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.

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.

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.

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）.

POWERS OF AN INVERTIBLE(s,p)-w-HYPONORMAL OPERATOR

It is known that the square of a ω-hyponormal operator is also ω-hyponormal. For any 0〈 p 〈 1, there exists a special invertible operator such that all of its integer powers are all p - ω-hyponormal. In this article, the author introduces the class of （s, p） -ω-hyponormal operators on the basis of the class of p- ω-hyponormal operators. For s 〉0, 0 〈 p 〈 1, the author gives a characterization of （s,p） -ω-hyponormal operatots; the author shows that all integer powers of special （s, p） -ω-hyponormal operators are （s,p） -ω-hyzponormal.

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 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.

Log-ω-hyponormal Operators

Let T be an operator on a separable Hilbert space H and T = U｜T｜ be the polar decomposition. T is said to be log-ω-hyponormal if log ｜~T｜ ≥ log｜T｜≥ log ｜~T^＊｜. In this paper we prove that the point spectrum of T is equal to its joint point spectrum if T is log-ω-hyponormal. We also prove that a log-ω-hyponormal operator is normaloid, i.e., r（T） =｜｜T｜｜. Finally, we obtain Putnam＇s theorem for log-ω-hyponormal operators.

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].

全Fock空间中生成子的谱研究

借助hyponormal算子的谱的性质,对全Fock空间中生成子的谱的相关性质进行研究,得出生成子的近似点谱包含在复平面中以原点为心半径1的方体内.

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.

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.

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。

广义弱亚正规算子

对0<t<1,引进了广义弱亚正规算子的概念,这类算子包含所有的弱亚正规算子。证出若T是广义弱亚正规算子,则T的谱半径与它的范数相等,并证出广义弱亚正规算子的点谱与联合点谱,近似点谱与联合近似点谱的非零点分别相同。

部分Hardy空间上的Toeplitz算子

探讨了部分Hardy空间上的Toeplitz算子的性质.特别地,刻画了部分Hardy空间上的Toeplitz算子及其共轭的核,得到了一类Toeplitz算子为部分Hardy空间上的正规算子的充要条件.

关于k—可换性与k—正规性

本文证明了对实自共轭迹类算子有Ｔｒ（（ＡＢ）２ｎ）≤Ｔｒ（Ａ２ｎＢ２ｎ），即ｋ＝２ｎ情形下的Ｈｉｌｂｅｒｔ空间中Ｂｅｌｌｍａｎ不等式；定义了ｋ－换位子，讨论其若干性质；并给出Ｔｒ（ＡＡ）２＝Ｔｒ（Ａ２Ａ）的充要条件，等价定义了正常这类算子。

亚正规对偶截断Toeplitz算子

记H^(2)是开单位圆盘D={z∈C:|z|<1}上的经典Hardy空间,L^(2)=L^(2)(T)是单位圆周T={z∈C:|z|=1}上的平方可积函数构成的空间.令u是非常值内函数,研究了定义在模型空间的正交补空间上的对偶截断Toeplitz算子,这是一类新的Toeplitz算子.证明了不存在解析的次正规对偶截断Toeplitz算子,并运用经典的Toeplitz算子理论完全刻画了亚正规对偶截断Toeplitz算子的符号.

log-弱亚正规算子的Riesz幂等元和Weyl定理

研究了log-弱亚正规算子T的Riesz幂等元Eλ和T的Aluthge变换T的Riesz幂等元Eλ的性质,其中λ∈isoσ(T)。证明了EλH=EλH,Eλ是自伴算子,Eλ=Eλ和EλH=ker(T-λ)=ker(T-λ),而且证出了Weyl定理对T及f(T),f∈H(σ(T))都适合。