一类不可分解Banach空间上线性算子的谱性质

给出一类不可分解的∑_e~1型Banach空间上线性算子(不一定有界)的谱结构,并讨论这种空间上生成C_0群或C_0半群的线性算子的有界性、特殊的谱性质和谱结构,还给出这种空间上闭算子是有界算子的一个充分条件。     

Spectral theory of linear relations on real Banach spaces

We consider the relationship between the spectral properties of linear relations (multivalued linear operators) on real Banach spaces and their complexifications.     

Some remarks on Birkhoff–James orthogonality of linear operators

We study Birkhoff–James orthogonality of compact (bounded) linear operators between Hilbert spaces and Banach spaces. Applying the notion of semi-inner-products in normed linear spaces and some related geometric ideas, we generalize and improve some of the recent results in this context. In particular, we obtain a characterization of Euclidean spaces and also prove that it is possible to retrieve the norm of a compact (bounded) linear operator (functional) in terms of its Birkhoff–James orthogonality set. We also present some best approximation type results in the space of bounded linear operators.     

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.     

On typical properties of Hilbert space operators

We study the typical behavior of bounded linear operators on infinite-dimensional complex separable Hilbert spaces in the norm, strong-star, strong, weak polynomial and weak topologies. In particular, we investigate typical spectral properties, the problem of unitary equivalence of typical operators, and their embeddability into C ₀-semigroups. Our results provide information on the applicability of Baire category methods in the theory of Hilbert space operators.     

商不可分解Banach空间上线性算子的谱

This paper discusses the special properties of the spectrum of linear operators (in particular, bounded linear operators) on quotient indecomposable Banach spaces; shows that in such spaces generators of Co-groups are always bounded linear operators, and that generators of Co-semigroups satisfy the spectral mapping theorem; and gives an example to show that the generators of Co-semigroups in quotient indecomposable spaces are not necessarily bounded.     

THE STRUCTURE OF $\[\Pi \]$ SPACES (I)

The spectral theory of selfadjoint operators and unitary operators in Hilbert space has been successfully generalized to $\[{\Pi _k}\]$ space. However, there are only a few results for the spectral theory of selfadjoint operators and unitary operators in $\[\Pi \]$ space. One of the important reasons is that the structure of $\[\Pi \]$ space is more complex than that of $\[{\Pi _k}\]$ space. This paper and the forthcoming paper "The structure of $\[\Pi \]$ space (II)" will mainly be dealt with the structure of $\[\Pi \]$ spaces, which will be used to further study the operators in $\[\Pi \]$ spaces.     

完备赋准范空间上一类无界线性随机算子的谱分解定理

本文利用随机内积空间方法给出了完备赋准范空间上一类无界线性随机算子的谱分解定理。此结果不仅推广了对称随机线性算子的谱分解定理，而且限于原情形也使其处理简明、清晰。     

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.     

Non-Archimedean Analogues of Orthogonal and Symmetric Operators and p-adic Quantization

We study orthogonal and symmetric operators in non-Archimedean Hilbert spaces in the connection with p-adic quantization. This quantization describes measurements with finite precision. Symmetric (bounded) operators in the p-adic Hilbert spaces represent physical observables. We study spectral properties of one of the most important quantum operators, namely, the operator of the position (which is represented in the p-adic Hilbert L₂-space with respect to the p-adic Gaussian measure). Orthogonal isometric isomorphisms of p-adic Hilbert spaces preserve precisions of measurements. We study properties of orthogonal operators. It is proved that each orthogonal operator in the non-Archimedean Hilbert space is continuous. However, there exist discontinuous operators with the dense domain of definition which preserve the inner product. There also exist nonisometric orthogonal operators. We describe some classes of orthogonal isometric operators and we study some general questions of the theory of non-Archimedean Hilbert spaces (in particular, general connections between topology, norm and inner product).     

