Families of Compact Operators on Hilbert Spaces and Fundamental Properties

2017 ◽  
pp. 197-249
Author(s):  
Valter Moretti
Keyword(s):  
Hilbert Spaces ◽  
Compact Operators ◽  
Fundamental Properties

scholarly journals The spectra of compact operators in Hilbert spaces

1965 ◽  
Vol 7 (1) ◽  
pp. 34-38
Author(s):  
T. T. West
Keyword(s):  
Hilbert Space ◽  
Unit Circle ◽  
Real Axis ◽  
Complex Plane ◽  
Hilbert Spaces ◽  
Compact Operators ◽  
Conformal Mappings ◽  
The Real ◽  
Finite Set

In [2] a condition, originally due to Olagunju, was given for the spectra of certain compact operators to be on the real axis of the complex plane. Here, by using conformal mappings, this result is extended to more general curves. The problem divides naturally into two cases depending on whether or not the curve under consideration passes through the origin. Discussion is confined to the prototype curves C0 and C1. The case of C0, the unit circle of centre the origin, is considered in § 3; this problem is a simple one as the spectrum is a finite set. In § 4 results are given for C1 the unit circle of centre the point 1, and some results on ideals of compact operators, given in § 2, are needed. No attempt has been made to state results in complete generality (see [2]); this paper is kept within the framework of Hilbert space, and particularly simple conditions may be given if the operators are normal.

STRUCTURE THEOREM FOR -OPERATORS

2014 ◽  
Vol 96 (3) ◽  
pp. 386-395 ◽  
Author(s):  
G. RAMESH
Keyword(s):  
Singular Value Decomposition ◽  
Compact Operator ◽  
Hilbert Spaces ◽  
Structure Theorem ◽  
Bounded Operator ◽  
Compact Operators ◽  
Singular Value ◽  
Scalar Multiple ◽  
Value Decomposition

AbstractIn this paper we prove a structure theorem for the class of $\mathcal{AN}$-operators between separable, complex Hilbert spaces which is similar to that of the singular value decomposition of a compact operator. Apart from this, we show that a bounded operator is $\mathcal{AN}$ if and only if it is either compact or a sum of a compact operator and scalar multiple of an isometry satisfying some condition. We obtain characterizations of these operators as a consequence of this structure theorem and deduce several properties which are similar to those of compact operators.

scholarly journals The irreducibility of C*-algebras acting on Hilbert C*-modules

Filomat ◽  
2016 ◽  
Vol 30 (9) ◽  
pp. 2425-2433
Author(s):  
Runliang Jiang
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Compact Operators ◽  
C Algebra ◽  
C Algebras ◽  
The Difference

Let B be a C*-algebra, E be a Hilbert B module and L(E) be the set of adjointable operators on E. Let A be a non-zero C*-subalgebra of L(E). In this paper, some new kinds of irreducibilities of A acting on E are introduced, which are all the generalizations of those associated to Hilbert spaces. The difference between these irreducibilities are illustrated by a number of counterexamples. It is concluded that for a full Hilbert B-module, these irreducibilities are all equivalent if and only if the underlying C*-algebra B is isomorphic to the C*-algebra of all compact operators on a Hilbert space.

scholarly journals Compact Operators on Hilbert Spaces

OALib ◽  
2014 ◽  
Vol 01 (06) ◽  
pp. 1-3 ◽  
Author(s):  
Sara Nozari
Keyword(s):  
Hilbert Spaces ◽  
Compact Operators

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.

scholarly journals On symmetry of strong Birkhoff orthogonality in $$B(\mathcal {H},\mathcal {K})$$ and $$K(\mathcal {H},\mathcal {K})$$

2020 ◽  
Vol 11 (3) ◽  
pp. 693-704 ◽  
Author(s):  
Ryotaro Tanaka ◽  
Debmalya Sain
Keyword(s):  
Hilbert Spaces ◽  
Compact Operators ◽  
Birkhoff Orthogonality ◽  
Bounded Linear ◽  
Symmetric Points

AbstractIn this paper, complete characterizations of left (or right) symmetric points for strong Birkhoff orthogonality in $$B(\mathcal {H},\mathcal {K})$$ B ( H , K ) and $$K(\mathcal {H},\mathcal {K})$$ K ( H , K ) are given, where $$\mathcal {H},\mathcal {K}$$ H , K are complex Hilbert spaces and $$B(\mathcal {H},\mathcal {K})$$ B ( H , K ) ($$K(\mathcal {H},\mathcal {K})$$ K ( H , K ) ) is the space of all bounded linear (compact) operators from $$\mathcal {H}$$ H into $$\mathcal {K}$$ K .

scholarly journals Products of Positive Operators

2021 ◽  
Vol 15 (2) ◽  
Author(s):  
Maximiliano Contino ◽  
Michael A. Dritschel ◽  
Alejandra Maestripieri ◽  
Stefania Marcantognini
Keyword(s):  
Positive Operator ◽  
Hilbert Spaces ◽  
Spectral Properties ◽  
Compact Operators ◽  
Positive Operators ◽  
Bounded Operators ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Special Classes

AbstractOn finite dimensional spaces, it is apparent that an operator is the product of two positive operators if and only if it is similar to a positive operator. Here, the class $${\mathcal {L}^{+\,2}}$$ L + 2 of bounded operators on separable infinite dimensional Hilbert spaces which can be written as the product of two bounded positive operators is studied. The structure is much richer, and connects (but is not equivalent to) quasi-similarity and quasi-affinity to a positive operator. The spectral properties of operators in $${\mathcal {L}^{+\,2}}$$ L + 2 are developed, and membership in $${\mathcal {L}^{+\,2}}$$ L + 2 among special classes, including algebraic and compact operators, is examined.

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.

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