On Compact and Fredholm Operators over C*-algebras and a New Topology in the Space of Compact Operators (original) (raw)

On a new topology in the space of Fredholm operators

Annals of Global Analysis and Geometry, 1989

Here Vdenotes a subspace of the Hilbert space 1 2 (C), dim V < o, a, ... , a, e 1 2 (C), and GL(1 2 (C)) denotes the group of invertible operators. Connections of this topology with other possible topologies are investigated. In one of the theorems it is proved that the identity map from the space of Fredholm operators, provided with the uniform topology, to the same space, provided with the F-topology, is a homotopy equivalence.

Semi-Fredholm theory in C*-algebras

arXiv (Cornell University), 2020

Kečkić and Lazović introduced an axiomatic approach to Fredholm theory by considering Fredholm type elements in a unital C*-algebra as a generalization of C*-Fredholm operators on the standard Hilbert C*module introduced by Mishchenko and Fomenko and of Fredholm operators on a properly infinite von Neumann algebra introduced by Breuer. In this paper we establish semi-Fredholm theory in unital C*-algebras as a continuation of the approach by Kečkić and Lazović. We introduce the notion of semi-Fredholm type elements and semi-Weyl type elements. We prove that the difference of the set of semi-Fredholm elements and the set of semi-Weyl elements is open in the norm topology, that the set of semi-Weyl elements is invariant under perturbations by finite type elements, and several other results generalizing their classical counterparts. Also, we illustrate applications of our results to the special case of properly infinite von Neumann algebras and we obtain a generalization of the punctured neighbourhood theorem in this setting.

On semi-Weyl and semi-B-Fredholm operators over C*-algebras

2020

We extend further semi-A-Fredholm theory by generalizing the results from [6], [39] such as the theorem about Riesz points. Moreover, we obtain an analogue of the results from [17] in the setting of non-adjointable operators. Finally, we provide several examples of semi-A-Fredholm and semi- A-Weyl operators. We give also the examples of semi-A-Fredholm operators with non-closed image.

On New Approach to Fredholm theory in unital C*-algebras

arXiv (Cornell University), 2023

Motivated by Fredholm theory on the standard Hilbert module over a unital C *-algebra introduced by Mishchenko and Fomenko, we provide a new approach to axiomatic Fredholm theory in unital C *-algebras established by Kečkić and Lazović in [28]. Our approach is equivalent to the approach introduced by Kečkić and Lazović, however, we provide new proofs which are motivated by the proofs given by Mishchenko and Fomenko in [39].

On non-adjointable semi-Weyl and semi-B-Fredholm operators over C*-algebras

arXiv: Functional Analysis, 2020

We extend further semi-A-Fredholm theory by generalizing the results from classical semi-Weyl theory on Hilbert spaces. Moreover, we obtain an analogue of the results from [17] in the setting of non-adjointable operators. Finally, we provide several examples on semi-A-Fredholm and semi- A-Weyl operators over a unital C*-algebra A. We give also the examples of semi-A-Fredholm operators with non-closed image.

On Operators with Closed Range and Semi-Fredholm Operators Over W*-Algebras

Russian Journal of Mathematical Physics, 2020

In this paper we consider A-Fredholm and semi-A-Fredholm operators on Hilbert C *-modules over a W *-algebra A defined in [3],[10]. Using the assumption that A is a W *-algebra (and not an arbitrary C *-algebra) such as a generalization of Schechter-Lebow characterization of semi-Fredholm operators and a generalization of "punctured neighbourhood" theorem, as well as some other results that generalize their classical counterparts. We consider both adjointable and non adjointable semi-Fredholm operators over W *-algebras. Moreover, we also work with general bounded, adjointable operators with closed ranges over C *-algebras and prove a generalization to Hilbert C *-modules of the result in [1] on Hilbert spaces.

Spectral Properties of Compact Operators

Asian Research Journal of Mathematics , 2022

The spectral properties of a compact operator T : X −→ Y on a normed linear space resemble those of square matrices. For a compact operator, the spectral properties can be treated fairly completely in the sense that Fredholm's famous theory of integral equations may be extended to linear functional equations T x − λx = y with a complex parameter λ. This paper has studied and investigated the spectral properties of compact operators in Hilbert spaces. The spectral properties of compact linear operators are relatively simple generalization of the eigenvalues of finite matrices. As a result, the paper has given a number of corresponding propositions and interesting facts which are used to prove basic properties of compact operators. The Fredholm theory has been introduced to investigate the solvability of linear integral equations involving compact operators.

On Various Generalizations of Semi-$${\mathcal {A}}$$-Fredholm Operators

Complex Analysis and Operator Theory, 2020

Starting from the definition of A-Fredholm and semi-A-Fredholm operator on the standard module over a unital C * algebra A, introduced in [8] and [4], we construct various generalizations of these operators and obtain several results as an analogue or a generalization of some of the results in [1], [2], [3], [7]. Moreover, we study also non-adjointable semi-A-Fredholm operators as a natural continuation of the work in [6] on non-adjointable A-Fredholm operators and obtain an analogue or a generalization in this setting of the results in [4], [5].