A Note on the Alexander Theorem on the Complex Plane

Sylwester Zając

Displaying similar documents to “A Note on the Alexander Theorem on the Complex Plane”

Ascent, descent and roots of Fredholm operators

Bertram Yood (2003)

Studia Mathematica

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Let T be a Fredholm operator on a Banach space. Say T is rootless if there is no bounded linear operator S and no positive integer m ≥ 2 such that $S^{m} = T$ . Criteria and examples of rootlessness are given. This leads to a study of ascent and descent whether finite or infinite for T with examples having infinite ascent and descent.

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...

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...

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 “A Note on the Alexander Theorem on the Complex Plane”

Ascent, descent and roots of Fredholm operators

A note on the index of B -Fredholm operators

Extended Weyl type theorems

A note on preservation of spectra for two given operators

On Fredholm alternative for certain quasilinear boundary value problems

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