COMPLEX SYMMETRIC TOEPLITZ OPERATORS ON THE UNIT POLYDISK AND THE UNIT BALL

In this article, we study complex symmetric Toeplitz operators on the Bergman space and the pluriharmonic Bergman space in several variables. Surprisingly, the necessary and sufficient conditions for Toeplitz operators to be complex symmetric on these two spaces with certain conjugations are just the same. Also, some interesting symmetry properties of complex symmetric Toeplitz operators are obtained.     

Interpolation on Symmetric Spaces Via the Generalized Polar Decomposition

We construct interpolation operators for functions taking values in a symmetric space—a smooth manifold with an inversion symmetry about every point. Key to our construction is the observation that every symmetric space can be realized as a homogeneous space whose cosets have canonical representatives by virtue of the generalized polar decomposition—a generalization of the well-known factorization of a real nonsingular matrix into the product of a symmetric positive-definite matrix times an orthogonal matrix. By interpolating these canonical coset representatives, we derive a family of structure-preserving interpolation operators for symmetric space-valued functions. As applications, we construct interpolation operators for the space of Lorentzian metrics, the space of symmetric positive-definite matrices, and the Grassmannian. In the case of Lorentzian metrics, our interpolation operators provide a family of finite elements for numerical relativity that are frame-invariant and have signature which is guaranteed to be Lorentzian pointwise. We illustrate their potential utility by interpolating the Schwarzschild metric numerically.     

Hardy空间上一类$m$-复对称Toeplitz算子

本文主要研究Toeplitz算子相对于一对置换的共轭算子是2-复对称的充要条件. 首先在经典的Hardy空间上介绍一类被称为一对置换的共轭算子, 其次完整地刻画了在这类共轭算子下Toeplitz算子是2-复对称的结构, 利用Toeplitz算子在Hardy空间的经典正规正交基下的矩阵表示来刻画2-复对称Toeplitz算子. 最后对于Toeplitz算子分别补充前提$f_n=-f_{-n}$和$f_n=f_{-n}$, 得到了更简化的结果. 在第二个前提下, 研究Toeplitz算子的3-复对称性, 得到$T_f$关于$C_{(i,j)}$是3-CSO的结果和是2-CSO相同.     

复对称算子的一些等价性质

证明了在复对称算子的前提下,对数-亚正规算子与正规算子是等价的,并且给出了复对称算子的一些等价性质;最后通过给出例子来说明我们的结论.     

Symmetric anti-eigenvalue and symmetric anti-eigenvector

The idea of symmetric anti-eigenvalue and symmetric anti-eigenvector of a bounded linear operator T on a Hilbert space H is introduced. The structure of symmetric anti-eigenvectors of a self-adjoint and certain classes of normal operators is found in terms of eigenvectors. The Kantorovich inequality for self-adjoint operators and bounds for symmetric anti-eigenvalues for certain classes of normal operators are also discussed.     

Unbounded commutants and intertwining spaces of unbounded symmetric operators and 1-representations

This paper investigates the strong commutant, the weak commutant and the form commutant of an unbounded symmetric (nonself-adjoint) operator and of an unbounded 1-representation on a Hilbert space. For two examples of unbounded symmetric operators these commutants are described in terms of singular integral operators and of Toeplitz operators, respectively.     

对称Fock空间上的加权复合算子

本文研究了可分Hilbert空间上的对称Fock空间上有界(线性和反线性)加权复合算子. 完全刻画了酉加权复合算子和自伴加权复合算子, 并考虑了一类正规加权复合算子.     

Generalized hermitian operators

In this paper, the concept of generalized hermitian operators defined on a complex Hilbert space is introduced. It is shown that the spectrums and the Fredholm fields of generalized hermitian operators are both symmetric with respect to the real axis. Some other results on generalized hermitian operators are obtained.     

Fock空间上的一类加权复合算子

This paper studies a class of weighted composition operators and their spectrum on the Fock space.As an application,bounded self-adjoint,a class of complex symmetric weighted composition operators on the Fock space are characterized.     

Isometric Equivalence of Certain Operators on Banach Spaces

A pair of operators on a Banach space X are isometrically equivalent if they are intertwined by a surjective isometry of X. We investigate the isometric equivalence problem for pairs of operators on specific types of Banach spaces. We study weighted shifts on symmetric sequence spaces, elementary operators acting on an ideal I of Hilbert space operators, and composition operators on the Bloch space. This last case requires an extension of known results about surjective isometries of the Bloch space.     

