ₁ and T ₂. This approach, call it (P), contrasts with a previous one developed by Ptak and Vrbova in 1988, call it (PV), based on the existence of a previously defined generalized Toeplitz operator. There seemed to be a strong but somewhat hidden connection between the theories (P) and (PV) and we clarify that connection by proving that (P) is more general than (PV), even strictly more general for some T ₁ and T ₂, and by studying when they coincide. Then we characterize the existence of Hankel operators, Hankel symbols and analytic Hankel symbols, solving in this way some open problems proposed by Ptak.     

Operators of Hankel type

Hankel operators and their symbols, as generalized by V. Pták and P. Vrbová, are considered. The present note provides a parametric labeling of all the Hankel symbols of a given Hankel operator X by means of Schur class functions. The result includes uniqueness criteria and a Schur like formula. As a by-product, a new proof of the existence of Hankel symbols is obtained. The proof is established by associating to the data of the problem a suitable isometry V so that there is a bijective correspondence between the symbols of X and the minimal unitary extensions of V.     

A Note on the Operator Equation Generalizing the Notion of Slant Hankel Operators

The operator equation λM_zX = XM_(zk), for k ≥ 2, λ∈ C, is completely solved.Further, some algebraic and spectral properties of the solutions of the equation are discussed.     

一类不变的Hankel算子

对于Ｌ^a,2（Ｄ）的两类Ｍｏｅｂｉｕｓ不变子空间Ａ_l^a,2（Ｄ）与Ａ_l^-a,2（Ｄ）,定义了对解析的记号函数ｂ（ｚ）的大的和小的Ｈａｎｋｅｌ算子Ｈ_b^ll′与ｈ_b^ll′,研究了它们的有界性、紧性及其Ｓｃｈａｔｔｅｎ－ｖｏｎ Ｎｅｕｍａｎｎ类的Ｓ_P估计．性质。     

Mixed Product of Hankel and Toeplitz Operators on Fock-Sobolev Spaces

Let f and g be functions in Fock-Sobolev space F^2,m. In this paper, we completely characterize the boundedness and compactness of H_f T_g.     

Finite Rank Hankel Operators over the Complex Wiener Space

This work studies finite rank Hankel operators H _b on a Hilbert space of holomorphic, square integrable Wiener functionals. The main tool to investigate these operators is their unitary equivalent representation on the Hilbert space of skeletons. The finite rank property is characterized in terms of a functional equation for the symbol b, which generalizes the well known equation b(z+w)=b(z)b(w). Also finite rank symbols of polynomial type are characterized in terms of their chaos expansions.     

Hankel Operators over the Complex Wiener Space

This work introduces and investigates (small) Hankel operators H _b on the Hilbert space of holomorphic, square integrable Wiener functionals. A regularity condition on the symbol b, which guarantees the boundedness of H _b , is provided. The symbols b for which H _b is of Hilbert–Schmidt type are characterized, and a representation of H _b by an integral operator is given. The proofs employ the hypercontractivity of the Ornstein–Uhlenbeck semigroup, together with approximations by finitely many variables. These results extend known results from a finite-dimensional context.     

广义Fock空间上的Hankel算子

利用有界(消失)平均振荡函数的性质,本文刻画了一类广义Fock空间上的Hankel算子的有界性(紧性),同时,还刻画了换位子[M_f,P]的有界性和紧性,其中P是一个Toeplitz投影算子,而M_f表示符号为f的乘子.最后,应用Berezin变换来研究了BMO空间和VMO空间的几何性质.     

Asymptotic and Partial Asymptotic Hankel Operators on $$H^2(\mathbb{D}^n)$$ H 2 ( D n ) )

In this paper, we generalize the concept of asymptotic Hankel operators on H²(D) to the Hardy space H²(Dⁿ) (over polydisk) in terms of asymptotic Hankel and partial asymptotic Hankel operators and investigate some properties in case of its weak and strong convergence. Meanwhile, we introduce i^th-partial Hankel operators on H²(Dⁿ) and obtain a characterization of its compactness for n > 1. Our main results include the containment of Toeplitz algebra in the collection of all strong partial asymptotic Hankel operators on H²(Dⁿ). It is also shown that a Toeplitz operator with symbol φ is asymptotic Hankel if and only if φ is holomorphic function in L^∞(Tⁿ).     

微分方程递归算子的一种推广

微分方程的多数重要的递归算子都是积微分算子。在试图获得整个对称族时人们常常遇到困难。有时甚至由于缺乏精确性而导致伪对称。本文给出递归算子一种推广,其在某种程度上消除了这些问题。文中还给出了若干重要例子说明这种推广及其重要性。     

加权Bergman空间上Toeplitz与Hankel算子紧性的一个新特征

本文研究加权Ｂｅｒｇｍａｎ空间Ａ^（ｐ,α）（Ｂ）（ｐ＞１）上Ｔｏｅｐｌｉｔｚ与Ｈａｎｋｅｌ算子的紧性．证明了符号属于Ｌ^∞（Ｂ）的Ｔｏｅｐｌｉｔｚ与Ｈａｎｋｅｌ算子的紧性与作用空间Ａ^ｐ,α（Ｂ）无关     

Bergman空间上可交换的Hankel算子和Toeplitz算子

证明了Bergman空间上的两个小Hankel算子如果是可交换的且其中一个是拟齐次的小Hankel算子,则另一个也是拟齐次的.还研究了Toeplitz算子和小Hankel算子的交换性.     

一类Hankel算子与Toeplitz算子的Sp性质

对于L~(α,2)(D))的两类Moebius不变子空间A~(α,2)(D)和A~(β,2)(D),我们定义了它们之间的Toeplitz算子T_f~s与其乘积空间上的Hankel算子H_f~r,并且研究了它们的有界性、紧性及Schatten-von Neumann性质。     

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.

     

双圆盘上Hardy空间上Hankel 算子和Toeplitz 算子的交换性

In this paper, we characterize when the Toeplitz operator Tf and the Hankel operator Hg commute on the Hardy space of the bidisk. For certain types of bounded symbols f and g, we give a necessary and sufficient condition on the symbols to guarantee TfHg = HgTf.     

调和Dirichlet空间上小Hankel算子的乘积(英文)

研究了调和Dirichlet空间上调和符号的小Hankel算子的乘积,给出了此类小Hankel算子交换性和乘积为零的完全刻画.     

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.     

球上加权Bergman空间上的Schatten类Hankel算子

本文给出了超球上加权Ｂｅｒｇｍａｎ空间上Ｈａｎｋｅｌ算子属于Ｓｃｈａｔｔｅｎｐ－类（２≤ｐ＜＋∞）的一个充要条件，推广了文［１０］中的主要结果。     

Little Hankel Operators on the Weighted Bergman Space of the Unit Ball

In this paper we mainly consider little Hankel operators with squareintegrable symbols on the weighted Bergman spaces of the unit ball.We obtain that Schatten class of little Hankel operators is equivalent to Schatten class of positive Toeplitz operators under the conditions that SMO（f） ∈ L p/2 （B n,dλ） and 2 ≤ p ∞,which is very important to research the relation between Toeplitz operators and little Hankel operators.     

Toeplitz plus Hankel Operators with Infinite Index

We study Toeplitz plus Hankel operators acting between Lebesgue spaces on the unit circle, and having symbols which contain standard almost periodic discontinuities. Conditions are obtained under which these operators are right-invertible and with infinite kernel dimension, left-invertible and with infinite cokernel dimension or simply not normally solvable.