Extreme contractions on finite-dimensional polygonal Banach spaces

Abstract

We explore extreme contractions between finite-dimensional polygonal Banach spaces, from the point of view of attainment of norm of a linear operator. We prove that if $X$ is an $n-$dimensional polygonal Banach space and $Y$ is any Banach space and $T \in L(X,Y)$ is an extreme contraction, then $T$ attains norm at $n$ linearly independent extreme points of $B_{X}.$ Moreover, if $T$ attains norm at $n$ linearly independent extreme points $x_1, x_2, \ldots, x_n$ of $B_X$ and does not attain norm at any other extreme point of $B_X,$ then each $Tx_i$ is an extreme point of $B_Y.$ We completely characterize extreme contractions between a finite-dimensional polygonal Banach space and a strictly convex Banach space. We introduce L-P property for a pair of Banach spaces and show that it has natural connections with our present study. We also prove that for any strictly convex Banach space $X$ and any finite-dimensional polygonal Banach space $Y,$ the pair $(X,Y)$ does not have L-P property. Finally, we obtain a characterization of Hilbert spaces among strictly convex Banach spaces in terms of L-P property.Comment: 9 page