Ergodic theorems in Banach ideals of compact operators

Abstract

Let H be an innite-dimensional Hilbert space, and let B(H) (K(H)) be the C algebra of all bounded (compact) linear operators in H. Let (E, • E ) be a fully symmetric sequence space. If {sn(x)} ∞ n=1 are the singular values of x ∈ K(H), let CE = {x ∈ K(H) : {sn(x)} ∈ E} with x C E = {sn(x)} E , x ∈ CE, be the Banach ideal of compact operators generated by E. We show that the averages An(T )(converge uniformly in CE for any Dunford-Schwartz operator T and x ∈ CE. Besides, if 0 ≤ x ∈ B(H)\K(H), there exists a Dunford-Schwartz operator T such that the sequence {An(T )(x)} does not converge uniformly. We also show that the averages An(T ) converge strongly in (CE, • C E ) if and only if E is separable and E = l 1 as sets.