Birkhoff–James orthogonality and smoothness of bounded linear operators

We present a sufficient condition for smoothness of bounded linear operators on Banach spaces for the first time. Let T,A∈B(X,Y)T,A∈B(X,Y), where XX is a real Banach space and YY is a real normed linear space. We find sufficient condition for T⊥BA⇔Tx⊥BAxT⊥BA⇔Tx⊥BAx for some x∈SXx∈SX with ‖Tx‖=‖T‖‖Tx‖=‖T‖, and use it to show that T   is a smooth point in B(X,Y)B(X,Y) if T   attains its norm at unique (upto multiplication by scalar) vector x∈SXx∈SX, Tx   is a smooth point of YY and supy∈C‖Ty‖<‖T‖supy∈C‖Ty‖<‖T‖ for all closed subsets C   of SXSX with d(±x,C)>0d(±x,C)>0. For operators on a Hilbert space HH we show that T⊥BA⇔Tx⊥BAxT⊥BA⇔Tx⊥BAx for some x∈SHx∈SH with ‖Tx‖=‖T‖‖Tx‖=‖T‖ if and only if the norm attaining set MT={x∈SH:‖Tx‖=‖T‖}=SH0MT={x∈SH:‖Tx‖=‖T‖}=SH0 for some finite dimensional subspace H0H0 and ‖T‖Ho⊥<‖T‖‖T‖Ho⊥<‖T‖. We also characterize smoothness of compact operators on normed spaces and bounded linear operators on Hilbert spaces.