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Analytic operator functions. (English) Zbl 0608.47023
We generalize a number of interesting results obtained by K. Fan [Math. Z. 160, 275-290 (1978; Zbl 0441.30060)], such as Fan’s theorem, operator analogs of Schwarz’s lemma, Pick’s theorem, etc., as well as a remarkable theorem of J. von Neumann [Math. Nachr. 4, 258-281 (1951; Zbl 0042.123)].
Our main result is as follows. Let \(H\) be a complex Hilbert space, and let \(\Delta\) denote the open unit disc in the complex plane, i.e., \(\Delta =\{z:| z| <1\}\). Suppose \(f(z)=\sum^{\infty}_{n=0}B_ nz^ n\) satisfies the following conditions:
(i) \(\{B_ n:\) \(n=0,1,2,...\}\) is a sequence included in a commutative (non-commutative) von Neumann algebra \({\mathcal A}\) acting on H,
(ii) the series \(\sum^{\infty}_{n=0}B_ nz^ n\) converges in the norm for each z in \(\Delta\),
(iii) \(\| f(z)\| <1\), \(\forall z\in \Delta.\)
Then \(\| f(A)\| <1\) holds for every A in the commutant \({\mathcal A}'\) of \({\mathcal A}\) (A\(\in {\mathcal A}\) and \(A\) is normal, respectively) with \(\| A\| <1\). Here \(f(A)\) is defined by \(f(A)=\sum^{\infty}_{n=0}B_ nA^ n\).

MSC:
47A60 Functional calculus for linear operators
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