Abstract

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 XX is an nn-dimensional polygonal Banach space and YY is any normed linear space and T∈L(X,Y)T \in L(X,Y) is an extreme contraction, then TT attains norm at nn linearly independent extreme points of BXB_{X}. Moreover, if TT attains norm at nn linearly independent extreme points x1,x2,…,xnx_1, x_2, \ldots, x_n of BXB_X and does not attain norm at any other extreme point of BXB_X, then each TxiTx_i is an extreme point of BY.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 XX and any finite-dimensional polygonal Banach space YY, the pair (X,Y)(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.

Author information

Contact details are reproduced from the original publication and may be historical.

Debmalya Sain

Dept. of Mathematics, Indian Institute of Science, Bengaluru 560012, Karnataka, India

saindebmalya@gmail.com

Kallol Paul

Dept. of Mathematics, Jadavpur University, Kolkata 700032, West Bengal, India

kalloldada@gmail.com

Keywords

  • Extreme contractions
  • polygonal Banach spaces
  • strict convexity
  • Hilbert spaces

Mathematics Subject Classification

46B20; 47L05

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Suggested citation

D. Sain, A. Ray, K. Paul. “Extreme Contractions on Finite-Dimensional Polygonal Banach Spaces.” Journal of Convex Analysis 26 (2019), No. 3, 877–885. https://doi.org/10.68381/jca26046

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Published by Heldermann Verlag, 2019. Rights now held by Banach Press LLC.