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Sharpened forms of an inequality of von Neumann. (English) Zbl 0631.47004

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).
Reviewer: J.R.L.Webb

MSC:

47A30 Norms (inequalities, more than one norm, etc.) of linear operators
47A60 Functional calculus for linear operators
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References:

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