Smooth, compact operators

Abstract:

It is a result of Holub’s [Math. Ann. 201 (1973), 157-163], that for T a compact operator on a real Hilbert space, T is smooth $\Leftrightarrow \left \| {T{x_1}} \right \| = \left \| {T{x_2}} \right \| = \left \| T \right \|$ for some $\left \| {{x_1}} \right \| = \left \| {{x_2}} \right \| = 1$ implies ${x_1} = \pm \;{x_2}$. We extend this characterization of smooth, compact operators to a large class of Banach spaces, including ${l_p},{L_p}[0,1]$, and $d(a,p)$, with $1 < p < \infty$. We show that for this same class of Banach spaces, one dimensional, norm one functionals in $K{(X)^\ast }$ must be extremal. We also present examples of spaces for which Holub’s condition does not characterize smooth, compact operators.