Ideals of Operators on Complete Inner Product Spaces and on Banach Spaces

Abstract

It is well known that if the composition of operators is taken as multiplication then the space of all bounded linear operators on a Banach space becomes an algebra. The study of this algebra and its subalgebras when the underlying Banach space is a complete inner product space was initiated in the monumental work of F. J. Murray and John von Neumann in 1941. J. W. Calkin’s studies of the two-sided ideals in that algebra, and their related congruences, has supplied the foundation for much of the work on such ideals. A fundamental contribution to the study of operators on Banach spaces was made in the papers of A. Grothendieck where the facts about absolutely summing and p-absolutely summing operators were obtained.