ERGODIC THEOREMS IN BANACH IDEALS OF COMPACT OPERATORS

Let H be an infinite-dimensional Hilbert space, and let B(H) (K(H)) be the C*-algebra of all bounded (compact) linear operators in H. Let (E, parallel to . parallel to(E)) be a fully symmetric sequence space. If {s(n)(x)}(n=1)(infinity) are the singular values of x is an element of K(H), let C-E = {x is an element of K(H) : {s(n)(x)} is an element of E} with parallel to x parallel to(CE) = parallel to{s(n)(x)}parallel to(E), x is an element of C-E, be the Banach ideal of compact operators generated by E. We show that the averages A(n)(T)(x) = 1/n+1 Sigma(n)(k=0) T-k(x) converge uniformly in C-E for any Dunford-Schwartz operator T and x is an element of C-E. Besides, if 0 <= x is an element of B(H)\K(H), there exists a Dunford-Schwartz operator T such that the sequence {A(n)(T)(x)} does not converge uniformly. We also show that the averages A(n)(T) converge strongly in (CE, parallel to.parallel to(CE)) if and only if E is separable and E not equal l(1) as sets.