Sharpened forms of an inequality of von Neumann (Q1094648)

Let H be a complex Hilbert space. The inequality of von Neumann asserts that for a function f analytic on a neighbourhood of the closed unit disk \({\bar \Delta}\), if f(\({\bar \Delta}\))\(\subset \bar D\) then \(\| f(A)\| \leq 1\) for every bounded linear operator A on H. An equivalent result holds for the open unit disk with \(\leq\) replaced by \(<\). This paper proves a number of inequalities which improve on the latter version. One corollary yields a sharpened form of Schwarz's lemma (for operators).