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Inequivalent measures of noncompactness and the radius of the essential spectrum

Authors: John Mallet-Paret and Roger D. Nussbaum
Journal: Proc. Amer. Math. Soc. 139 (2011), 917-930
MSC (2010): Primary 47H08, 46B20; Secondary 46B25, 46B45, 47A10, 47H10
Published electronically: October 29, 2010
MathSciNet review: 2745644
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Abstract: The Kuratowski measure of noncompactness $ \alpha$ on an infinite dimensional Banach space $ (X,\Vert\cdot\Vert)$ assigns to each bounded set $ S$ in $ X$ a nonnegative real number $ \alpha (S)$ by the formula

\begin{equation*} \begin{aligned} \alpha (S)= & \inf \{\delta >0 \mid S=\textsty... ...(S_i)\leq \delta,\hbox{ for }1\le i\le n<\infty \}. \end{aligned}\end{equation*}

In general a map $ \beta$ which assigns to each bounded set $ S$ in $ X$ a nonnegative real number and which shares most of the properties of $ \alpha$ is called a homogeneous measure of noncompactness or homogeneous MNC. Two homogeneous MNC's $ \beta$ and $ \gamma$ on $ X$ are called equivalent if there exist positive constants $ b$ and $ c$ with $ b\beta (S)\leq \gamma (S)\leq c\beta (S)$ for all bounded sets $ S\subset X$. There are many results which prove the equivalence of various homogeneous MNC's. Working with $ X=\ell^p (\mathbb{N})$ where $ 1\leq p\leq \infty$, we give the first examples of homogeneous MNC's which are not equivalent.

Further, if $ X$ is any complex, infinite dimensional Banach space and $ L:X\rightarrow X$ is a bounded linear map, one can define $ \rho (L)=\sup \{\vert\lambda\vert \mid \lambda \in \textrm{ess}(L)\}$, where $ \textrm{ess}(L)$ denotes the essential spectrum of $ L$. One can also define

$\displaystyle \beta (L)=\inf\{\lambda>0 \mid \beta(LS) \le\lambda\beta(S)\hbox{ for every }S\in{\mathcal{B}(X)}\}. $

The formula $ \rho (L)=\displaystyle{\lim_{m\rightarrow \infty}} \beta (L^m)^{1/m}$ is known to be true if $ \beta$ is equivalent to $ \alpha$, the Kuratowski MNC; however, as we show, it is in general false for MNC's which are not equivalent to $ \alpha$. On the other hand, if $ B$ denotes the unit ball in $ X$ and $ \beta$ is any homogeneous MNC, we prove that

$\displaystyle \rho (L)=\limsup_{m\to\infty}\beta(L^mB)^{1/m} =\inf \{\lambda>0 \mid \lim_{m\to \infty} \lambda^{-m} \beta (L^mB)=0\}. $

Our motivation for this study comes from questions concerning eigenvectors of linear and nonlinear cone-preserving maps.

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Additional Information

John Mallet-Paret
Affiliation: Division of Applied Mathematics, Brown University, Providence, Rhode Island 02912

Roger D. Nussbaum
Affiliation: Department of Mathematics, Rutgers University, Piscataway, New Jersey 08854

Keywords: Measure of noncompactness, essential spectral radius, cone map.
Received by editor(s): September 21, 2009
Received by editor(s) in revised form: January 16, 2010
Published electronically: October 29, 2010
Additional Notes: The first author was partially supported by NSF Grant DMS-0500674
The second author was partially supported by NSF Grant DMS-0701171
Communicated by: Nigel J. Kalton
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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