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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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References:
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