### On the local spectral radius in partially ordered Banach spaces

Mirosława Zima (1999)

Czechoslovak Mathematical Journal

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Mirosława Zima (1999)

Czechoslovak Mathematical Journal

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Jan Ligęza (2005)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

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We study the existence of positive solutions of the integral equation $$x\left(t\right)=\mu {\int}_{0}^{1}k(t,s)f(s,x\left(s\right),{x}^{\text{'}}\left(s\right),...,{x}^{(n-1)}\left(s\right))\phantom{\rule{0.166667em}{0ex}}ds,\phantom{\rule{1.0em}{0ex}}n\ge 2$$ in both ${C}^{n-1}[0,1]$ and ${W}^{n-1,p}[0,1]$ spaces, where $p\ge 1$ and $\mu >0$. Throughout this paper $k$ is nonnegative but the nonlinearity $f$ may take negative values. The Krasnosielski fixed point theorem on cone is used.

Vladimír Müller (1997)

Annales Polonici Mathematici

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We prove the ${}_{p}$-spectral radius formula for n-tuples of commuting Banach algebra elements

Mirosława Zima (2014)

Czechoslovak Mathematical Journal

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We establish several inequalities for the spectral radius of a positive commutator of positive operators in a Banach space ordered by a normal and generating cone. The main purpose of this paper is to show that in order to prove the quasi-nilpotency of the commutator we do not have to impose any compactness condition on the operators under consideration. In this way we give a partial answer to the open problem posed in the paper by J. Bračič, R. Drnovšek, Y. B. Farforovskaya, E. L. Rabkin,...

Trung Dinh Tran (2002)

Applications of Mathematics

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The paper defines and studies the Drazin inverse for a closed linear operator $A$ in a Banach space $X$ in the case that $0$ belongs to a spectral set of the spectrum of $A$. Results are applied to extend a result of Krein on a nonhomogeneous second order differential equation in a Banach space.

M. T. Karaev (2006)

Colloquium Mathematicae

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We calculate the spectral multiplicity of the direct sum T⊕ A of a weighted shift operator T on a Banach space Y which is continuously embedded in ${l}^{p}$ and a suitable bounded linear operator A on a Banach space X.

Nikolai Nikolov, Pascal J. Thomas (2008)

Annales Polonici Mathematici

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Given A∈ Ωₙ, the n²-dimensional spectral unit ball, we show that if B is an n×n complex matrix, then B is a “generalized” tangent vector at A to an entire curve in Ωₙ if and only if B is in the tangent cone ${C}_{A}$ to the isospectral variety at A. In the case of Ω₃, the zero set of the Kobayashi-Royden pseudometric is completely described.

Victor S. Shulman, Yuriĭ V. Turovskii (2002)

Studia Mathematica

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The formula $\varrho \left(M\right)=max{\varrho}_{\chi}\left(M\right),r\left(M\right)$ is proved for precompact sets M of weakly compact operators on a Banach space. Here ϱ(M) is the joint spectral radius (the Rota-Strang radius), ${\varrho}_{\chi}\left(M\right)$ is the Hausdorff spectral radius (connected with the Hausdorff measure of noncompactness) and r(M) is the Berger-Wang radius.