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.

     

On the essential spectra of some matrix of linear relations

In this paper, we investigate a detailed treatment of some subsets of essential spectra of a closed multivalued linear operator. On the following, we will establish some results on perturbation theory of 2 × 2 matrix of multivalued linear operators. Copyright © 2013 John Wiley & Sons, Ltd.     

On K_0-groups of operator algebras on Banach spaces

This paper merges some classifications of G-M-type Banach spaces simplifically, discusses the condition of K0(B(X)) = 0 for operator algebra B(X) on a Banach space X, and obtains a result to improve Laustsen's sufficient condition, gives an example to show that X ≈ X2 is not a sufficient condition of K0(B(X)) = 0.     

Banach空间中线性算子的Drazin广义逆的表示

1981年,1985年,2000年,乔三正、蔡东汉、魏益民分别给出Drazin广义逆不同形式的表示．本文将对上述结果进行推广,给出Drazin广义逆的统一表示,使上述三个结果均成为本文主要结果的特例．在本文主要结果的基础上,利用算子谱理论,给出Drazin广义逆的一种逼近形式的表示,同时给出逼近解的估计．此结果推广了蔡东汉、魏益民的相应结果．     

On hypercyclic operators on Banach spaces

We provide in this paper a direct and constructive proof of the following fact: for a Banach space there are bounded linear operators having hypercyclic vectors if and only if is separable and dim. This is a special case of a recent result, which in turn solves a problem proposed by S. Rolewicz.

     

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。     

Laurent series for inversion of linearly perturbed bounded linear operators on Banach space

In this paper we find necessary and sufficient conditions for the existence of a Laurent series expansion with a finite order pole at the origin for the inverse of a linearly perturbed bounded linear operator mapping one Banach space to another. In particular we show that the inversion defines linear projections that separate the Banach spaces into corresponding complementary subspaces. We present two pertinent applications.     

Periodic groups of operators in Banach spaces

Spectral operators of scalar type in the sense of Dunford often occur in connection with unconditionally convergent series expansions, whereas conditionally convergent expansions under similar conditions may be described with the help of operators having a more general type of spectral decomposition. We show that under certain conditions even in the latter case we can restrict our considerations to a dense linear submanifold of the original Banach space with a stronger topology, where the convergence of the expansion under study will be unconditional. Though our conditions could be formulated in terms of a single operator, it seems to be more natural to state them in terms of (the generator of) a periodic group of operators.

     

Functions from the Schoenberg classT act in the cone of dissipative elements of a Banach algebra. IIact in the cone of dissipative elements of a Banach algebra. II

The functional calculus of several commuting dissipative elements of a complex Banach algebra with identity, first introduced in the preceding work by the author, is developed. Uniqueness, continuity, and stability theorems, composite function theorems, and a formula for the resolvent are established. Applications to the theory of sectorial operators in Hilbert space are given. Translated fromMatematickeskie Zametki, Vol. 64, No. 3, pp. 423–430, September, 1998.     

An approximation problem of the finite rank operator in Banach spaces

By the method of geometry of Banach spaces, we have proven that a bounded linear operator in Banach space is a compact linear one iff it can be uniformly approximated by a sequence of the finite rank bounded homogeneous operators, which reveals the essence of the counter example given by Enflo.     

Limit distributions and one-parameter groups of linear operators on Banach spaces

Let = {U_t: t > 0} be a strongly continuous one-parameter group of operators on a Banach space X and Q be any subset of a set (X) of all probability measures on X. By (Q; ) we denote the class of all limit measures of {U_tn(μ₁ * μ₂*…*μ_n)*δ_xn}, where {μ_n}Q, {x_n}X and measures U_tnμ_j (j=1, 2,…, n; N=1, 2,…) form an infinitesimal triangular array. We define classes L_m( ) as follows: L₀( )= ( (X); ), L_m( )= (L_m−1( ); ) for m=1, 2,… and L_∞( )=_m=0^∞L_m( ). These classes are analogous to those defined earlier by Urbanik on the real line. Probability distributions from L_m( ), m=0, 1, 2,…, ∞, are described in terms of their characteristic functionals and their generalized Poisson exponents and Gaussian covariance operators.     

