HYPONORMALITY OF BLOCK TOEPLITZ OPERATORS ON THE WEIGHTED BERGMAN SPACES

In this paper we consider the block Toeplitz operators TФ on the weighted Bergman space A2α（D, Cn） and we give a necessary and sufficient condition for the hyponor-mality of block Toeplitz operators with symbol in the class of functions Ф=F ＋ G＊ withmatrix-valued polynomial functions F and G with degree 2.

ON REFLEXIVITY OF HYPONORMAL AND WEIGHTED SHIFT OPERATORS

By an elementary proof, we use a result of Conway and Dudziak to show that if A is a hyponormal operator with spectral radius r（A） such that its spectrum is the closed disc {z：｜z｜ ≤ r（A）} then A is reflexive. Using this result, we give a simple proof of a result of Bercovici, Foias, and Pearcy on reflexivity of shift operators. Also, it is shown that every power of an invertible bilateral weighted shift is reflexive.

APPLICATION OF TWO OPERATORS TRANSFORM FROM CLASS A OPERATOR TO THE CLASS OF HYPONORMAL OPERATOR

In this article, we give an operator transform T （＊） from class A operator to the class of hyponormal operators. It is different from the operator transform T defined by M. Ch and T. Yamazaki. Then, we show that σ（T ） = σ（ T （＊）） and σa（T ）／{0} = σa（ T （＊））／{0}, in case T belongs to class A. Next, we obtain some relations between T and T （9）.

PROPERTIES OF p-ω-HYPONORMAL OPERATORS

The approximate point spectrum properties of p-ω-hyponormal operators are given and proved. In faet, it is a generalization of approximate point speetrum properties of ω- hyponormal operators. The relation of spectra and numerical range of p-ω-hyponormal operators is obtained, On the other hand, for p-ω-hyponormal operators T,it is showed that if Y is normal,then T is also normal.

Some Classes of Operators Related to <i>p</i>-Hyponormal Operator

We introduce a new family of classes of operators termed as *p-paranormal operator, classes *A(p,p);p > 0 and *A(p,q);p, q > 0, parallel to p-paranormal operator and classes A(p,p);p> 0 and A(p,q);p, q > 0 introduced by M. Fujii, D. Jung, S. H. Lee, M. Y. Lee and R. Nakamoto [1]. We present a necessary and sufficient condition for p-hyponormal operator T∈B(H)to be *p-paranormal and the monotonicity of *A(p,q). We also present an alternative proof of a result of M. Fujii, et al. [1, Theorem 3.4].

The Interpolation of Operators between Weighted Hardy Spaces and Weighted L^p Spaces

The paper is given the interpolation of operators between weighted Hardy spaces and weighted L p spaces when w∈A 1 by Calderon Zygmund decomposition.

Spectral Characterization of Hyponormal Weighted Shifts

The complete characterizations of the spectra and their various parts of hyponormal unilateral and bilateral weighted shifts are presented respectively in this paper. The results obtained here generalize the corresponding work of the references.

TOEPLITZ OPERATORS ASSOCIATED WITH SEMIFINITE VON NEUMANN ALGEBRA

Let H^2（M） be a noncommutative Hardy space associated with semifinite von Neumann algebra M, we get the connection between numerical spectrum and the spectrum of Toeplitz operator Tt acting on H^2（M）, and the norm of Toeplitz operator Tt is equivalent to ｜｜t｜｜ when t is hyponormal operator in M.

Some Notes on M-hyponormal Weighted Shifts and Hyponormalizable Weighted Shifts

Let {an}∞n=0be a weight sequence and let W denote the associated unilateral weighted shift on H. In this paper, we consider the connection between the M-hyponormal and hyponormalizable weighted shift operator. Main results are Theorems 4.1 and Theorems4.2. Theorem 4.1 gives the sufficient condition that a weighted shifts M-hyponormal operator is hyponormalizable. Theorem 4.2 gives the sufficient condition that a hyponormalizable weighted shift operator is M-hyponormal. Finally, invariant subspaces of such operators are discussed.

关于(s,p)-w-亚正常算子类

本文在w-亚正常算子类的基础上,引入(s,p)-w-亚正常算子类,进而讨论了该类算子的特征,包含关系,对一类特殊的该类算子还考虑了其平方性质.

