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Operator-valued analytic functions of constant norm. (English) Zbl 0686.47013
Let $${\mathcal L}(H,K)$$ be the space of bounded linear operators on a Hilbert space H into a Hilbert space K, endowed with the operator norm $$\| \cdot \|$$. For $$A\in {\mathcal L}(H,K)$$ defne E(A) to be the set of all $$B\in {\mathcal L}(H,K)$$ such that $$\| A+\lambda B\| =\| A\|$$ for all complex numbers $$\lambda$$ in some nonempty open disk about the origin. The author shows that E(A) is precisely the set of all operators $$B\in {\mathcal L}(H,K)$$ of the form $B=(I-AA^*)^{1/2}C(I- A^*A)^{1/2},$ where C belongs to $${\mathcal L}(H,K)$$. The result has features in common with the theorem on completing two-by-two operator matrix contradictions [cf., Ch. Davis, W. M. Kahan and H. F. Weinberger, SIAM J. Numer. Anal. 19, 445-469 (1982; Zbl 0491.47003)]. A possible generalization of the result to $$C^*$$-algebras is discussed. The set E(A), which has been defined in an arbitrary Banach space setting [see J. Globevnik, Stud. Math. 53, 29-37 (1975; Zbl 0267.30031)], plays a role in the analysis of operator-valued analytic functions.
Reviewer: M.A.Kaashoek

##### MSC:
 47A20 Dilations, extensions, compressions of linear operators 47A56 Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones)
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##### References:
 [1] R. G. Douglas: On majorization, factorization, and range inclusion of operators on Hilbert space. Proc. Amer. Math. Soc. 17 (1966), 413-415. · Zbl 0146.12503 [2] J. Globevník: On vector-valued analytic functions with constant norm. Studia Math. 53 (1975),29-37. · Zbl 0267.30031 [3] L. A. Harris: Schwarz’s lemma in normed linear spaces. Proc. Nat. Acad. Sci. 62 (1969), 1014-1017. · Zbl 0199.19401 [4] V. Pták, P. Vrbová: Lifting intertwining relations. Integral Equations and Operator Theory, 11 (1988), 128-147. · Zbl 0639.47022 [5] E. Thorp, R. Whitley: The strong maximum modulus theorem for analytic functions into a Banach space. Proc. Amer. Math. Soc. 18 (1967), 640-646. · Zbl 0185.20102
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