Linear maps preserving the set of Fredholm operators

Let be an infinite-dimensional separable complex Hilbert space and the algebra of all bounded linear operators on . In this paper we characterize surjective linear maps preserving the set of Fredholm operators in both directions. As an application we prove that preserves the essential spectrum if and only if the ideal of all compact operators is invariant under and the induced linear map on the Calkin algebra is either an automorphism, or an anti-automorphism. Moreover, we have, either or for every Fredholm operator .

     

A proof of the Borsuk Antipodal Theorem for Fredholm maps

With only the means of elementary analysis and the Smale-Sard lemma, a direct and self-contained proof of the Borsuk Antipodal Theorem for Fredholm maps is given. The discussion extends the arguments of J. C. Alexander and J. A. Yorke used in their analytical proof of the Antipodal Theorem in $\text{R}$ⁿ.     

Linear maps preserving Fredholm and Atkinson elements of C *-algebras

Let A be a unital C*-algebra of real rank zero and B be a unital semisimple complex Banach algebra. We characterize linear maps from A onto B preserving different essential spectral sets and quantities such as the essential spectrum, the (left, right) essential spectrum, the Weyl spectrum, the index and the essential spectral radius.     

A continuation principle for Fredholm maps I: theory and basics

We prove an abstract and flexible continuation theorem for zeros of parametrized Fredholm maps between Banach spaces. It guarantees not only the existence of zeros to corresponding equations, but also that they form a continuum for parameters from a connected manifold. Our basic tools are transfer maps and a specific topological degree. The main result is tailor-made to solve boundary value problems over infinite time-intervals and for the (global) continuation of homoclinic solutions.     

A continuation principle for Fredholm maps II: application to homoclinic solutions

When studying the behavior of autonomous ordinary differential equations under time-dependent perturbations vanishing for $t\to \pm \infty$, their equilibria generically persist locally as homoclinic solutions. Using an abstract and flexible continuation theorem, we find even global branches of such homoclinic solutions for parametrized nonautonomous ordinary differential equations. Our approach is based on degree-theoretical arguments. In particular, Landesman–Lazer conditions are proposed to obtain the existence of homoclinic solutions by means of a nonzero degree.     

Geometrical properties of nonlinear maps and their application, Part I: The topological degree of quasiruled Fredholm maps on quasicylindrical domains

In this paper we introduce the notion of quasicylindrical domains in Banach spaces and develop a concept of a degree for quasiruled Fredholm mappings on quasicylindrical domains. Note that this quasicylindrical structure appears in a rather natural way, whenever “analytically” given nonlinear pseudodifferential operators are investigated in spaces of sufficiently smooth functions. Moreover, the class of quasiruled Fredholm mappings on quasicylindrical domains is sufficiently large, so that within this framework one can study a quite large class of nonlinear boundary value problems which are related to pseudodifferential operators.     

Fredholm determinants

The article provides with a down to earth exposition of the Fredholm theory with applications to Brownian motion and KdV equation.     

Large Fredholm triples

We give several equivalent condition for Busby extensions of a given algebra to be absorbing, considerably improving our earlier results [G.A. Elliott, D. Kucerovsky, An abstract Brown-Douglas-Fillmore absorption theorem, Pacific J. Math. 198 (2001) 385-409], and establish sufficient conditions for Fredholm triples to be absorbing in a suitable sense. As an application of one of our criteria, we prove a multivariable Brown-Douglas-Fillmore type theorem.     

Fredholm composition operators

Fredholm composition operators on a variety of Hilbert spaces of analytic functions on domains in , are characterized.

     

Fredholm integro-differential equation

We consider numerical solution of an integro-differential equation with nonsmooth initspaial values. Unique solvability in Sobolev spaceW ₂ (0, 1), =1,2, is proved. We establish the rate of convergence of the approximate solution to the exact solution in fractional spacesW ₂ ⁺¹ , 01, with approximation order O(h ^++1/2 ) for 01/2 andO(h ⁺¹ |ln h|1/2, for 1/2 #x2264;1.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 64, pp. 8–16, 1988.     

Fredholm quasilinear manifolds and degree of a Fredholm quasilinear mapping between them

A new class of Banach manifolds and a new class of mappings between them are presented, and the theory of degree of such mappings is given.     

Les operateurs quasi Fredholm: Une generalisation des operateurs semi Fredholm

The present paper is concerned with the study of a new class of linear operators on a Hilbert space: the class of quasi-Fredholm operators, which contains many operators already studied in the litterature (in particular semi-Fredholm operators). An operatorA is said to be quasi-Fredholm of degreed, if the following conditions are satisfied:

1. For alln greater thand, R(A ⁿ )∩N(A)=R(A ^d )∩N(A);
2. N(A)∩R(A ^d ) is closed inH;
3. R(A)+N(A ^d ) is closed inH.

Two characterisations of quasi-Fredholm operators are given:

1. A is quasi-Fredholm iff there exists a direct decomposition ofH into the sum of two subspacesH ₁ andH ₂ which are invariant underA and such that the restriction ofA toH ₁ is quasi-Fredholm of degree 0 and the restriction ofA toH ₂ is nilpotent (Kato decomposition).
2. A is quasi-Fredholm iff there exists a neighborhoodD of 0 in C such that for all λ≠0 in that neighborhoodA?λI has a generalized inverse which is meromorphic inD?{0} (The generalized inverse is holomorphic inD iffA is of degree 0).

The bulk of the paper is devoted to the proofs of these characterizations and of related results, making use of the theory of operators ranges and of generalized inverses. Most of the results extend easily to the Banach case. The rest of the paper deals with the class of quasi-normal operators, which is closely related to the class of spectral operators. Some applications of the first part of the paper are given in this context.     

Fredholm operator families -I

Integral Equations and Operator Theory - First, we give a characterization of semi-Fredholm operators (i.e. those which are left or right invertible modulo compact operators) on Hausdorff tvs which...     

Fredholm weighted composition operators

We characterize the Fredholm weighted composition operators onC(X). In particular, ifX is a set with some regular property like intervals or balls inR ⁿ , our characterization implies that a weighted composition operator is Fredholm if and only if it is invertible. This equivalence is true for weighted composition operators onL ^p (), where is a nonatomic measure (1p<).