Journal of Convex Analysis 26 (2019), No. 3, 877--885
Copyright Heldermann Verlag 2019
Extreme Contractions on Finite-Dimensional Polygonal Banach Spaces
Debmalya Sain
Dept. of Mathematics, Indian Institute of Science, Bengaluru 560012, Karnataka, India
saindebmalya@gmail.com
Anubhab Ray
Dept. of Mathematics, Jadavpur University, Kolkata 700032, West Bengal, India
anubhab.jumath@gmail.com
Kallol Paul
Dept. of Mathematics, Jadavpur University, Kolkata 700032, West Bengal, India
kalloldada@gmail.com
[Abstract-pdf]
We explore extreme contractions on 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 normed linear 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 normed linear 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.
Keywords: Extreme contractions, polygonal Banach spaces, strict convexity, Hilbert spaces.
MSC: 46B20; 47L05
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