Schwarz norms for operators

James Patrick Williams

Vol. 24 (1968), No. 1, 181–188

DOI: 10.2140/pjm.1968.24.181

Abstract

Let H be a complex Hilbert space and let ℒ(H) be the algebra of bounded linear operators from H into itself. A norm | | on ℒ(H) which is equivalent to the usual norm will be called a Schwarz norm if the following version of the Schwarz lemma is valid for | |:

SCHWARZ LEMMA If f is analytic and bounded by 1 in |z| < 1, and if f(0) = 0, then |f(T) ≦|T| for each operator T with |T| < 1.

Mathematical Subject Classification

Primary: 46.50

Milestones

Received: 17 March 1967

Published: 1 January 1968