On the instability of non-semi-Fredholm closed operators under compact perturbations with applications to ordinary differential operators

1988 ◽  
Vol 109 (1-2) ◽  
pp. 97-108 ◽  
Author(s):  
L. E. Labuschagne
Keyword(s):  
Banach Spaces ◽  
Hilbert Spaces ◽  
Differential Operators ◽  
Compact Operators ◽  
Fredholm Operators ◽  
Closed Operators ◽  
Compact Perturbations ◽  
Ordinary Differential Operators ◽  
The Stability

SynopsisThe stability of several natural subsets of the bounded non-semi-Fredholm operators undercompact perturbations were studied by R. Bouldin [2] in separable Hilbert spaces and by M. Gonzales and V. M. Onieva [6] in Banach spaces. The aim of this paper is to study this problem for closed operators in operator ranges. The main results are a characterisation of the non-semi-Fredholm operators with respect to α-closed and α-compact operators as well as a generalisation of a result of M. Goldman [5]. We also give some applications of the theory developed to ordinary differential operators.

Unbounded Linear Operators

2018 ◽  
Author(s):  
D. E. Edmunds ◽  
W. D. Evans
Keyword(s):  
Hilbert Spaces ◽  
Linear Operators ◽  
Von Neumann ◽  
Limit Circle ◽  
Sectorial Operators ◽  
Closed Operators ◽  
The Stability ◽  
Symmetric Operators ◽  
Unbounded Linear Operators ◽  
Special Case

This chapter is concerned with closable and closed operators in Hilbert spaces, especially with the special classes of symmetric, J-symmetric, accretive and sectorial operators. The Stone–von Neumann theory of extensions of symmetric operators is treated as a special case of results for compatible adjoint pairs of closed operators. Also discussed in detail is the stability of closedness and self-adjointness under perturbations. The abstract results are applied to operators defined by second-order differential expressions, and Sims’ generalization of the Weyl limit-point, limit-circle characterization for symmetric expressions to J-symmetric expressions is proved.

Regularly solvable extensions of non-self-adjoint ordinary differential operators

1984 ◽  
Vol 97 ◽  
pp. 79-95 ◽  
Author(s):  
W. D. Evans
Keyword(s):  
Hilbert Space ◽  
Weighted Space ◽  
Differential Operators ◽  
Linear Operators ◽  
Order Linear ◽  
Adjoint Pair ◽  
Closed Operators ◽  
Ordinary Differential Operators ◽  
Linear Differential ◽  
Densely Defined

SynopsisLetL0,M0be closed densely defined linear operators in a Hilbert spaceHwhich form an adjoint pair, i.e.. In this paper, we study closed operatorsSwhich satisfyand are regularly solvable in the sense of Višik. The abstract results obtained are applied to operators generated by second-order linear differential expressions in a weighted spaceL2(a, b; w).

scholarly journals Range-kernel complementation

2018 ◽  
Vol 55 (3) ◽  
pp. 327-344
Author(s):  
Carlos. S. Kubrusly
Keyword(s):  
Banach Space ◽  
Normed Space ◽  
Compact Operators ◽  
Space Operator ◽  
Fredholm Operators ◽  
Compact Perturbations

If a Banach-space operator has a complemented range, then its normed-space adjoint has a complemented kernel and the converse holds on a re exive Banach space. It is also shown when complemented kernel for an operator is equivalent to complemented range for its normed-space adjoint. This is applied to compact operators and to compact perturbations. In particular, compact perturbations of semi-Fredholm operators have complemented range and kernel for both the perturbed operator and its normed-space adjoint.

scholarly journals Some results about spectral continuity and compact perturbations

Filomat ◽  
2020 ◽  
Vol 34 (14) ◽  
pp. 4837-4845
Author(s):  
S. Sánchez-Perales ◽  
S.V. Djordjevic ◽  
S. Palafox
Keyword(s):  
Hilbert Spaces ◽  
Compact Operators ◽  
Compact Perturbations ◽  
Spectral Continuity ◽  
Continuity Of The Spectrum ◽  
The Ideal

In this paper, we are interested in the continuity of the spectrum and some of its parts in the setting of Hilbert spaces. We study the continuity of the spectrum in the class of operators {T}+K(H), where K(H) denote the ideal of compact operators. Also, we give conditions in order to transfer the continuity of spectrum from T to T + K, where K ? K(H). Then, we characterize those operators for which the continuity of spectrum is stable under compact perturbations.

The perturbation classes problem for closed operators

Filomat ◽  
2017 ◽  
Vol 31 (3) ◽  
pp. 621-627
Author(s):  
Pietro Aiena ◽  
Muneo Chō ◽  
Manuel González
Keyword(s):  
Banach Spaces ◽  
Fredholm Operators ◽  
Open Problems ◽  
Dense Domain ◽  
Closed Operators

We compare the perturbation classes for closed semi-Fredholm and Fredholm operators with dense domain acting between Banach spaces with the corresponding perturbation classes for bounded semi-Fredholm and Fredholm operators. We show that they coincide in some cases, but they are different in general. We describe several relevant examples and point out some open problems.

scholarly journals SOME RESULTS ON ESSENTIAL SPECTRA OF DIFFERENTIAL OPERATORS IN BANACH SPACES

2003 ◽  
Vol 8 (3) ◽  
pp. 203-216
Author(s):  
V. A. Erovenko
Keyword(s):  
Banach Spaces ◽  
Differential Operators ◽  
Lebesgue Spaces ◽  
Essential Spectra ◽  
Polynomial Coefficients ◽  
Ordinary Differential Operators

In this paper we investigate spectral and semi‐Predholm properties of maximum and minimum Puchsian differential operators on Lebesgue spaces on a semi‐axis. These results are applied for determination of various essential spectra and spectrum of ordinary differential operators with polynomial coefficients, which order does not exceed the order of the corresponding derivative.

scholarly journals New Decompositions for Classes of Operators with Topological Uniform Descent

2020 ◽  
Vol 55 (6) ◽  
Author(s):  
Orlando García ◽  
Carlos Carpintero ◽  
José Sanabria ◽  
Osmin Ferrer
Keyword(s):  
Banach Spaces ◽  
Hilbert Spaces ◽  
Finite Range ◽  
Weyl Operator ◽  
Decomposition Property ◽  
Large Classes ◽  
Lower Semi ◽  
Topological Uniform Descent ◽  
The Stability

The article describes a new decomposition property for operators with topological uniform descent, like Kato type operators, as well as new results on the stability of this class of operators under perturbations by operators with finite-range power based on topological descent notion, from which we can generalize many perturbation results for a large classes of operators by extending to Banach spaces known techniques on Hilbert spaces. As application of our resuts we obtain that is a lower semi B-Weyl operator if and only if , where is a lower semi B-Browder operator and , for some . Our methods generalize to Banach spaces some results obtained by Aiena for operators acting on Hilbert spaces.

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