For a continuous complex-valued function g on the real line without zeros, several notions of a mean winding number are introduced. We give necessary conditions for a Toeplitz operator with matrix-valued symbol G to be semi-Fredholm in terms of mean winding numbers of detG. The matrix function G is assumed to be continuous on the real line, and no other apriori assumptions on it are made.

Let H1 and H2 be Hilbert spaces and let B(H1,H2) be the space of bounded linear operators A : H1 → H2. An operator A ∈ B(H1,H2) is said to be Fredholm if first, the kernel ofA is finite-dimensional, and second the image ofA is closed and has finite codimension. An application of the open mapping theorem shows that the closedness requirement on the image is redundant. A well-known example of Fre...

in this paper‎, ‎we show the stability of gustafson‎, ‎weidmann‎, ‎kato‎, ‎wolf‎, ‎schechter and browder essential spectrum of bounded linear operators on banach spaces which remain invariant under additive perturbations‎ ‎belonging to a broad classes of operators $u$ such $gamma(u^m)

Let A be a linear bounded operator in a Hilbert space H, N(A) and R(A) its null-space and range, and A∗ its adjoint. The operator A is called Fredholm iff dim N(A) = dim N(A∗) := n < ∞ and R(A) and R(A∗) are closed subspaces of H. A simple and short proof is given of the following known result: A is Fredholm iff A = B + F , where B is an isomorphism and F is a finite-rank operator. The proof co...

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 introduce and study some operational quantities associated to a space ideal A. These quantities are used to define generalized semi-Fredholm operators associated to A, and the corresponding perturbation classes which extend the strictly singular and strictly cosingular operators, and we study the generalized Fredholm theory obtained in this way. Finally we present some examples and show that...

We introduce a new concept of perturbation of closed linear subspaces and operators in Banach spaces called uniform λ−adjustment which is weaker than perturbations by small gap, operator norm, q−norm, and K2−approximation. In arbitrary Banach spaces some of the classical Fredholm stability theorems remain true under uniform λ−adjustment, while other fail. However, uniformly λ−adjusted subspaces...