Let A be a bounded operator on a separable, infinite dimensional Hilbert space H. Moreover, assume that A is not in the set C + K(H) of operators expressible as a sum of a scalar multiple of identity and a compact operator. What is the smallest similarity invariant semigroup containing operator A? Equivalently, which operators can be expressed as products of operators, similar to A? A partial a...

Co-author: Lúıs Castro We will present a study of the Fredholm property for matrix Wiener-Hopf plus/minus Hankel operators with semi-almost periodic Fourier symbols. The Fredholm property will be described based on certain factorizations of the representatives at infinity of the original symbols, and on the geometrical mean values of the corresponding factors. The Fredholm indices of those oper...

In this paper the concept of unbounded Fredholm operators on Hilbert C∗modules over an arbitrary C∗-algebra is discussed and the Atkinson theorem is generalized for bounded and unbounded Feredholm operators on Hilbert C∗-modules over C∗-algebras of compact operators. In the framework of Hilbert C∗-modules over C∗-algebras of compact operators, the index of an unbounded Fredholm operator and the...

1.1 (Fredholm operators). Let X, Y be real Banach spaces and denote their dual spaces by X, Y . A bounded linear operator D : X → Y is called Fredholm if it has a closed image and if its kernel and cokernel (the quotient space Y/imD) are finite dimensional. Equivalently, there exists a bounded linear operator T : Y → X such that the operators TD− idX and DT − idY are compact. The Fredholm index...

We revisit the computation of (2-modified) Fredholm determinants for operators with matrix-valued semi-separable integral kernels. The latter occur, for instance, in the form of Green’s functions associated with closed ordinary differential operators on arbitrary intervals on the real line. Our approach determines the (2-modified) Fredholm determinants in terms of solutions of closely associate...

We define a new topology, weaker than the gap topology, on the space of selfadjoint unbounded operators on a separable Hilbert space. We show that the subspace of selfadjoint Fredholm operators represents the functor K from the category of compact spaces to the category of abelian groups and prove a similar result for K. We define the spectral flow of a continuous path of selfadjoint Fredholm o...

We study the C∗-algebra generated by Teoplitz operators with symbols of vanishing (mean) oscillation on the Bergman space of the unit ball. We show that the index calculation for Fredholm operators in this C∗-algebra can be easily and completely reduced to the classic case of Toeplitz operators with symbols that are continuous on the closed unit ball. Moreover, in addition to a number of other ...

We discuss several natural metrics on spaces of unbounded self– adjoint operators and their relations, among them the Riesz and the graph metric. We show that the topologies of the spaces of Fredholm operators resp. invertible operators depend heavily on the metric. Nevertheless we prove that in all cases the spectral flow is up to a normalization the only integer invariant of paths which is pa...