The Fredholm Elements of a Ring

1969 ◽  
Vol 21 ◽  
pp. 84-95 ◽  
Author(s):  
Bruce Alan Barnes
Keyword(s):  
Banach Space ◽  
Compact Operators ◽  
Bounded Operators ◽  
Fredholm Operators ◽  
Finite Dimensional ◽  
Dimensional Range

In (1), Atkinson characterized the set of Fredholm operators on a Banach space X as those bounded operators invertible modulo the two-sided ideal of compact operators on X. It follows from this characterization that the Fredholm operators can also be described as those bounded operators which are invertible modulo the two-sided ideal of bounded operators on X which have finite-dimensional range. This ideal is the socle of the algebra of all bounded operators on X. Now, if A is any ring with no nilpotent left or right ideals, then the concept of socle makes sense (the socle of A in this case is the algebraic sum of the minimal left ideals of A, or 0 if A has no minimal left ideals). Also, in this case, the socle is a two-sided ideal of A. In this paper we study the elements in a ring A which are invertible modulo the socle. We call these elements the Fredholm elements of A.

Near Triangularizability Implies Triangularizability

2004 ◽  
Vol 47 (2) ◽  
pp. 298-313 ◽  
Author(s):  
Bamdad R. Yahaghi
Keyword(s):  
Banach Space ◽  
Complex Banach Space ◽  
Compact Operators ◽  
Linear Operators ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Bounded Operators ◽  
Ground Field ◽  
Subspace Theorem ◽  
Finite Dimensional

AbstractIn this paper we consider collections of compact operators on a real or complex Banach space including linear operators on finite-dimensional vector spaces. We show that such a collection is simultaneously triangularizable if and only if it is arbitrarily close to a simultaneously triangularizable collection of compact operators. As an application of these results we obtain an invariant subspace theorem for certain bounded operators. We further prove that in finite dimensions near reducibility implies reducibility whenever the ground field is or .

scholarly journals Representations of normed algebras with minimal left ideals

1974 ◽  
Vol 19 (2) ◽  
pp. 173-190 ◽  
Author(s):  
Bruce A. Barnes
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Representation Theory ◽  
Banach Spaces ◽  
Banach Algebras ◽  
Normed Algebra ◽  
Bounded Operators ◽  
General Representation ◽  
Finite Dimensional ◽  
Dimensional Range

The theory of *-representations of Banach *-algebras on Hilbert space is one of the most useful and most successful parts of the theory of Banach algebras. However, there are only scattered results concerning the representations of general Banach algebras on Banach spaces. It may be that a comprehensive representation theory is impossible. Nevertheless, for some special algebras interesting and worthwhile results can be proved. This is true for (Y), the algebra of all bounded operators on a Banach space Y, and for (Y), the subalgebra of (Y) consisting of operators with finite dimensional range. The representations of (Y) are studied in a recent paper by H. Porta and E. Berkson (6), and in another recent paper (8), P. Chernoff determines the structure of the representations of (Y) (and also of some more general algebras of operators). In both these papers, (Y), which is the socle of the algebras under consideration, plays an important role in the theory. This suggests the possibility that a more general representation theory can be formulated in the case of a normed algebra with a nontrivial socle. This we attempt to do in this paper.

scholarly journals Some special characterisations of Fredholm operators in Banach space

BIBECHANA ◽  
2014 ◽  
Vol 11 ◽  
pp. 169-174
Author(s):  
Mahendra Shahi
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Essential Spectrum ◽  
Fredholm Operator ◽  
Bounded Linear Operator ◽  
Global Analysis ◽  
Spectral Theorem ◽  
Bounded Operators ◽  
Fredholm Operators ◽  
Bounded Linear

A bounded linear operator which has a finite index and which is defined on a Banach space is often referred to in the literature as a Fredholm operator. Fredholm operators are important for a variety of reasons, one being the role that their index plays in global analysis. The aim of this paper is to prove the spectral theorem for compact operators in refined form and to describe some properties of the essential spectrum of general bounded operators by the use of the theorem of Fredholm operators. For this, we have analysed the Fredholm operator which is defined in a Banach space for some special characterisations. DOI: http://dx.doi.org/10.3126/bibechana.v11i0.10399 BIBECHANA 11(1) (2014) 169-174

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 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.

Spectral characterization of the socle in Jordan–Banach algebras

1995 ◽  
Vol 117 (3) ◽  
pp. 479-489 ◽  
Author(s):  
Bernard Aupetit
Keyword(s):  
Banach Space ◽  
Banach Algebra ◽  
Finite Rank ◽  
Banach Algebras ◽  
Spectral Characterization ◽  
Bounded Operators ◽  
Finite Rank Operators ◽  
Finite Dimensional ◽  
Minimal Idempotent

If A is a complex Banach algebra the socle, denoted by Soc A, is by definition the sum of all minimal left (resp. right) ideals of A. Equivalently the socle is the sum of all left ideals (resp. right ideals) of the form Ap (resp. pA) where p is a minimal idempotent, that is p2 = p and pAp = ℂp. If A is finite-dimensional then A coincides with its socle. If A = B(X), the algebra of bounded operators on a Banach space X, the socle of A consists of finite-rank operators. For more details about the socle see [1], pp. 78–87 and [3], pp. 110–113.

