ACTA issues

Which operators approximately annihilate orthonormal bases?

Damir Bakić, Boris Guljaš

Acta Sci. Math. (Szeged) 64:3-4(1998), 601-607

Abstract. A theorem of Fillmore and Williams implies that a bounded operator $A$ on a separable Hilbert space $H$ is compact if and only if it satisfies $\lim_n Ae_n=0$, or equivalently, $\lim_n (Ae_n|e_n)=0$ for each orthonormal basis $(e_n)$ for $H$. In the present note this theorem is reproved using the fact observed by Halmos that each sequence of unit vectors weakly converging to $0$ approximately contains an orthonormal subsequence. It is also noted that the stronger version of the theorem remains true, namely without the continuity assumption on $A$. In the second part of the note the bounded operators for which there exists an orthonormal basis such that either of the above equalities holds are exhibited and completely described.

AMS Subject Classification (1991): 47A10, 47B07

Received February 16, 1998 and in revised form July 27, 1998. (Registered under 3311/2009.)