Birkhoff–James orthogonality and smoothness of bounded linear operators

Abstract

We present a sufficient condition for smoothness of bounded linear operators on Banach spaces for the first time. Let T,A∈B(X,Y), where X is a real Banach space and Y is a real normed linear space. We find sufficient condition for T⊥BA⇔Tx⊥BAx for some x∈SX with ‖Tx‖=‖T‖, and use it to show that T is a smooth point in B(X,Y) if T attains its norm at unique (upto multiplication by scalar) vector x∈SX, Tx is a smooth point of Y and supy∈C‖Ty‖<‖T‖ for all closed subsets C of SX with d(±x,C)>0. For operators on a Hilbert space H we show that T⊥BA⇔Tx⊥BAx for some x∈SH with ‖Tx‖=‖T‖ if and only if the norm attaining set MT={x∈SH:‖Tx‖=‖T‖}=SH0 for some finite dimensional subspace H0 and ‖T‖Ho⊥<‖T‖. We also characterize smoothness of compact operators on normed spaces and bounded linear operators on Hilbert spaces.