Operators Which Do Not Have the Single Valued Extension Property

In this paper we shall consider the relationships between a local version of the single valued extension property of a bounded operator T  L(X) on a Banach space X and some quantities associated with T which play an important role in Fredholm theory. In particular, we shall consider some conditions for which T does not have the single valued extension property at a point λ_o  C.     

Kato type operators and Weyl's theorem

A Banach space operator T satisfies Weyl's theorem if and only if T or T^∗ has SVEP at all complex numbers λ in the complement of the Weyl spectrum of T and T is Kato type at all λ which are isolated eigenvalues of T of finite algebraic multiplicity. If T^∗ (respectively, T) has SVEP and T is Kato type at all λ which are isolated eigenvalues of T of finite algebraic multiplicity (respectively, T is Kato type at all λ∈isoσ(T)), then T satisfies a-Weyl's theorem (respectively, T^∗ satisfies a-Weyl's theorem).     

Spectral mapping theorem for generalized Kato spectrum

In this paper, we give an affirmative answer to Mbekhta's conjecture (Mbekhta, 1990 [13]) about the pseudo Fredholm operators in Hilbert space. As a consequence, we characterize pseudo Fredholm operators and we prove that the generalized Kato spectrum satisfies the spectral mapping theorem in the Hilbert spaces setting.     

Components of generalized Kato resolvent set and single-valued extension property

In this paper, we use the constancy of certain subspace valued mappings on the components of the generalized Kato resolvent set and the equivalences of the single-valued extension property at a point 0 for operators which admit a generalized Kato decomposition to obtain a classification of the components of the generalized Kato resolvent set of operators. We also give some applications of these results.     

Hilbert空间加权移位算子的局部

本文考察Hilbert空间$H$上单射单     

On Central Endomorphisms of a Group with Operators

Let G be a group with a set of operators such that Z(G) is -admissible. Central -automorphisms occur in the Krull-Remak-Schmidt Theorem. We discuss the existence of a central -endomorphism of G that is not an automorphism.2000 Mathematics Subject Classification: 20E36     

Generalized Kato decomposition, single-valued extension property and approximate point spectrum

In this paper, we define the generalized Kato spectrum of an operator, and obtain that the generalized Kato spectrum differs from the semi-regular spectrum on at most countably many points. We study the localized version of the single-valued extension property at the points which are not limit points of the approximate point spectrum, as well as of the surjectivity spectrum. In particular, we shall characterize the single-valued extension property at a point λ₀∈C in the case that λ₀I−T admits a generalized Kato decomposition. From this characterization we shall deduce several results on cluster points of some distinguished parts of the spectrum.     

Helton类算子及单值扩张性质

本文研究了Helton类算子在紧摄动下单值扩张性质的稳定性, 同时研究了2×2上三角算子矩阵在紧摄动下单值扩张性质的稳定性. 利用半Fredholm域的特点, 获得了2×2上三角算子矩阵具有单值扩张性质的稳定性的充分必要条件.     

$(\omega_1)$性质与单值延拓性质

In this note we study the property（ω1）,a variant of Weyl＇s theorem by means of the single valued extension property,and establish for a bounded linear operator defined on a Banach space the necessary and sufficient condition for which property（ω1） holds.As a consequence of the main result,the stability of property（ω1） is discussed.     

On an Algebra of a Certain Class of Operators in a Slab Domain in R2

In this paper a Dirichlet problem for the Laplacian in a domain with a corner in R² is treated and a new method of construction of a local parametrix for that Dirichlet problem at a corner point is given. In order to construct a parametrix a certain class of operators in a slab domain is introduced and it is shown that that operator class is an algebra.     

A note on Weyl's theorem for operator matrices

When and are given we denote by an operator acting on the Banach space of the form

In this note we examine the relation of Weyl's theorem for and through local spectral theory.

     

Hardy空间上解析Toeplitz算子的局部谱

考察Hardy空间H^2（T）上的解析Toeplitz算子的局部谱,得到的主要结果是：当φ∈H^∞（T）时,A↓∈H^2（T）,x≠0,σTφ（x）=σ（Tφ）．     

Weyl's theorem for algebraically totally hereditarily normaloid operators

A Banach space operator T∈B(X) is said to be totally hereditarily normaloid, T∈THN, if every part of T is normaloid and every invertible part of T has a normaloid inverse. The operator T is said to be an H(q) operator for some integer q?1, T∈H(q), if the quasi-nilpotent part H₀(T−λ)=(T−λ)^−q(0) for every complex number λ. It is proved that if T is algebraically H(q), or T is algebraically THN and X is separable, then f(T) satisfies Weyl's theorem for every function f analytic in an open neighborhood of σ(T), and T^∗ satisfies a-Weyl's theorem. If also T^∗ has the single valued extension property, then f(T) satisfies a-Weyl's theorem for every analytic function f which is non-constant on the connected components of the open neighborhood of σ(T) on which it is defined.     

Generalized Resolvents of Symmetric Operators

Mathematical Notes -     

Null controllability for the parabolic equation with a complex principal part

The paper is devoted to a study of the null controllability for the semilinear parabolic equation with a complex principal part. For this purpose, we establish a key weighted identity for partial differential operators (with real functions α and β), by which we develop a universal approach, based on global Carleman estimate, to deduce not only the desired explicit observability estimate for the linearized complex Ginzburg-Landau equation, but also all the known controllability/observability results for the parabolic, hyperbolic, Schrödinger and plate equations that are derived via Carleman estimates.     

Solving linear systems of differential and difference equations with respect to a part of the unknowns

The existence of solutions y to a linear differential or difference system whose components $y_{i_1 } ,...,y_{i_m } $ belong to a given class of functions and the problem of constructing such components are considered.     

Boundeness of a Class of Maximal Operators on H~p Spaces on Compact Lie Groups

In this paper we discuss the weak type(H^p,L^p) boundedness of a class of maximal operators T*^ψ and themaximal strong mean boundedness of a family of the operators ｛T^ψ｝ on the atomic H^p spaces on compact Lic groups.Also,we obtain the correspoding convergent results.     

关于Sz\''{a}sz算子的强逆不等式

本文应用新的$K$-泛函$K_{\lambda}^{\alpha}(f,t^2)=\inf_{g\in C_{\lambda}^2}\{\|f-g\|_0+t^2\|g\|_2\}, ~~0\leq \lambda\leq 1, 0<\alpha<2,$得到了Sz\'{a}sz算子关于$K$-泛函的强逆不等式,其中$\|\cdot\|_{0}, \|\cdot\|_2, C_\lambda^2 $定义在文中给出. 作为其应用, 我们推广了以前的结果.     

On Some Expectation and Derivative Operators Related to Integral Representations of Random Variables with Respect to a PII Process

Given a process with independent increments X (not necessarily a martingale) and a large class of square integrable r.v. H = f(X _T ), f being the Fourier transform of a finite measure μ, we provide a direct expression for Kunita-Watanabe and Föllmer-Schweizer decompositions of H. The representation is expressed by means of two significant maps: the expectation and derivative operators related to the characteristics of X. We also evaluate the expression for the variance optimal error when hedging the claim H with underlying process X. Those questions are motivated by finding the solution of the celebrated problem of global and local quadratic risk minimization in mathematical finance.