Inner functions of matrix argument and conjugacy classes in unitary groups

Yu. A. Neretin

Sbornik. Mathematics, Tome 213 (2022) no. 8, pp. 1041-1057

Résumé

Let $\mathrm{B}_n$ denote the set of complex square matrices of order $n$ whose Euclidean operator norms are less than one. Its Shilov boundary is the set $\operatorname{U}(n)$ of all unitary matrices. A holomorphic map $\mathrm{B}_m\to\mathrm{B}_n$ is inner if it sends $\operatorname{U}(m)$ to $\operatorname{U}(n)$. On the other hand we consider the group $\operatorname{U}(n+mj)$ and its subgroup $\operatorname{U}(j)$ that is embedded in $\operatorname{U}(n+mj)$ in the block-diagonal way ($m$ blocks $\operatorname{U}(j)$ and a unit block of size $n$). To any conjugacy class of $\operatorname{U}(n+mj)$ with respect to $\operatorname{U}(j)$ we assign a ‘characteristic function’, which is a rational inner map $\mathrm{B}_m\to\mathrm{B}_n$. We show that the class of inner functions that can be obtained as ‘characteristic functions’ is closed with respect to such natural operations as pointwise direct sums, pointwise products, compositions, substitutions into finite-dimensional representations of general linear groups and so on. We also describe explicitly the corresponding operations on conjugacy classes. Bibliography: 24 titles.

Détail

Keywords: inner functions, operator colligations, classical complex domains, characteristic operator functions, transfer functions.

Bibliographie

[1] A. B. Aleksandrov, “The existence of inner functions in the ball”, Mat. Sb. (N.S.), 118(160):2(6) (1982), 147–163 ; English transl. in Sb. Math., 46:2 (1983), 143–159 | MR | Zbl | DOI

[2] A. B. Aleksandrov, “Inner functions on compact spaces”, Funktsional. Anal. i Prilozhen., 18:2 (1984), 1–13 ; English transl. in Funct. Anal. Appl., 18:2 (1984), 87–98 | MR | Zbl | DOI

[3] D. Alpay, An advanced complex analysis problem book. Topological vector spaces, functional analysis, and Hilbert spaces of analytic functions, Birkhäuser/Springer, Cham, 2015, ix+520 pp. | DOI | MR | Zbl

[4] J. A. Ball and V. Bolotnikov, “Canonical transfer-function realization for Schur-Agler-class functions of the polydisk”, A panorama of modern operator theory and related topics, Birkhäuser/Springer Basel AG, Basel, 2012, 75–122 | DOI | MR | Zbl

[5] H. Bart, “Transfer functions and operator theory”, Linear Algebra Appl., 84 (1986), 33–61 | DOI | MR | Zbl

[6] M. S. Brodskii, “Unitary operator colligations and their characteristic functions”, Uspekhi Mat. Nauk, 33:4(202) (1978), 141–168 ; English transl. in Russian Math. Surveys, 33:4 (1978), 159–191 | MR | Zbl | DOI

[7] V. M. Brodskiĭ, “On operator nodes and their characteristic functions”, Dokl. Akad. Nauk SSSR, 198:1 (1971), 16–19 ; English transl. in Soviet Math. Dokl., 12 (1971), 696–700 | MR | Zbl

[8] J. B. Garnett, Bounded analytic functions, Pure Appl. Math., 96, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York–London, 1981, xvi+467 pp. | MR | Zbl

[9] G. Knese, “Rational inner functions in the Schur-Agler class of the polydisk”, Publ. Mat., 55:2 (2011), 343–357 | DOI | MR | Zbl

[10] M. G. Kreĭn and Ju. L. Šmul'jan (Shmul'yan), “On linear-fractional transformations with operator coefficients”, Mat. Issled. (Kishinev), 2:3 (1967), 64–96 ; English transl. in Amer. Math. Soc. Transl. Ser. 2, 103, Amer. Math. Soc., Providence, RI, 1974, 125–152 | MR | Zbl | DOI

