Extreme Contractions on Finite-Dimensional Polygonal Banach Spaces

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

is an

n n

-dimensional polygonal Banach space and

Y Y

is any normed linear space and

T \in L(X,Y) T ∈ L ( X , Y )

is an extreme contraction, then

T T

attains norm at

n n

linearly independent extreme points of

B_{X} B X

. Moreover, if

T T

attains norm at

n n

linearly independent extreme points

x_1, x_2, \ldots, x_n x 1 , x 2 , … , x n

of

B_X B X

and does not attain norm at any other extreme point of

B_X B X

, then each

Tx_i T x i

is an extreme point of

B_Y. 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 X

and any finite-dimensional polygonal Banach space

Y Y

, 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.