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1
Content available remote

Perturbation theory relative to a Banach algebra of operators

100%
Barnes B. A.
Studia Mathematica
|
1993
|
tom 106
|
nr 2
153-174
EN
Let ℬ be a Banach algebra of bounded linear operators on a Banach space X. Let S be a closed linear operator in X, and let R be a linear operator in X. In this paper the spectral and Fredholm theory relative to ℬ of the perturbed operator S + R is developed. In particular, the situation where R is S-inessential relative to ℬ is studied. Several examples are given to illustrate the usefulness of these concepts.
2
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The coincidence index for fundamentally contractible multivalued maps with nonconvex values

88%
Gabor D.
Annales Polonici Mathematici
|
2000
|
tom 75
|
nr 2
143-166
EN
We study a coincidence problem of the form A(x) ∈ ϕ (x), where A is a linear Fredholm operator with nonnegative index between Banach spaces and ϕ is a multivalued A-fundamentally contractible map (in particular, it is not necessarily compact). The main tool is a coincidence index, which becomes the well known Leray-Schauder fixed point index when A=id and ϕ is a compact singlevalued map. An application to boundary value problems for differential equations in Banach spaces is given.
3
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Closed operators affiliated with a Banach algebra of operators

88%
Barnes B. A.
Studia Mathematica
|
1991-1992
|
tom 101
|
nr 3
215-240
EN
Let ℬ be a Banach algebra of bounded linear operators on a Banach space X. If S is a closed operator in X such that (λ - S)^{-1} ∈ ℬ for some number λ, then S is affiliated with ℬ. The object of this paper is to study the spectral theory and Fredholm theory relative to ℬ of an operator which is affiliated with ℬ. Also, applications are given to semigroups of operators which are contained in ℬ.
4
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Local properties of the solution set of the operator equation in Banach spaces in a neighbourhood of a bifurcation point

75%
Janczewska J.
Open Mathematics
|
2004
|
tom 2
|
nr 4
561-572
EN
In this work we study the problem of the existence of bifurcation in the solution set of the equation F(x, λ)=0, where F: X×R k →Y is a C 2-smooth operator, X and Y are Banach spaces such that X⊂Y. Moreover, there is given a scalar product 〈·,·〉: Y×Y→R 1 that is continuous with respect to the norms in X and Y. We show that under some conditions there is bifurcation at a point (0, λ0)∈X×R k and we describe the solution set of the studied equation in a small neighbourhood of this point.
5
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The geometry of Kato Grassmannians

63%
Bojarski B., Khimshiashvili G.
Open Mathematics
|
2005
|
tom 3
|
nr 4
705-717
EN
We discuss Fredholm pairs of subspaces and associated Grassmannians in a Hilbert space. Relations between several existing definitions of Fredholm pairs are established as well as some basic geometric properties of the Kato Grassmannian. It is also shown that the so-called restricted Grassmannian can be endowed with a natural Fredholm structure making it into a Fredholm Hilbert manifold.
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