Operator theory and the Carathéodory metric
type: Item
instance of: scholarly article
title: Operator theory and the Carathéodory metric
zbMATH Open document ID: 0722.32013
DOI: 10.1007/BF01231512
author: Jim Agler
published in: Inventiones Mathematicae
publication date: 1990-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143813
review text: Let z and w be distinct points of the bounded domain of holomorphy U in \({\mathbb{C}}^ d\). For each d-tuple M of commuting \(2\times 2\) complex matrices with joint spectrum \(\{\) z,w\(\}\), let \(\theta_ M\) denote the angle between its two (one-dimensional) eigenspaces. The author proves that the Carathéodory distance is given by \[ c_ U(z,w)=\inf \tanh^{-1}(\sin \theta_ M), \] where the infimum is taken over all such M satisfying \(\| h(M)\| \leq \| h\|_{\infty}\) whenever h is a bounded holomorphic function on U. Here \(\| h\|_{\infty}\) denotes the supremum norm, and \(\| h(M)\|\) denotes the operator norm of the \(2\times 2\) matrix h(M) regarded as a linear transformation of \({\mathbb{C}}^ 2.\) The author also provides an analogous formula for the Carathéodory- Reiffen metric. These formulas allow the author to apply complex geometry to operator theory and vice versa. Thus he uses the Carathéodory distance on \(n\times n\) matrix balls to prove that certain contractive unital representations of function algebras are completely contractive, generalizing a result of \textit{V. Paulsen} [J. Oper. Theory 18, No.2, 249- 263 (1987; Zbl 0655.47008)]. Conversely, this result and the dilation techniques of operator theory lead to a new proof for \textit{L. Lempert}'s theorem [Anal. Math. 8, 257-261 (1982; Zbl 0509.32015)] about equality of the Carathéodory and Kobayashi distances when U is convex. Finally the author indicates how Lempert's theorem can be regarded as the case \(n=2\) for interpolating n distinct points of a bounded convex domain by a totally geodesic analytic disc.
Mathematics Subject Classification ID: 32F45
Mathematics Subject Classification ID: 47A20
Mathematics Subject Classification ID: 47A25
zbMATH DE Number: 4190139
zbMATH Keywords: Kobayashi distance
zbMATH Keywords: complete spectral domain
zbMATH Keywords: Carathéodory distance
zbMATH Keywords: Carathéodory-Reiffen metric
reviewed by: Theodore J. Barth
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4190139