A GUESS ON HYPONORMAL OPERATOR TUPLES

For a hyponormal operator T if there is a polynomial p(·) such that σ(p(T))= {0}, then p(T) =0. In general, it is proved in this way: from σ(p(T)) =0, we know that σ(T) consists of finite points, and therefore T must be normal and so is p(T), hence p(T) =0.

Hyponormal Toeplitz Operators on the Polydisk

By bounded vector-valued functions and block matrix representations of Hankel operators, we completely characterize the hyponormality of Toeplitz operators on the Hardy space of the polydisk.

Hyponormality of Toeplitz Operators on the Dirichlet Space

In this paper,we prove that the necessary and sufficient condition for a Toeplitz operator Tu on the Dirichlet space to be hyponormal is that the symbol u is constant for the case that the projection of u in the Dirichlet space is a polynomial and for the case that u is a class of special symbols,respectively.We also prove that a Toeplitz operator with harmonic polynomial symbol on the harmonic Dirichlet space is hyponormal if and only if its symbol is constant.

Weakly k-hyponormal and polynomially hyponormal commuting operator pairs

We introduce the notion of weak k-hyponormality and polynomial hyponormality for commuting operator pairs on a Hilbert space and investigate their relationship with k-hyponormality and subnormality.We provide examples of 2-variable weighted shifts which are weakly 1-hyponormal but not hyponormal.By relating the weak k-hyponormality and k-hyponormality of a commuting operator pair to positivity of restriction of some linear functionals to corresponding cones of functions,we prove that there is an operator pair that is polynomially hyponormal but not 2-hyponormal,generalizing Curto and Putinar’s result(1991,1993)to the two-variable case.

Hyponormal Dual Toeplitz Operators on the Orthogonal Complement of the Harmonic Bergman Space

In this paper,we mainly study the hyponormality of dual Toeplitz operators on the orthogonal complement of the harmonic Bergman space.First we show that the dual Toeplitz operator with the bounded symbol is hyponormal if and only if it is normal.Then we obtain a necessary and sufficient condition for the dual Toeplitz operator S_(φ) with the symbol φ(z)=az^(n1zm1)+bz^(n2zm2)(n1,n2,m1,m2∈N and a,b∈C)to be hyponormal.Finally,we show that the rank of the commutator of two dual Toeplitz operators must be an even number if the commutator has a finite rank.

Fock正交补空间上亚正规对偶Toeplitz算子

Fock空间是由整函数构成的一类重要解析函数空间.由于与量子力学中的海森堡群表示理论相关,所以不仅在函数空间算子理论有着重要研究意义,还与量子力学的研究紧密相关.对偶Toeplitz算子也是算子理论研究内容之一,且与Toeplitz算子相互影响.算子的亚正规性,起源于Halmos的Hilbert空间问题集中的第5个问题,一直都是算子理论的重点研究内容之一.主要利用函数空间的结构和对偶Toeplitz算子的性质,刻画Fock空间的正交补空间上以(z-a)(z-b)为符号的对偶Toeplitz算子亚正规性,并得出其亚正规性等价于正规性,同时也探究了以调和多项式为符号的对偶Toeplitz算子是亚正规的一些必要条件.

亚正规算子的性质

研究了亚正规算子A∈B(H)的性质,给出了A∈B(H)的约化子空间的充分条件,并且讨论了其与正规算子的关系.得到了定理1设A∈B(H)是一个亚正规算子,则以下结论成立:(1)若0<p≤1,则A为p-亚正规算子.(2)对于每一个正整数n,An∈B(H)为1n-亚正规算子.(3)若0<p≤1,则|A|2p≥|A*|2p.(4)对每一个正整数n,有‖An‖=‖A‖n.定理2设A∈B(H)是一个亚正规算子,W是A的不变子空间,且A在W上的限制A|W为正规算子,则W是A的约化子空间.定理3设A∈B(H)是一个亚正规算子,且A*下有界,则A是可逆的,且A-1也为亚正规算子.

ωA(s,t)类算子的推广及其性质

作为ωA(s,t)类算子的推广形式,给出了ωA(s,t,p)类算子的定义及其相关性质,例如ωA(s,t,p)类算子有关参数s,t,p的单调性,ωA(s,t,p)类算子的正规性.

算子的拟正规性及亚正规性

设H是复可分H ilbert空间。L(H)是H上有界线性算子全体构成的C*代数。讨论算子的拟正规性与亚正规性的关系,并以单侧加权移位算子为例证明了并非所有的亚正规算子是拟正规的。证明了紧的亚正规算子是拟正规的。