[11] M. S. Livšic (Livshits), “On a class of linear operators in Hilbert space”, Mat. Sb., 19(61):2 (1946), 239–262 ; English transl. in Amer. Math. Soc. Transl. Ser. 2, 13, Amer. Math. Soc., Providence, RI, 1960, 61–83 | MR | Zbl | DOI | MR | Zbl

[12] M. S. Livšic (Livshits), “On the spectral decomposition of linear non-selfadjoint operators”, Mat. Sb., 34(76):1 (1954), 145–199 ; English transl. in Amer. Math. Soc. Transl. Ser. 2, 5, Amer. Math. Soc., Providence, RI, 1957, 67–114 | MR | Zbl | DOI | MR | Zbl

[13] E. Løw, “A construction of inner functions on the unit ball in $C^p$”, Invent. Math., 67:2 (1982), 223–229 | DOI | MR | Zbl

[14] Yu. A. Neretin, Categories of symmetries and infinite-dimensional groups, Èditorial URSS, Moscow, 1998, 431 pp.; English transl. London Math. Soc. Monogr. (N.S.), 16, The Clarendon Press, Oxford Univ. Press, New York, 1996, xiv+417 pp. | MR | Zbl

[15] Yu. A. Neretin, Lectures on Gaussian integral operators and classical groups, EMS Ser. Lect. Math., Eur. Math. Soc. (EMS), Zürich, 2011, xii+559 pp. | DOI | MR | Zbl

[16] Yu. A. Neretin, “Multi-operator colligations and multivariate characteristic functions”, Anal. Math. Phys., 1:2–3 (2011), 121–138 | DOI | MR | Zbl

[17] Yu. A. Neretin, “Sphericity and multiplication of double cosets for infinite-dimensional classical groups”, Funktsional. Anal. i Prilozhen., 45:3 (2011), 79–96 ; English transl. in Funct. Anal. Appl., 45:3 (2011), 225–239 | DOI | MR | Zbl | DOI

[18] Yu. A. Neretin, “Multiplication of conjugacy classes, colligations, and characteristic functions of matrix argument”, Funktsional. Anal. i Prilozhen., 51:2 (2017), 25–41 ; English transl. in Funct. Anal. Appl., 51:2 (2017), 98–111 | DOI | MR | Zbl | DOI

[19] N. I. Nessonov, “Factor-representation of the group $GL(\infty)$ and admissible representations $GL(\infty)^X$”, Mat. Fiz. Anal. Geom., 10:2 (2003), 167–187 (Russian) | MR | Zbl

[20] G. I. Ol'shanskiĭ (Olshanski), “Unitary representations of infinite-dimensional pairs $(G,K)$ and the formalism of R. Howe”, Representation of Lie groups and related topics, Adv. Stud. Contemp. Math., 7, Gordon and Breach, New York, 1990, 269–463 | MR | Zbl

[21] I. I. Piatetskii-Shapiro, Geometry of classical domains and the theory of discrete groups, Fizmatlit, Moscow, 1961, 191 pp. ; English version I. I. Piatetskii-Shapiro, Automorphic functions and the geometry of classical domains, Math. Appl., 8, Gordon and Breach Science Publishers, New York–London–Paris, 1969, viii+264 pp. | MR | Zbl | MR | Zbl

[22] V. P. Potapov, “The multiplicative structure of $J$-contractive matrix functions”, Tr. Mosk. Mat. Obshch., 4, GITTL, Moscow, 1955, 125–236 ; English transl. in Amer. Math. Soc. Transl. Ser. 2, 15, Amer. Math. Soc., Providence, RI, 1960, 131–243 | MR | Zbl | DOI | MR | Zbl

[23] B. Sz.-Nagy and C. Foiaş, Analyse harmonique des opérateurs de l'espace de Hilbert, Akadémiaí Kiadó, Budapest; Masson et Cie, Paris, 1967, xi+373 pp. | MR | Zbl

[24] D. P. Želobenko, Compact Lie groups and their representations, Nauka, Moscow, 1970, 664 pp. ; English transl. in Transl. Math. Monogr., 40, Amer. Math. Soc., Providence, RI, 1973, viii+448 pp. | MR | Zbl | MR | Zbl