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1.
A bounded operator defined on a Banach space is said to be polaroid if every isolated point of the spectrum is a pole of the
resolvent. In this paper we consider the two related notions of left and right polaroid, and explore them together with the
condition of being a-polaroid. Moreover, the equivalences of Weyl type theorems and generalized Weyl type theorems are investigated for left and
a-polaroid operators. As a consequence, we obtain a general framework which allows us to derive in a unified way many recent
results, concerning Weyl type theorems (generalized or not) for important classes of operators. 相似文献
2.
If \(A\in B(\mathcal{X})\) is an upper triangular Banach space operator with diagonal \((A_1,A_2)\), \(A_1\) invertible and \(A_2\) quasinilpotent, then \(A_1^{-1}\oplus A_2\) satisfies either of the single-valued extension property, Dunford’s condition (C), Bishop’s property \((\beta )\), decomposition property \((\delta )\) or is decomposable if and only if \(A_1\) has the property. The operator \(A^{-1}_1\oplus 0\) is subscalar (resp., left polaroid, right polaroid) if and only if \(A_1\) is subscalar (resp., left polaroid, right polaroid). For Drazin invertible operators A, with Drazin inverse B, this implies that B satisfies any one of these properties if and only if A satisfies the property. 相似文献
3.
On the Equivalence and Generalized of Weyl Theorem Weyl Theorem 总被引:3,自引:0,他引:3
M. BERKANI 《数学学报(英文版)》2007,23(1):103-110
We know that an operator T acting on a Banach space satisfying generalized Weyl's theorem also satisfies Weyl's theorem. Conversely we show that if all isolated eigenvalues of T are poles of its resolvent and if T satisfies Weyl's theorem, then it also satisfies generalized Weyl's theorem. We give also a sinlilar result for the equivalence of a-Weyl's theorem and generalized a-Weyl's theorem. Using these results, we study the case of polaroid operators, and in particular paranormal operators. 相似文献
4.
The property (w) is a variant of Weyl's theorem, for a bounded operator T acting on a Banach space. In this note we consider the preservation of property (w) under a finite rank perturbation commuting with T, whenever T is polaroid, or T has analytical core K(λ0I−T)={0} for some λ0∈C. The preservation of property (w) is also studied under commuting nilpotent or under injective quasi-nilpotent perturbations. The theory is exemplified in the case of some special classes of operators. 相似文献
5.
B. P. Duggal 《Rendiconti del Circolo Matematico di Palermo》2007,56(3):317-330
A Banach space operatorT ɛB(X) is polaroid,T ɛP, if the isolated points of the spectrum ofT are poles of the resolvent ofT. LetPS denote the class of operators inP which have have SVEP, the single-valued extension property. It is proved that ifT is polynomiallyPS andA ɛB(X) is an algebraic operator which commutes withT, thenf(T+A) satisfies Weyl’s theorem andf(T
*+A
*) satisfiesa-Weyl’s theorem for everyf which is holomorphic on a neighbourhood of σ(T+A). 相似文献
6.
M.H.M. Rashid 《Journal of Mathematical Analysis and Applications》2011,383(1):82-94
An operator T acting on a Banach space X possesses property (gb) if , where σa(T) is the approximate point spectrum of T, is the essential semi-B-Fredholm spectrum of T and π(T) is the set of all poles of the resolvent of T. In this paper we study property (gb) in connection with Weyl type theorems, which is analogous to generalized Browder?s theorem. Several sufficient and necessary conditions for which property (gb) holds are given. We also study the stability of property (gb) for a polaroid operator T acting on a Banach space, under perturbations by finite rank operators, by nilpotent operators and, more generally, by algebraic and Riesz operators commuting with T. 相似文献
7.
Åsvald Lima 《Israel Journal of Mathematics》1993,84(3):451-475
We show that ifE is a Banach space with the Radon-Nikodym property thenE has the metric approximation property if and only if the space of finite rank operators is locally complemented in the space
of bounded operators. 相似文献
8.
T. Len Miller Vivien G. Miller Michael M. Neumann 《Mediterranean Journal of Mathematics》2009,6(2):149-168
This paper centers on local spectral conditions that are both necessary and sufficient for the equality of the essential spectra
of two bounded linear operators on complex Banach spaces that are intertwined by a pair of bounded linear mappings. In particular,
if the operators T and S are intertwined by a pair of injective operators, then S is Fredholm provided that T is Fredholm and S has property (δ) in a neighborhood of 0. In this case, ind(T) ≤ ind(S), and equality holds precisely when the eigenvalues of the adjoint T* do not cluster at 0. By duality, we obtain refinements of results due to Putinar, Takahashi, and Yang concerning operators
with Bishop’s property (β) intertwined by pairs of operators with dense range. Moreover, we establish an extension of a result due to Eschmeier that,
under appropriate assumptions regarding the single-valued extension property, leads to necessary and sufficient conditions
for quasi-similar operators to have equal essential spectra. In particular it turns out that the single-valued extension property
plays an essential role in the preservation of the index in this context.
相似文献
9.
10.
This paper studies Weyl's theorems, and some related results for operators with Dunford's property (C). Weyl's theorem in some classes of operators (e.g.M-hyponormal,p-hyponormal and totally paranormal operators) is considered. 相似文献
11.
