Propiedades en norma de los operadores en relación con la sumabilidad absoluta en el espacio de Hilbert

Abstract

Let B(H) be the algebra of all bounded linear operators acting on the real separable Hilbert space H and let S={an}∞n=1 be a sequence in H. Consider the linear manifolds CS={AB(H):∑∞n=1||Aan||<∞}and DS={AB(H):{Aan}∞n=1 is summable} of B(H), and MS={xH:∑∞ n=1|(an,x)|<∞} of H. The author proves that CS is not closed, in general, and characterizes the cases when CS is closed in terms of the domain of weak summability of S. If dimMS is finite, or dim(linear spanS) is finite, then CS=DS, but the converse is false.