Local forms of Bishop-Phelps-Bollobás type properties for bilinear maps

2022/10/15 by Uday Shankar Chakraborty, Chakraborty, Uday Shankar
Mathematics · #Advanced Banach Space Theory #Advanced Harmonic Analysis Research #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2210.08192

Abstract

In this paper, we characterize the so called property L_o,o as defined by Dantas and Rueda Zoca, for compact, weak-weak continuous bilinear maps. Motivated by this we weaken this property by defining the weak L_o,o for bilinear maps. We provide equivalence of the weak L_o,o property of (X⊗_πY,ℝ) for linear functionals and that of (X,Y;ℝ) for bilinear forms under certain conditions on X and Y. Moreover, we have also established that under certain preassigned conditions, if a bilnear map T belongs to a class which satisfies the property L_o,o (resp. the weak L_o,o) for bilinear maps, then T^* is a member of a class of operators which satisfy the property L_o,o (resp. the weak L_o,o) for operators.

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