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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:
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