We show the existence of Banach spaces X, Y such that the set of strictly singular operators ᏿(X,Y) (resp., the set of strictly cosingular operators Ꮿ᏿(X,Y)) would be strictly included in F + (X,Y) (resp., F − (X,Y)) for the nonempty class of closed densely defined upper semi-Fredholm operators Φ + (X,Y) (resp., for the nonempty class of closed densely defined lower semi-Fredholm operators Φ − ...

(2) If K ∈ B(X) is compact, then for all λ ∈ C \ {0}, K − λ1 is Fredholm with index zero. (3) The shift operator S± ∈ B(`p) for 1 ≤ p ≤ ∞ defined by (S±x)n = xn±1 is Fredholm with index ±1. (4) If X,Y are finite dimensional and T ∈ B(X,Y ), then by the Rank-Nullity Theorem, ind(T ) = dim(X)− dim(Y ). Lemma 3. Suppose E,F ⊆ X are closed subspaces with F finite dimensional. (1) The subspace E + F...

We propose a new method, called the textit{the weighted space method}, for the study of the generalized Hyers-Ulam-Rassias stability. We use this method for a nonlinear functional equation, for Volterra and Fredholm integral operators.

We consider Dirichlet-to-Neumann maps associated with (not necessarily self-adjoint) Schrödinger operators in L2(Ω; dnx), where Ω ⊂ Rn, n = 2, 3, are open sets with a compact, nonempty boundary ∂Ω satisfying certain regularity conditions. As an application we describe a reduction of a certain ratio of modified Fredholm perturbation determinants associated with operators in L2(Ω; dnx) to modifie...

The index formula of Douglas is a formula which expresses the index of a Fredholm Toeplitz operator with a discontinuous symbol as the limit of the indices of a family of Fredholm Toeplitz operators with continuous symbols. This paper is concerned with Toeplitz operators on the Bergman space A of the unit ball of C. The symbols are supposed to be matrix functions with entries in C+H∞ or to be c...

We compare various topologies on the space of (possibly unbounded) Fredholm selfadjoint operators and explain their K-theoretic relevance.∗ Introduction The work of Atiyah and Singer on the index of elliptic operators on manifolds has singled out the role of the space of bounded Fredholm operators in topology. It is a classifying space for a very useful functor, the topological K-theory. This m...