Proceedings of the American Mathematical Society

Abstract:

The Kuratowski measure of noncompactness $\alpha$ on an infinite dimensional Banach space $(X,\|\cdot \|)$ assigns to each bounded set $S$ in $X$ a nonnegative real number $\alpha (S)$ by the formula \[ \begin {aligned} \alpha (S)= & \inf \{\delta >0 \mid S=\textstyle {\bigcup ^n_{i=1}} S_i \hbox { for some }S_i\\ & \hbox {with }\textrm {diam}(S_i)\leq \delta ,\hbox { for }1\le i\le n<\infty \}. \end {aligned} \] 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 $\rho (L)=\sup \{|\lambda | \mid \lambda \in \textrm {ess}(L)\}$, where $\textrm {ess}(L)$ denotes the essential spectrum of $L$. One can also define \[ \beta (L)=\inf \{\lambda >0 \mid \beta (LS) \le \lambda \beta (S)\hbox { for every }S\in {\mathcal {B}(X)}\}. \] 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 \[ \rho (L)=\limsup _{m\to \infty }\beta (L^mB)^{1/m} =\inf \{\lambda >0 \mid \lim _{m\to \infty } \lambda ^{-m} \beta (L^mB)=0\}. \]

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