Sharpened forms of a von Neumann inequality for $\rho$-contractions

Abstract

The purpose of this paper is to give some sharpened forms of the von Neumann inequality for strict $\rho$-contractions which were obtained earlier. Also, some sharpened forms of the Schwarz inequality for strict $\rho$-contractions will be given, and as applications, corresponding inequalities for strict contractions and for uniformly stable operators will be derived. In particular, we recover the results of K. Fan covering the strict contractions, and in the scalar context we find an improved form of the interior Schwarz inequality quoted by R. Osserman .