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An application of the Parrott’s theorem to the geometry of the unit sphere. (English) Zbl 0936.47006
Let \(\mathcal H, \mathcal K\) be Hilbert spaces and \(T\) be a \(2\times 2\) operator matrix acting on \(\mathcal H \oplus \mathcal K,\) with three entries, \(A\in \mathcal B(\mathcal H)\), \(B\in \mathcal B(\mathcal K, \mathcal H)\), \(C\in \mathcal B(\mathcal H, \mathcal K),\) specified and one unknown entry. S. Parrott’s theorem [J. Func. Anal. 30, 311-325 (1978; Zbl 0409.47004)] asserts that \(T\) has a contraction extension iff \(AA^* + BB^* \leq 1\) and \( AA^* + CC^* \leq 1.\) Let \(\mathcal H_i\) be Hilbert spaces, \(\widetilde H = \oplus_{i=1}^n \mathcal H_i\), and \( T = (T_{jk}), S= (S_{jk})\), \(1\leq j,k\leq n,\) be operator matrices acting on \(\widetilde {\mathcal H}.\) If \( Z=(z_{jk})\), \(1\leq j,k \leq n,\) is a scalar matrix let \(Z * S =(z_{jk} S_{jk})\), \(1\leq j,k\leq n.\) An operator matrix \( T=(T_{jk})\) with \(\|T\|=1,\) is called a matrix extreme point of the unit sphere of \(\mathcal B(\widetilde{\mathcal H})\) if \(\|T+Z*S\|\leq 1,\) for any scalar matrix \(Z=(z_{jk})\) with \(|z_{jk}|\leq 1,\) implies \(S=0.\) This notion extends that of complex extreme point [see V. I. Istrăţescu, “Strict convexity and complex strict convexity”, M. Dekker, New York (1984; Zbl 0538.46012)].
The author gives several characterizations of matrix extreme points whose proofs are based on Parrott’s theorem mentioned above. As application he gives a matrix operator version of a theorem of E. Thorp and R. Whitley [Proc. Am. Math. Soc. 18, 640-646 (1967; Zbl 0185.20102)], on strong maximum modulus theorem for analytic functions into Banach spaces.
MSC:
47A56 Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones)
47A20 Dilations, extensions, compressions of linear operators
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