Unbounded Toeplitz Operators

This partly expository article develops the basic theory of unbounded Toeplitz operators on the Hardy space H ², with emphasis on operators whose symbols are not square integrable. Unbounded truncated Toeplitz operators on coinvariant subspaces of H ² are also studied. In memory of Paul R. Halmos     

Unbounded Quasinormal Operators Revisited

Various characterizations of unbounded closed densely defined operators commuting with the spectral measures of their moduli are established. In particular, Kaufman’s definition of an unbounded quasinormal operator is shown to coincide with that given by the third-named author and Szafraniec. Examples demonstrating the sharpness of results are constructed.     

Unbounded Hankel Operators and Moment Problems

This note presents a matricial generalization of the concept of rigid functions in the Hardy space H¹, and a spectral characterization of complete non-determinacy for multivariate stationary processes.     

Automorphic Invariant Isometric Operators

We consider isometric operators with finite defect numbers that are unitarily equivalent to their M?bius transformations. A functional characterization of such isometries is given in terms of their characteristic operator-functions. We show that any such isometry admits a contractive extension that is also unitarily equivalent to its M?bius transformation and a unitary extension in a larger space that has the same property. Examples of automorphic invariant operators are considered.     

Quasinilpotent Operators and the Invariant Subspace Problem

In this paper we construct quasinilpotent operators withoutnontrivial invariant subspaces. As well as being interestingin its own right, we believe that this is a sensible (perhapsnecessary) first step for addressing the difficult problem ofconstructing topologically simple Banach algebras.     

Ideals in Algebras of Unbounded Operators

A general procedure is given to get ideals in algebras of unbounded operators starting with ideals in ??(??). Algebraical and topological properties of ideals obtained in this manner from the well-known symmetrically-normed ideals S_?(??) are described.     

基于二次算子族的无界Hamilton算子根向量组的基性质

利用无界Hamilton算子导出的二次算子族,本文研究了一类无界Hamilton算子根向量组的Schauder基性质.首先,建立了无界Hamilton算子的根向量与相应的二次算子族的根向量之间的关系.其次,借助二次算子族谱的相关性质,刻画了无界Hamilton算子的本征值分布以及本征值的代数指标,并得到了无界Hamilton算子的根向量组是某个Hilbert空间的一个块状Schauder基的充要条件.最后,将所得结果应用于矩形薄板弯曲问题.     

Elliptic Operators in Subspaces and the Eta Invariant

In this paper we obtain a formula for the fractional part of the -invariant for elliptic self-adjoint operators in topological terms. The computation of the -invariant is based on the index theorem for elliptic operators in subspaces obtained by Savin and Sternin. We also apply the K-theory with coefficients _n . In particular, it is shown that the group K(T ^* M, _n ) is realized by elliptic operators (symbols) acting in appropriate subspaces.     

Some Algebraic Operators and the Invariant Subspace Problem

We consider the resolvent algebra ${R_A=\{T\in\mathcal{L} (X):\sup_{m \geq 0}\|(1+\,mA)T(1+\,mA)^{-1}\| < \infty \}}$ , and Deddens?? algebra ${B_A= \{T\in \mathcal{B}(H) : \sup_{n\geq 0}\|A^nTA^{-n}\|<\infty\}}$ . It is shown that both R _A and B _A?CI possess non-trivial invariant subspaces when A is an algebraic operator of degree 2. This assertion becomes stronger than the existence of a hyper-invariant subspace for R _A whenever R _A ?? {A}??. Investigation of the relationship between these two algebras is addressed for different classes of operators A. Also, a complete characterization of the algebra R _A when A is an algebraic operator is given. For the finite dimensional case, we present an elementary example showing that R _A contains properly {A}?? whenever A has an eigenvalue other than zero.     

The Density Problem for Unbounded Bergman Operators

The unbounded Bergman operator, the operator of multiplication by on an unbounded open subset of the plane, is considered. We give a complete answer regarding the density problem of unbounded Bergman operators in terms of its equivalence to the problem of bounded point evaluations for the Bergman spaces. Using this equivalence and the notion of Wiener capacity, we obtain simple geometric conditions that classify almost those open subsets of the plane for which the corresponding Bergman operators are densely defined. With the aid of an analytic approach, we are also able to give condition for a large collection of open subsets of the plane for which all the positive integer powers of the corresponding Bergman operators are densely defined. Submitted: December 14, 2001? Revised: January 14, 2001.     

Unbounded Weighted Composition Operators on Fock space

Potential Analysis - The pluripolar hull of a pluripolar set E in ?n is the intersection of all complete pluripolar sets in ?n that contain E. We prove that the pluripolar hull of each...     

Spectral Inclusion Properties of Unbounded Hamiltonian Operators

In this paper, the authors investigate the spectral inclusion properties of the quadratic numerical range for unbounded Hamiltonian operators. Moreover, some examples are presented to illustrate the main results.     

广义Segal-Bargmann空间上无界Toeplitz算子的交换

本文给出了复平面C上广义Fock空间中两个Toeplitz算子T_u和T_v的性质.假设u是一个径向函数,两算子是可交换的.在一定的增长条件之下,我们证明出u也是一个径向函数.最后还构造了一个具有本性无界符号的S_p紧,Toeplitz算子.     

Identities and Invariant Operators on Homogeneous Spaces

We study identities (functional relations between the generators of the transformation group) and also algebras of invariant operators on homogeneous spaces using the method of orbits of the coadjoint representation (coadjoint orbits). This method permits establishing the relation between these two objects and elaborating an algorithm for their construction. A classification of homogeneous spaces is introduced based on the coadjoint orbit method.     

一类无界算子的Weyl定理

研究了一类无界算子的Weyl定理,刻画了具有扰动的无界自伴算子的Weyl定理.