Local spectral theory for normal operators in Krein spaces

Sign type spectra are an important tool in the investigation of spectral properties of selfadjoint operators in Krein spaces. It is our aim to show that also sign type spectra for normal operators in Krein spaces provide insight in the spectral nature of the operator: If the real part and the imaginary part of a normal operator in a Krein space have real spectra only and if the growth of the resolvent of the imaginary part (close to the real axis) is of finite order, then the normal operator possesses a local spectral function defined for Borel subsets of the spectrum which belong to positive (negative) type spectrum. Moreover, the restriction of the normal operator to the spectral subspace corresponding to such a Borel subset is a normal operator in some Hilbert space. In particular, if the spectrum consists entirely out of positive and negative type spectrum, then the operator is similar to a normal operator in some Hilbert space. We use this result to show the existence of operator roots of a class of quadratic operator polynomials with normal coefficients.     

Calderón–Zygmund operators with operator‐valued kernel on homogeneous Besov spaces

We consider generalized Calderón–Zygmund operators whose kernel takes values in the space of all continuous linear operators between two Banach spaces. In the spirit of the T (1) theorem of David and Journé we prove boundedness results for such operators on vector‐valued Besov spaces. (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Spectral Theory of Toeplitz Operators Applied to Approximation Problems in Hilbert Spaces

The solution to a particular constrained approximation problem, in an abstract Hilbert space setting, may be interpreted in terms of a generalised Toeplitz operator. We consider concrete versions of this problem, in settings which involve generalised Hardy spaces, Paley–Wiener spaces and the Segal–Bargmann space, and derive spectral representations of the associated Toeplitz operators.     

Partial inner product spaces. II. Operators

We study linear operators between nondegenerate partial inner product spaces and their relationships to selfadjoint operators in a “middle” Hilbert space.     

Best approximation by polynomial operators in banach spaces

The problem of approximation in the space of bounded linear operators ? (E;G) between normed spaces E and G by compact operators has been extensively studied in the last few years.

Recently Deutsch, Mach and Saatkamp ([2]) have considered the problem of approximating elements of ?(E;G) by the subset K _N(E;G) of operators whose range is at most N dimensional. We consider in this paper the problem of approximating operators (not necessarily linear) beteen normed spaces E and G by continuous homogeneous polynomials, and in particular by such polynomials which have finite-dimensional range.     

复随机内积模上一类算子的谱定理

本文给出复随机内积模上几乎处处有界线性算子谱的几个基本定理，这些定理不但为随机内积模上几乎处处有界线性算子的进一步讨论有基本重要性，而且也为Ｈｉｌｂｅｒｔ空间上连续随机算子的谱研究提供了一个新途径。     

Compact adjoint operators and approximation properties

This paper is concerned with the space of all compact adjoint operators from dual spaces of Banach spaces into dual spaces of Banach spaces and approximation properties. For some topology on the space of all bounded linear operators from separable dual spaces of Banach spaces into dual spaces of Banach spaces, it is shown that if a bounded linear operator is approximated by a net of compact adjoint operators, then the operator can be approximated by a sequence of compact adjoint operators whose operator norms are less than or equal to the operator norm of the operator. Also we obtain applications of the theory and, in particular, apply the theory to approximation properties.     

An overview of semi-continuity results on the spectral radius and positivity

Our aim is to provide a novelly comprehensive and unifying approach to showing the continuous dependence of the spectral radius of compact linear operators defined on Banach spaces (with specific attention to positive operators defined on normal Banach spaces) and emphasizing that the upper semi-continuity generally holds unlike the lower semi-continuity.     

Weighted Moore-Penrose inverses and fundamental theorem of even-order tensors with Einstein product

We treat even-order tensors with Einstein product as linear operators from tensor space to tensor space, define the null spaces and the ranges of tensors, and study their relationship. We extend the fundamental theorem of linear algebra for matrix spaces to tensor spaces. Using the new relationship, we characterize the least-squares (?) solutions to a multilinear system and establish the relationship between the minimum-norm (N) least-squares (?) solution of a multilinear system and the weighted Moore-Penrose inverse of its coefficient tensor. We also investigate a class of even-order tensors induced by matrices and obtain some interesting properties.     

Well-bounded operators on nonreflexive Banach spaces

Every well-bounded operator on a reflexive Banach space is of type (B), and hence has a nice integral representation with respect to a spectral family of projections. A longstanding open question in the theory of well-bounded operators is whether there are any nonreflexive Banach spaces with this property. In this paper we extend the known results to show that on a very large class of nonreflexive spaces, one can always find a well-bounded operator which is not of type (B). We also prove that on any Banach space, compact well-bounded operators have a simple representation as a combination of disjoint projections.