scholarly journals Uniformly factoring weakly compact operators and parametrised dualisation

2021 ◽  
Vol 9 ◽  
Author(s):  
L. Antunes ◽  
K. Beanland ◽  
B. M. Braga
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Borel Subset ◽  
Compact Operators ◽  
Schauder Basis ◽  
Bounded Operators ◽  
Weakly Compact ◽  
Weakly Compact Operator ◽  
Borel Map ◽  
Weakly Compact Operators

Abstract This article deals with the problem of when, given a collection $\mathcal {C}$ of weakly compact operators between separable Banach spaces, there exists a separable reflexive Banach space Z with a Schauder basis so that every element in $\mathcal {C}$ factors through Z (or through a subspace of Z). In particular, we show that there exists a reflexive space Z with a Schauder basis so that for each separable Banach space X, each weakly compact operator from X to $L_1[0,1]$ factors through Z. We also prove the following descriptive set theoretical result: Let $\mathcal {L}$ be the standard Borel space of bounded operators between separable Banach spaces. We show that if $\mathcal {B}$ is a Borel subset of weakly compact operators between Banach spaces with separable duals, then for $A \in \mathcal {B}$ , the assignment $A \to A^*$ can be realised by a Borel map $\mathcal {B}\to \mathcal {L}$ .

scholarly journals On Compact and Fredholm Operators over C*-algebras and a New Topology in the Space of Compact Operators

2008 ◽  
Vol 2 (2) ◽  
pp. 329-351 ◽  
Author(s):  
Anwar A. Irmatov ◽  
Alexandr S. Mishchenko
Keyword(s):  
Fredholm Operator ◽  
Main Property ◽  
Compact Operators ◽  
Continuous Functions ◽  
Bounded Operators ◽  
Fredholm Operators ◽  
C Algebras ◽  
Compact Topological Space ◽  
Space Of Compact Operators ◽  
The Ideal

AbstractIt is well-known that bounded operators in Hilbert C*-modules over C*-algebras may not be adjointable and the same is true for compact operators. So, there are two analogs for classical compact operators in Hilbert C*-modules: adjointable compact operators and all compact operators, i.e. those not necessarily having an adjoint.Classical Fredholm operators are those that are invertible modulo compact operators. When the notion of a Fredholm operator in a Hilbert C*-module was developed in [6], the first analog was used: Fredholm operators were defined as operators that are invertible modulo adjointable compact operators.In this paper we use the second analog and develop a more general version of Fredholm operators over C*-algebras. Such operators are defined as bounded operators that are invertible modulo the ideal of all compact operators. The main property of this new class is that a Fredholm operator still has a decomposition into a direct sum of an isomorphism and a finitely generated operator.The special case of Fredholm operators (in the sense of [6]) over the commutative C*-algebra C(K) of continuous functions on a compact topological space K was also considered in [2]. In order to describe general Fredholm operators (invertible modulo all compact operators over C(K)) we construct a new IM-topology on the space of compact operators on a Hilbert space such that continuous families of compact operators generate the ideal of all compact operators over C(K).

Global Perturbation of Nonlinear Eigenvalues

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Julián López-Gómez ◽  
Juan Carlos Sampedro
Keyword(s):  
Perturbation Theory ◽  
Spectral Parameter ◽  
Classical Theory ◽  
General Setting ◽  
Perturbation Parameter ◽  
Linear Operators ◽  
Fredholm Operators ◽  
Index Zero ◽  
Finite Dimensional ◽  
Nonlinear Eigenvalues

Abstract This paper generalizes the classical theory of perturbation of eigenvalues up to cover the most general setting where the operator surface 𝔏 : [ a , b ] × [ c , d ] → Φ 0 ⁢ ( U , V ) {\mathfrak{L}:[a,b]\times[c,d]\to\Phi_{0}(U,V)} , ( λ , μ ) ↦ 𝔏 ⁢ ( λ , μ ) {(\lambda,\mu)\mapsto\mathfrak{L}(\lambda,\mu)} , depends continuously on the perturbation parameter, μ, and holomorphically, as well as nonlinearly, on the spectral parameter, λ, where Φ 0 ⁢ ( U , V ) {\Phi_{0}(U,V)} stands for the set of Fredholm operators of index zero between U and V. The main result is a substantial extension of a classical finite-dimensional theorem of T. Kato (see [T. Kato, Perturbation Theory for Linear Operators, 2nd ed., Class. Math., Springer, Berlin, 1995, Chapter 2, Section 5]).

scholarly journals The dual of the space of bounded operators on a Banach space

2021 ◽  
Vol 8 (1) ◽  
pp. 48-59
Author(s):  
Fernanda Botelho ◽  
Richard J. Fleming
Keyword(s):  
Banach Space ◽  
Tensor Product ◽  
Banach Spaces ◽  
Dual Space ◽  
Tensor Products ◽  
Bounded Operators ◽  
Projective Tensor Product ◽  
Projective Tensor

Abstract Given Banach spaces X and Y, we ask about the dual space of the 𝒧(X, Y). This paper surveys results on tensor products of Banach spaces with the main objective of describing the dual of spaces of bounded operators. In several cases and under a variety of assumptions on X and Y, the answer can best be given as the projective tensor product of X ** and Y *.

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