Factorizing operators on Banach function spaces through spaces of multiplication operators

In order to extend the theory of optimal domains for continuous operators on a Banach function space X(μ) over a finite measure μ, we consider operators T satisfying other type of inequalities than the one given by the continuity which occur in several well-known factorization theorems (for instance, Pisier Factorization Theorem through Lorentz spaces, pth-power factorable operators …). We prove that such a T factorizes through a space of multiplication operators which can be understood in a certain sense as the optimal domain for T. Our extended optimal domain technique does not need necessarily the equivalence between μ and the measure defined by the operator T and, by using δ-rings, μ is allowed to be infinite. Classical and new examples and applications of our results are also given, including some new results on the Hardy operator and a factorization theorem through Hilbert spaces.     

Commuting relations for Hausdorff operators and Hilbert transforms on real Hardy spaces

We consider Hausdorff operators generated by a function ϕ integrable in Lebesgue"s sense on either R or R ², and acting on the real Hardy space H ¹(R), or the product Hardy space H ¹¹(R×R), or one of the hybrid Hardy spaces H ¹⁰(R ²) and H ⁰¹(R ²), respectively. We give a necessary and sufficient condition in terms of ϕ that the Hausdorff operator generated by it commutes with the corresponding Hilbert transform. This revised version was published online in June 2006 with corrections to the Cover Date.     

Some G-M-type Banach spaces and K-groups of operator algebras on them

By providing several new varieties of G-M-type Banach spaces according todecomposable and compoundable properties, this paper discusses the operator structuresof these spaces and the K-theory of the algebra of the operators on these G-M-type Ba-nach spaces through calculation of the K-groups of the operator ideals contained in theclass of Riesz operators.     

Interpolation properties of scales of Banach spaces

It is shown that any interpolation scales joining weight spaces L _p or similar spaces have many remarkable properties. Not only are such scales intrinsically interpolation scales, but an analog of the Arazy-Cwikel theorem describing interpolation spaces between the spaces from the scale is valid.     

Spectral Analysis of Powers of the Operator (Vf)(x)=q(x)∫0 xw(t)f(t),dt

Mathematical Notes -     

Spectral theory of discrete linear Hamiltonian systems

This paper is concerned with spectral problems for a class of discrete linear Hamiltonian systems with self-adjoint boundary conditions, where the existence and uniqueness of solutions of initial value problems may not hold. A suitable admissible function space and a difference operator are constructed so that the operator is self-adjoint in the space. Then a series of spectral results are obtained: the reality of eigenvalues, the completeness of the orthogonal normalized eigenfunction system, Rayleigh's principle, the minimax theorem and the dual orthogonality. Especially, the number of eigenvalues including multiplicities and the number of linearly independent eigenfunctions are calculated.     

Criteria for the single-valued metric generalized inverses of multi-valued linear operators in banach spaces

Let X, Y be Banach spaces and M be a linear subspace in X × Y = {{x, y}|x ∈ X, y ∈ Y }. We may view M as a multi-valued linear operator from X to Y by taking M (x) = {y|{x, y} ∈ M }. In this paper, we give several criteria for a single-valued operator from Y to X to be the metric generalized inverse of the multi-valued linear operator M . The principal tool in this paper is also the generalized orthogonal decomposition theorem in Banach spaces.     

Necessary and sufficient conditions for the inversion of linearly-perturbed bounded linear operators on Banach space using Laurent series

Using a Laurent series representation, we present a detailed discussion of necessary and sufficient conditions for the inversion of linearly-perturbed bounded linear operators on Banach space that are singular in the unperturbed state.     

Improving the rate of convergence of Newton methods on Banach spaces with a convergence structure and applications

In this note, we use inexact Newton-like methods to find solutions of nonlinear operator equations on Banach spaces with a convergence structure. Our technique involves the introduction of a generalized norm as an operator from a linear space into a partially ordered Banach space. In this way, the metric properties of the examined problem can be analyzed more precisely. Moreover, this approach allows us to derive from the same theorem, on the one hand, semilocal results of Kantorovich-type, and on the other hand, global results based on monotonicity considerations. By imposing very general Lipschitz-like conditions on the operators involved, on the one hand, we cover a wider range of problems, and on the other hand, by choosing our operators appropriately, we can find sharper error bounds on the distances involved than before. Furthermore, we show that special cases of our results reduce to the corresponding ones already in the literature. Finally, several examples are being provided where our results compare favorably with earlier ones.