# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] Ando, T.; Hara, T., Another approach to the strong perrott theorem, J. math. anal. appl., 171, 125-130, (1992) · Zbl 0782.47003 [2] Constantinescu, T., Schur parameters, factorization and dilation problems, Operator theory: advances and applications, (1996), Birkhäuser Boston · Zbl 0872.47008 [3] Douglas, R.G., On majorization, factorization and range inclusion of operators in Hilbert space, Proc. amer. math. soc., 17, 413-416, (1966) · Zbl 0146.12503 [4] Fillmore, P.A.; Williams, J.P., On operator ranges, Adv. math., 7, 254-281, (1971) · Zbl 0224.47009 [5] Lang, S., Complex analysis, Graduate texts in mathematics, 103, (1985), Springer-Verlag New York [6] Parrott, S., On a quotient norm and the sz. Nagy-foias lifting theorem, J. funct. anal., 30, 311-328, (1978) · Zbl 0409.47004 [7] Thorp, E.; Whitley, R., The strong maximum modulus theorem for analytic functions into a Banach space, Proc. amer. math. soc., 18, 640-646, (1967) · Zbl 0185.20102
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.