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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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##### References:
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