Ascent, descent and roots of Fredholm operators

Bertram Yood

Displaying similar documents to “Ascent, descent and roots of Fredholm operators”

A note on the index of $B$ -Fredholm operators

M. Berkani, Dagmar Medková (2004)

Mathematica Bohemica

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From Corollary 3.5 in [Berkani, M; Sarih, M.; Studia Math. 148 (2001), 251–257] we know that if $S$ , $T$ are commuting $B$ -Fredholm operators acting on a Banach space $X$ , then $S T$ is a $B$ -Fredholm operator. In this note we show that in general we do not have $error (S T) = error (S) + error (T)$ , contrarily to what has been announced in Theorem 3.2 in [Berkani, M; Proc. Amer. Math. Soc. 130 (2002), 1717–1723]. However, if there exist $U, V \in L (X)$ such that $S$ , $T$ , $U$ , $V$ are commuting and $U S + V T = I$ , then $error (S T) = error (S) + error (T)$ , where $error$ stands for the index of a $B$ -Fredholm...

A Note on the Alexander Theorem on the Complex Plane

Sylwester Zając (2012)

Bulletin of the Polish Academy of Sciences. Mathematics

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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.

Extended Weyl type theorems

M. Berkani, H. Zariouh (2009)

Mathematica Bohemica

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An operator $T$ acting on a Banach space $X$ possesses property $(gw)$ if $σ_{a} (T) ∖ σ_{{SBF}_{+}^{-}} (T) = E (T),$ where $σ_{a} (T)$ is the approximate point spectrum of $T$ , $σ_{{SBF}_{+}^{-}} (T)$ is the essential semi-B-Fredholm spectrum of $T$ and $E (T)$ is the set of all isolated eigenvalues of $T .$ In this paper we introduce and study two new properties $(b)$ and $(gb)$ in connection with Weyl type theorems, which are analogous respectively to Browder’s theorem and generalized Browder’s theorem. Among other, we prove that if $T$ is a bounded linear operator acting on a Banach space...

A note on preservation of spectra for two given operators

Carlos Carpintero, Alexander Gutiérrez, Ennis Rosas, José Sanabria (2020)

Mathematica Bohemica

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We study the relationships between the spectra derived from Fredholm theory corresponding to two given bounded linear operators acting on the same space. The main goal of this paper is to obtain sufficient conditions for which the spectra derived from Fredholm theory and other parts of the spectra corresponding to two given operators are preserved. As an application of our results, we give conditions for which the above mentioned spectra corresponding to two multiplication operators...

Fredholm theory and transversality for the parametrized and for the $S^{1}$ -invariant symplectic action

Frédéric Bourgeois, Alexandru Oancea (2010)

Journal of the European Mathematical Society

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We study the parametrized Hamiltonian action functional for finite-dimensional families of Hamiltonians. We show that the linearized operator for the $L^{2}$ -gradient lines is Fredholm and surjective, for a generic choice of Hamiltonian and almost complex structure. We also establish the Fredholm property and transversality for generic $S^{1}$ -invariant families of Hamiltonians and almost complex structures, parametrized by odd-dimensional spheres. This is a foundational result used to define $S^{1}$ -equivariant...

On Fredholm alternative for certain quasilinear boundary value problems

Pavel Drábek (2002)

Mathematica Bohemica

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We study the Dirichlet boundary value problem for the $p$ -Laplacian of the form $- Δ_{p} u - λ_{1} {| u |}^{p - 2} u = f in Ω, u = 0 on \partial Ω,$ where $Ω \subset ℝ^{N}$ is a bounded domain with smooth boundary $\partial Ω$ , $N \geq 1$ , $p > 1$ , $f \in C (\bar{Ω})$ and $λ_{1} > 0$ is the first eigenvalue of $Δ_{p}$ . We study the geometry of the energy functional $E_{p} (u) = \frac{1}{p} \int_{Ω} {| \nabla u |}^{p} - \frac{λ_{1}}{p} \int_{Ω} {| u |}^{p} - \int_{Ω} f u$ and show the difference between the case $1 < p < 2$ and the case $p > 2$ . We also give the characterization of the right hand sides $f$ for which the above Dirichlet problem is solvable and has multiple solutions.

Displaying similar documents to “Ascent, descent and roots of Fredholm operators”

A note on the index of B -Fredholm operators

A Note on the Alexander Theorem on the Complex Plane

Extended Weyl type theorems

A note on preservation of spectra for two given operators

Fredholm theory and transversality for the parametrized and for the S 1 -invariant symplectic action

On Fredholm alternative for certain quasilinear boundary value problems

A note on the index of $B$ -Fredholm operators

Fredholm theory and transversality for the parametrized and for the $S^{1}$ -invariant symplectic action