Abstract
We extend the classical construction of operator colligations and characteristic functions. Consider the group G of finitary block unitary matrices of order α+∞+···+∞ (m times) and its subgroup K ≅ U(∞), which consists of block diagonal unitary matrices with the identity block of order α and a matrix u ∈ U(∞) repeated m times. It turns out that there is a natural multiplication on the space G//K of conjugacy classes. We construct “spectral data” of conjugacy classes, which visualize the multiplication and are sufficient for reconstructing a conjugacy class.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. Beauville, “Determinantal hypersurfaces,” Michigan Math. J., 48:1 (2000), 39–64.
V. M. Brodskij, “On operator nodes and their characteristic functions,” Dokl. Akad. Nauk SSSR, 198 (1971), 16–19; English transl.: Sov. Math. Dokl., 12 (1971), 696–700.
M. S. Brodskij, “Unitary operator colligations and their characteristic functions,” Uspekhi Mat. Nauk, 33:4(202) (1978), 141–168; English transl.: Russian Math. Surveys, 33:4 (1978), 159–191.
H. Dym, Linear Algebra in Action, Amer. Math. Soc., Providence, RI, 2007.
A. A. Gaifullin and Yu. A. Neretin, “Infinite symmetric group and bordisms of pseudomanifolds” J. Topol. Anal. (to appear); https://arxiv.org/abs/1501.04062.
J. B. Garnett, Bounded Analytic Functions, Academic Press, New York–London, 1981.
I. Gohberg, S. Goldberg, and M. A. Kaashoek, Classes of Linear Operators. Vol. II, Birkhauser, Basel, 1993.
Ph. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley Classics Library, John Wiley & Sons, New York, 1994.
M. Hazewinkel, “Lectures on invariants, representations and Lie algebras in systems and control theory” in: Séminaire d’algèbre P. Dubreil et M.-P. Malliavin, 35ème Année (Paris 1982), Lecture Notes in Math., vol. 1029, Springer-Verlag, Berlin–Heidelberg, 1983, 1–36.
N. Hitchin, “Riemann surfaces and integrable systems,” in: Integrable Systems, Clarendon Press, Oxford, 1999, 11–52.
R. S. Ismagilov, “Elementary spherical functions on the groups SL(2, P) over a field P, which is not locally compact with respect to the subgroup of matrices with integral elements,” Izv. Akad. Nauk SSSR, Ser. Mat., 31:2 (1967), 361–390; English transl.: Math. USSR Izv., 1:2 (1967), 349–380.
M. S. Livshits, “On a certain class of linear operators in Hilbert space,” Mat. Sb., 19(61):2 (1946), 239–262; English transl.: Amer. Math. Soc. Transl. (Ser. 2), 13 (1960), 61–83.
M. S. Livshits, “On spectral decomposition of linear nonself-adjoint operators,” Mat. Sb., 34:1 (1954), 145–199; English transl.: Amer. Math. Soc. Transl. (Ser. 2), 5 (1957), 67–114.
C. Martin and R. Hermann, “Applications of algebraic geometry to systems theory: the McMillan degree and Kronecker indices of transfer functions as topological and holomorphic system invariants” SIAM J. Control Optim., 16:5 (1978), 743–755.
Yu. A. Neretin, Categories of Symmetries and Infinite-Dimensional Groups, Oxford University Press, New York, 1996.
Yu. A. Neretin, “Multi-operator colligations and multivariate characteristic functions” Anal. Math. Phys., 1:2–3 (2011), 121–138.
Yu. A. Neretin, “Sphericity and multiplication of double cosets for infinite-dimensional classical groups,” Funkts. Anal. Prilozhen., 45:3 (2011), 79–96; English transl.: Functional Anal. Appl., 45:3 (2011), 225–239.
Yu. A. Neretin, On concentration of convolutions of double cosets at infinite-dimensional limit, http://arxiv.org/abs/1211.6149.
Yu. Neretin, “Infinite tri-symmetric group, multiplication of double cosets, and checker topological field theories,” Int. Math. Res. Not. IMRN, 3 (2012), 501–523.
Yu. A. Neretin, “Infinite-dimensional p-adic groups, semigroups of double cosets, and inner functions on Bruhat–Tits buildings,” Izv. Ross. Akad. Nauk, Ser. Mat., 79:3 (2015), 87–130; English transl.: Russian Acad. Sci. Izv. Math., 79:3 (2015), 512–553.
Yu. A. Neretin, “Infinite symmetric groups and combinatorial constructions of topological field theory type,” Uspekhi Mat. Nauk, 70:4(424) (2015), 143–204; English transl.: Russian Math. Surveys, 70:4 (2015), 715–773.
G. I. Olshanski, “Unitary representations of (G,K)-pairs that are connected with the infinite symmetric group S(∞),” Algebra i Analiz, 1:4 (1989), 178–209; English transl.: Leningrad Math. J., 1:4 (1990), 983–1014.
G. I. Olshanski, “Unitary representations of infinite dimensional pairs (G,K) and the formalism of R. Howe,” in: Representation of Lie Groups and Related Topics, Gordon and Breach, New York, 1990, 269–463.
G. I. Olshanskii, “Caractères généralisés du groupe U(∞) et fonctions intérieures” C. R. Acad. Sci. Paris. Sèr. 1, 313:1 (1991), 9–12.
V. L. Popov and E. B. Vinberg, “Invariant theory,” in: Algebraic Geometry IV, Itogi Nauki i Tekhniki, Sovremennye Problemy Matematiki, Fundamental’nye Napravleniya, VINITI, Moscow, 1989, 137–309; English transl.: Encyclopaedia of Math. Sci., vol. 55, Springer-Verlag, Berlin, 1994.
V. P. Potapov, “The multiplicative structure of J -contractive matrix functions,” Trudy Moskov. Mat. Obshch., 4 (1955), 125–236; English transl.: Amer. Math. Soc. Transl. (2), 15 (1960), 131–243.
C. Procesi, “The invariant theory of n × n matrices,” Adv. in Math., 19:3 (1976), 306–381.
C. Procesi, Lie groups. An Approach Through Invariants and Representations, Springer-Verlag, New York, 2007.
B. Sz.-Nagy and C. Foias, Harmonic analysis of operators on Hilbert space, Akademiai Kiado, Budapest, 1970.
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 51, No. 2, pp. 25–41, 2017 Original Russian Text Copyright © by Yu. A. Neretin
Supported by FWF grants P22122 and P25142.
Rights and permissions
About this article
Cite this article
Neretin, Y.A. Multiplication of conjugacy classes, colligations, and characteristic functions of matrix argument. Funct Anal Its Appl 51, 98–111 (2017). https://doi.org/10.1007/s10688-017-0172-5
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10688-017-0172-5