On Strongly Stable Normal Operators in Krein Spaces

We give a necessary and sufficient condition for strong stability of bounded normal operators in Krein spaces. A similar result was obtained for unitary operators by M. G. Krein [1], and for selfadjoint operators by H. Langer [2], [3].     

Analysis of Spectral Points of the Operators T [*] T and TT [*] in a Krein Space

Spectra and sets of regular and singular critical points of definitizable operators of the form T ^[*] T and TT ^[*] in a Krein space are compared. The relation between the Jordan chains of the above operators (corresponding to the same eigenvalue) is shown.      

A Variational Principle in Krein Space II

Let A be a self-adjoint operator in a Krein space Under certain natural assumptions, it is shown precisely which real eigenvalues of A can be given a max-inf characterization generalizing the usual one in Hilbert space. This result unifies several approaches in the recent literature.     

Boundary Relations and Generalized Resolvents of Symmetric Operators in Krein Spaces

The classical Krein-Naimark formula establishes a one-to-one correspondence between the generalized resolvents of a closed symmetric operator in a Hilbert space and the class of Nevanlinna families in a parameter space. Recently it was shown by V.A. Derkach, S. Hassi, M.M. Malamud and H.S.V. de Snoo that these parameter families can be interpreted as so-called Weyl families of boundary relations, and a new proof of the Krein-Naimark formula in the Hilbert space setting was given with the help of a coupling method. The main objective of this paper is to adapt the notion of boundary relations and their Weyl families to the Krein space case and to prove some variants of the Krein-Naimark formula in an indefinite setting.      

The Krein String and Characteristic Functions of Maximal Dissipative Operators

Theory of the Krein string is considered in the framework of a continual analog of the theory of unitary operator nodes. On this basis, a generalization of the M. G. Krein-A. A. Nudelman result on the completeness of the string operator is obtained. Bibliography: 28 titles.     

Spectral Theorem for Definitizable Normal Linear Operators on Krein Spaces

In the present note a spectral theorem for normal definitizable linear operators on Krein spaces is derived by developing a functional calculus \({\phi \mapsto \phi(N)}\) which is the proper analogue of \({\phi \mapsto \int \phi \, dE}\) in the Hilbert space situation. This paper is the first systematical study of definitizable normal operators on Krein spaces.     

On Finite Rank Perturbations of Selfadjoint Operators in Krein Spaces and Eigenvalues in Spectral Gaps

It is shown that the finiteness of eigenvalues in a spectral gap of a definitizable or locally definitizable selfadjoint operator in a Krein space is preserved under finite rank perturbations. This results is applied to a class of singular Sturm–Liouville operators with an indefinite weight function.     

Singular Rank One Perturbations of Self-Adjoint Operators and Krein Theory of Self-Adjoint Extensions

Gesztesy and Simon recently have proven the existence of the strong resolvent limit A_, for A_, = A + (·), where A is a self-adjoint positive operator, being the A-scale). In the present note it is remarked that the operator A_, also appears directly as the Friedrichs extension of the symmetric operator :=A \{f (A)| f,=0\}. It is also shown that Krein's resolvents formula: (A_{_b,}-z)^-1 =(A-z)^-1+ (·, ) _z, with b=b-(1+z) (_z,_-1),_z= (A-z)^-1 defines a self-adjoint operator A_b, for each and b R¹. Moreover it is proven that for any sequence _n which goes to in there exists a sequence _n0 such that A_b, in the strong resolvent sense.     

Increasing Monotone Operators in Banach Space

An operator mapping a separable reflexive Banach space X into the dual space X is called increasing if as . Necessary and sufficient conditions for the superposition operators to be increasing are obtained. The relationship between the increasing and coercive properties of monotone partial differential operators is studied. Additional conditions are imposed that imply the existence of a solution for the equation with an increasing operator A.     

从Bergman空间到Bloch空间的叠加算子

研究从Bergman空间到Bloch空间的叠加算子.从函数空间的性质出发,利用不同维数的函数空间之间的关系,讨论叠加算子,把单变量有关叠加算子的结果推广到多变量,得到了与单变量一致的结果,从而刻画了从Bergman空间到Bloch空间的叠加算子的特征,得到了从Bergman空间到Bloch空间的叠加算子的充要条件.     

调和Dirichlet空间上的Toeplitz算子

刻画了调和Dirichlet空间上Toeplitz算子的有界性、紧性,讨论了Toeplitz算子的代数性质,得到了Toeplitz代数与Hankel代数的短正合列.还计算了Toeplitz代数与Hankel代数中Fredholm算子的Fredholm指标,得到了Toeplitz代数与Hankel代数的K_0与K_1群.     

Fock空间上的乘法算子

Fock空间是由整函数组成的具有再生核的Hilbert空间.Fock空间上的乘法算子的定义域不是整个Fock空间,它在Fock空间上是稠定的.研究了Fock空间上的乘法算子的性质,对其值域进行了刻画,并得出了乘法算子作用在Fock空间的拟不变子空间上值域余一维的充要条件.     

Quadratic Inequalities for Hilbert Space Operators

Properties of sets of solutions to inequalities of the form

Toeplitz Operators on the Fock Space

We study Toeplitz operators on the Fock space with positive measures as symbols. Main results include characterizations of Fock–Carleson measures, bounded Toeplitz operators, compact Toeplitz operators, and Toeplitz operators in the Schatten p-classes.     

Toeplitz Operators on the Dirichlet Space

In this paper a decomposition of Sobolev space is obtained. Then we prove that a Toeplitz operator on the Dirichlet space is compact only when it is the zero operator. For two Toeplitz operators on the Dirichlet space, we obtain the conditions for that they commute, their product is a Toeplitz operator, and their commutator or semi-commutator has finite rank, respectively.     

Banach空间上的强混合算子

1999年,L.B.Gonzalez证明了任意无限维可分Banach空间上存在拓扑传递的有界线性算子.这个结果肯定地回答了S.Rolewicz提出的问题.本文证明了由L.B.Gonzalez所给出的算子实际上是强混合的,同时,对加权移位算子的混合性利用权序列进行了刻划并指出任意无限维可分Hilbert空间上存在弱混合而非强混合的有界线性算子.     

Banach空间上的强混合算子

1999年,L．B．Gonzalez 证明了任意无限维可分 Banach 空间上存在拓扑传递的有界线性算子．这个结果肯定地回答了 S．Rolewicz 提出的问题．本文证明了由 L．B．Gonzalez 所给出的算子实际上是强混合的,同时,对加权移位算子的混合性利用权序列进行了刻划并指出任意无限维可分 Hilbert 空间上存在弱混合而非强混合的有界线性算子．     

局部凸空间中的Hankel算子

将 Hardy空间中的一些概念推广到了局部凸空间 ,尤其是将 Hankel算子推广到了局部凸空间 ,得到了局部凸空间 Hankel算子的计算方法 .     

Boundedness of Multilinear Operators in Herz-type Hardy Space

Let κ∈ℕ. We prove that the multilinear operators of finite sums of products of singular integrals on ℝⁿ are bounded from HK ^α1,p1 _q1 (ℝⁿ) ×···×HK ^αk,pk _qk (ℝⁿ) into HK ^α,p _q (ℝⁿ) if they have vanishing moments up to a certain order dictated by the target spaces. These conditions on vanishing moments satisfied by the multilinear operators are also necessary when α_j≥ 0 and the singular integrals considered here include the Calderón-Zygmund singular integrals and the fractional integrals of any orders. Received September 6, 1999, Revised November 17, 1999, Accepted December 9, 1999