LE MATEMATICHE Vol. LIII (1998)  Fasc. II, pp. 305317 CONVERGENCE IN GENERALIZED SCHATTEN CLASSES OF OPERATORS J.M. ALMIRA - F. P ´ EREZ ACOSTA In this paper we give de�nitions for several generalized Schatten classes of operators and prove that several properties from the classes S p can be generalized to them. The main goal is characterization of convergence in these classes. We also will prove that they are quasi-Banach spaces. 1. Introduction. Let H be a Hilbert space and consider B( H , H ), the algebra of bounded linear operators T : H → H . Let T ∈ B( H , H ) be a compact operator. Then the numbers α n (T ) = inf x 1 ,..., x n−1 ∈H sup �x �=1,(x , x 1 )=...=(x , x n−1 )=0 �Tx � and λ n (T ) = inf rank( S )<n �T − S � both coincide with the n -th s -number of T (which is the n -th eigenvalue of (T ∗ T ) 1 2 ) ([3]). The Schatten class of operators S p (1 ≤ p < ∞) is then de�ned (see [3]) by S p ={T ∈ B( H , H ) : {λ n (T )} ∞ n=1 ∈ l p }. Entrato in Redazione il 30 aprile 1998. The �rst author was supported by Gobierno Aut´ onomo de Canarias. brought to you by CORE View metadata, citation and similar papers at core.ac.uk provided by Le Matematiche (Dipartimento di Matematica e Informatica, Università degli Studi di...