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A Note on the Alexander Theorem on the Complex Plane

Sylwester Zając (2012)

Bulletin of the Polish Academy of Sciences. Mathematics

We investigate the Banach manifold consisting of complex r functions on the unit disc having boundary values in a given one-dimensional submanifold of the plane. We show that ∂/∂λ̅ restricted to that submanifold is a Fredholm mapping. Moreover, for any such function we obtain a relation between its homotopy class and the Fredholm index.

Fredholm spectrum and growth of cohomology groups

Jörg Eschmeier (2008)

Studia Mathematica

Let T ∈ L(E)ⁿ be a commuting tuple of bounded linear operators on a complex Banach space E and let σ F ( T ) = σ ( T ) σ e ( T ) be the non-essential spectrum of T. We show that, for each connected component M of the manifold R e g ( σ F ( T ) ) of all smooth points of σ F ( T ) , there is a number p ∈ 0, ..., n such that, for each point z ∈ M, the dimensions of the cohomology groups H p ( ( z - T ) k , E ) grow at least like the sequence ( k d ) k 1 with d = dim M.

Sur la topologie de l’espace des opérateurs pseudodifférentiels inversibles d’ordre 0

Frédéric Rochon (2008)

Annales de l’institut Fourier

Les groupes d’homotopie du groupe (stabilisé) G 0 ( X ) des opérateurs pseudodifférentiels inversibles d’ordre zéro agissant sur une variété compacte sans bord X sont calculés en termes de la K -théorie du fibré cosphérique S * X . Du même coup, on montre que le sous-groupe des perturbations compactes inversibles de l’identité est faiblement rétractile dans G 0 ( X ) . Les résultats sont aussi adaptés au cas des opérateurs suspendus. Des applications à la théorie de l’indice et pour le déterminant résiduel de Simon Scott...

The geometry of Kato Grassmannians

Bogdan Bojarski, Giorgi Khimshiashvili (2005)

Open Mathematics

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.

The Morse-Sard-Brown Theorem for Functionals on Bounded Fréchet-Finsler Manifolds

Kaveh Eftekharinasab (2015)

Communications in Mathematics

In this paper we study Lipschitz-Fredholm vector fields on bounded Fréchet-Finsler manifolds. In this context we generalize the Morse-Sard-Brown theorem, asserting that if M is a connected smooth bounded Fréchet-Finsler manifold endowed with a connection 𝒦 and if ξ is a smooth Lipschitz-Fredholm vector field on M with respect to 𝒦 which satisfies condition (WCV), then, for any smooth functional l on M which is associated to ξ , the set of the critical values of l is of first category in . Therefore,...

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