In this paper we study conditions on a Banach spaceX that ensure that the Banach algebraК(X) of compact operators is amenable. We give a symmetrized approximation property ofX which is proved to be such a condition. This property is satisfied by a wide range of Banach spaces including all the classical
spaces. We then investigate which constructions of new Banach spaces from old ones preserve the property of carrying amenable
algebras of compact operators. Roughly speaking, dual spaces, predual spaces and certain tensor products do inherit this property
and direct sums do not. For direct sums this question is closely related to factorization of linear operators. In the final
section we discuss some open questions, in particular, the converse problem of what properties ofX are implied by the amenability ofК(X).
BEJ supported by MSRVP at Australian National University; GAW supported by SERC grant GR-F-74332. 相似文献
12.
Let E be a Hilbert C*-module,and ■ be an orthogonally complemented closed submodule of E.The authors generalize the definitions of ■-complementability and ■-compatibility for general(adjointable) operators from Hilbert space to Hilbert C*-module,and discuss the relationship between each other.Several equivalent statements about ■-complementability and ■-compatibility,and several representations of Schur complements of ■-complementable operators(especially,of ■-compatible operators and of positive ■-compatib... 相似文献
13.
Octavian Agratini 《Chaos, solitons, and fractals》2011,44(11):977-981
Starting from a general sequence of linear and positive operators of discrete type, we associate its r-th order generalization. This construction involves high order derivatives of a signal and it looses the positivity property. Considering that the initial approximation process is A-statistically uniform convergent, we prove that the property is inherited by the new sequence. Also, our result includes information about the uniform convergence. Two applications in q-Calculus are presented. We study q-analogues both of Meyer-König and Zeller operators and Stancu operators. 相似文献
14.
It is well known that not every summability property for multilinear operators leads to a factorization theorem. In this paper we undertake a detailed study of factorization schemes for summing linear and nonlinear operators. Our aim is to integrate under the same theory a wide family of classes of mappings for which a Pietsch type factorization theorem holds. Our construction includes the cases of absolutely p-summing linear operators, (p, σ)-absolutely continuous linear operators, factorable strongly p-summing multilinear operators, (p1,?. . .?, pn)-dominated multilinear operators and dominated (p1,?. . .?, pn; σ)-continuous multilinear operators. 相似文献
15.
Andrzej Kryczka 《Integral Equations and Operator Theory》2008,61(4):559-572
We introduce the arithmetic separation of a sequence—a geometric characteristic for bounded sequences in a Banach space which
describes the Banach-Saks property. We define an operator seminorm vanishing for operators with the Banach-Saks property.
We prove quantitative stability of the seminorm for a class of operators acting between l
p
-sums of Banach spaces. We show logarithmically convex-type estimates of the seminorm for operators interpolated by the real
method of Lions and Peetre.
相似文献
16.
Boris Yu. Sternin 《Central European Journal of Mathematics》2011,9(4):814-832
We consider a class of nonlocal operators associated with an action of a compact Lie group G on a smooth closed manifold. Ellipticity condition and Fredholm property for elliptic operators are obtained. This class
of operators is studied using pseudodifferential uniformization, which reduces the problem to a pseudodifferential operator
acting in sections of infinite-dimensional bundles. 相似文献
17.
Ya. V. Kurylev 《Journal of Mathematical Sciences》1995,73(3):385-388
Existence of wave operators for scattering on a line in R3 is proved. It is shown that the completeness of have operators is violated in the case of a line with periodic boundary conditions.
However, the asymptotic completeness and unitary property of wave operators are preserved. Bibliography:11 titles.
Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 186, pp. 136–141, 1990.
Translated by Ya. V. Kurylev. 相似文献
18.
Rubn A. Martínez-Avendao 《Journal of Functional Analysis》2002,190(2):418-446
We introduce a class of operators, called λ-Hankel operators, as those that satisfy the operator equation S*X−XS=λX, where S is the unilateral forward shift and λ is a complex number. We investigate some of the properties of λ-Hankel operators and show that much of their behaviour is similar to that of the classical Hankel operators (0-Hankel operators). In particular, we show that positivity of λ-Hankel operators is equivalent to a generalized Hamburger moment problem. We show that certain linear spaces of noninvertible operators have the property that every compact subset of the complex plane containing zero is the spectrum of an operator in the space. This theorem generalizes a known result for Hankel operators and applies to λ-Hankel operators for certain λ. We also study some other operator equations involving S. 相似文献
19.
A real square matrix A is called a P-matrix if all its principal minors are positive. Such a matrix can be characterized by the sign non-reversal property. Taking a cue from this, the notion of a P-operator is extended to infinite dimensional spaces as the first objective. Relationships between invertibility of some subsets of intervals of operators and certain P-operators are then established. These generalize the corresponding results in the matrix case. The inheritance of the property of a P-operator by the Schur complement and the principal pivot transform is also proved. If A is an invertible M-matrix, then there is a positive vector whose image under A is also positive. As the second goal, this and another result on intervals of M-matrices are generalized to operators over Banach spaces. Towards the third objective, the concept of a Q-operator is proposed, generalizing the well known Q-matrix property. An important result, which establishes connections between Q-operators and invertible M-operators, is proved for Hilbert space operators. 相似文献
20.
In the present paper we propose a generalization of the Baskakov operators, based on q integers. We also estimate the rate of convergence in the weighted norm. In the last section, we study some shape preserving
properties and the property of monotonicity of q-Baskakov operators. 相似文献