On Conformal Invariants Built from Spinor Fields

By means of a conformal covariant differentiation process we construct generating systems for conformally invariant symmetric (r, s)–spinors in an arbitrary curved space–time. Extending this method to conformally invariant linear differential operators acting on symmetric spinor fields some classes of such operators are derived.     

Complex symmetric operators and applications

We study a few classes of Hilbert space operators whose matrix representations are complex symmetric with respect to a preferred orthonormal basis. The existence of this additional symmetry has notable implications and, in particular, it explains from a unifying point of view some classical results. We explore applications of this symmetry to Jordan canonical models, self-adjoint extensions of symmetric operators, rank-one unitary perturbations of the compressed shift, Darlington synthesis and matrix-valued inner functions, and free bounded analytic interpolation in the disk.

     

Linear operators and positive semidefiniteness of symmetric tensor spaces

We study symmetric tensor spaces and cones arising from polynomial optimization and physical sciences.We prove a decomposition invariance theorem for linear operators over the symmetric tensor space,which leads to several other interesting properties in symmetric tensor spaces.We then consider the positive semidefiniteness of linear operators which deduces the convexity of the Frobenius norm function of a symmetric tensor.Furthermore,we characterize the symmetric positive semidefinite tensor(SDT)cone by employing the properties of linear operators,design some face structures of its dual cone,and analyze its relationship to many other tensor cones.In particular,we show that the cone is self-dual if and only if the polynomial is quadratic,give specific characterizations of tensors that are in the primal cone but not in the dual for higher order cases,and develop a complete relationship map among the tensor cones appeared in the literature.     

Bloch spaces on bounded symmetric domains in complex Banach spaces Dedicated to Professor Sheng GONG on the occasion of his 75th birthday

We give a definition of Bloch space on bounded symmetric domains in arbitrary complex Banach space and prove such function space is a Banach space. The properties such as boundedness, compactness and closed range of composition operators on such Bloch space are studied.     

SMALL HANKEL OPERATORS ON WEIGHTEDBERGMAN SPACES OF BOUNDED SYMMETRIC DOMAINS

1IntroductionLetnbeaboundedsynunetricdomaininC"withBerg1llankernelK(z,w),fldenotestheEuclideanclosureofninCnandoflistheTopologicalboundaryWeassumethatnisinitsstandardrepresentationandthevolumemeasuredVofflisuormalized.Itfollowsfrom[1],[2]thatthekernelfullctionsK(-,.)havethespecialproperties:(l):K(O,w)=K(z,o)=l,z,wEfl;(2):K(z,w)/o1zEfl,wEfl;(3):.13hK(z,z)=oo;(4):K(z,w)-'isasmoothfunctiononC',xC".ofcourse,K(z,w)=K(w,z).Thecomplexcol1jugateoffisdenotedbyf.By5.7of[3]andpolarcoordinates,th…     

Bloch spaces on bounded symmetric domains in complex Banach spaces

We give a definition of Bloch space on bounded symmetric domains in arbitrary complex Banach space and prove such function space is a Banach space. The properties such as boundedness, compactness and closed range of composition operators on such Bloch space are studied. Dedicated to Professor Sheng GONG on the occasion of his 75th birthday     

On the negative squares of indefinite Sturm-Liouville operators

The number of negative squares of all self-adjoint extensions of a simple symmetric operator of defect one with finitely many negative squares in a Krein space is characterized in terms of the behaviour of an abstract Titchmarsh-Weyl function near 0 and ∞. These results are applied to a large class of symmetric and self-adjoint indefinite Sturm-Liouville operators with indefinite weight functions.     

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.     

Symmetric relations and module spaces

We study module spaces for linear relations (multi-valued operators) in a Hilbert space. The defect spaces are not required to be finite-dimensional. In particular we pay attention to module spaces for symmetric relations.     

Self-adjoint extensions of symmetric operators

Let ? denote the Hilbert space of analytic functions on the unit disk which are square summable with respect to the usual area measure. In this paper we consider the formal differential exepressons of order two or greater having the form {fx321-1} and {fx321-2} which give rise to symmetric operators in ?. We show that these operators in ? admit self-adjoint extensions in ?.