Abstract
We survey various Nevanlinna-Pick type interpolation problems for contractive multipliers between two vector Fock spaces of noncommutative formal power series. An adaptation of Potapov’s method leads to a chain-matrix linear-fractional parametrization for the set of all solutions for the case where the Pick operator is invertible. The most general problem considered here is a noncommutative multivariable analogue of the Abstract Interpolation Problem formulated by Katsnelson, Kheifets and Yuditskii for the single-variable case; we obtain a Redheffer-type linear-fractional parametrization for the set of all solutions (including in degenerate cases) via an adaptation of ideas of Arov and Grossman.
Dedicated to Professor Victor Katsnelson on the occasion of his 75th birthday
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
We note that the term McCullough-Trent (McCT) inner rather than inner is used in [9] for additional emphasis of the distinction between these different notions of inner, but here for simplicity we contract McCT-inner to inner (see [26] where this notion of inner appears in the commutative context of multipliers on the Drury-Arveson space).
References
J. Agler and J.E. McCarthy, Pick interpolation for free holomorphic functions, Amer. J. Math. 137 (6) (2015), 241–285.
A. Arias and G. Popescu, Noncommutative interpolation and Poisson transforms, Israel J. Math. 115 (2000), 205–234.
N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc., 68 (1950), 337–404.
D.Z. Arov and L.Z. Grossman, Scattering matrices in the theory of unitary extensions of isometric operators, Soviet Math. Dokl. 270 (1983), 17–20.
D.Z. Arov and L.Z. Grossman, Scattering matrices in the theory of unitary extensions of isometric operators, Math. Nachr. 157 (1992), 105–123.
J.A. Ball and V. Bolotnikov, Interpolation problems for Schur multipliers on the Drury-Arveson space: from Nevanlinna-Pick to abstract interpolation problem, Integral Equations Operator Theory 62 (2008), no. 3, 301–349.
J.A. Ball and V. Bolotnikov, Nevanlinna-Pick interpolation for Schur-Agler-class functions on domains with matrix polynomial defining function inC n, New York J. Math. 11 (2005), 1–44.
J.A. Ball and V. Bolotnikov, Interpolation in the noncommutative Schur-Agler class, J. Operator Theory 58 (2007) no. 1, 83–126.
J.A. Ball and V. Bolotnikov, Hardy-space function theory, operator model theory, and dissipative linear systems: the multivariable, free-noncommutative, weighted Bergman-space setting, available at arXiv 1906.02814.
J.A. Ball, V. Bolotnikov and Q. Fang, Multivariable backward-shift invariant subspaces and observability operators, Multidimens. Syst. Signal Process. 18 (2007), no. 4, 191–248.
J.A. Ball, V. Bolotnikov, and Q. Fang, Schur-class multipliers on the Fock space: de Branges-Rovnyak reproducing kernel spaces and transfer-function realizations, in: Operator Theory, Structured Matrices, and Dilations: Tiberiu Constantinescu Memorial Volume (Ed. M. Bakonyi, A. Gheondea, M. Putinar, and J. Rovnyak), pp. 85–114, Theta Series in Advances Mathematics, Theta, Bucharest, 2007.
J.A. Ball, G. Marx, and V. Vinnikov, Noncommutative reproducing kernel Hilbert spaces, J. Func. Anal. 271 (2016), 1844–1920.
J.A. Ball, G. Marx, and V. Vinnikov, Interpolation and transfer-function realization for the noncommutative Schur-Agler class, in: Operator Theory in Different Settings and Related Applications (ed. R. Duduchava, M.A. Kaashoek, N. Vasilevski, V. Vinnikov) pp. 23–116, Oper. Theory Adv. Appl. 262, Birkhäuser/Springer, Cham, 2018.
J.A. Ball, T.T. Trent and V. Vinnikov, Interpolation and commutant lifting for multipliers on reproducing kernels Hilbert spaces, in: Operator Theory and Analysis (Ed. H. Bart, I. Gohberg and A.C.M. Ran), Oper. Theory Adv. Appl. 122, Birkhäuser, Basel, 2001, pp. 89–138.
J.A. Ball and V. Vinnikov, Formal reproducing kernel Hilbert spaces: the commutative and noncommutative settings, in: Reproducing Kernel Spaces and Applications (Ed. D. Alpay), pp. 77–134, OT 143, Birkhäuser, Basel, 2003.
L. de Branges and J. Rovnyak, Square summable power series, Holt, Rinehart and Winston, New-York, 1966.
K.R. Davidson and D. R. Pitts, Invariant subspaces and hyper-reflexivity for free semigroup algebras, Proc. London Math. Soc. 78 (1999) no. 2, 401–430.
M. A. Dritschel and J. Rovnyak, Extension theorems for contraction operators on Kreı̆n spaces, in: Extension and interpolation of linear operators and matrix functions, pp. 221–305, Oper. Theory Adv. Appl., 47, Birkhäuser, Basel, 1990.
Y. P. Ginzburg, OnJ-contractive operator functions, Dokl. Akad. Nauk SSSR 117 (1957), 171–173.
V. Katsnelson, A. Kheifets and P. Yuditskii, An abstract interpolation problem and extension theory of isometric operators, in: Operators in Spaces of Functions and Problems in Function Theory (V.A. Marchenko, ed.), 146, Naukova Dumka, Kiev, 1987, pp. 83–96. English transl. in: Topics in Interpolation Theory (H. Dym, B. Fritzsche, V. Katsnelson and B. Kirstein, eds.), Oper. Theory Adv. Appl. 95, Birkhäuser, Basel, 1997, pp. 283–298.
A. Kheifets, The Parseval equality in an abstract problem of interpolation, and the union of open systems, Teor. Funktsii Funktsional. Anal. i Prilozhen., 49, 1988, 112–120, and 50, 1988, 98–103. English transl. in: J. Soviet Math., 49(4), 1990, 114–1120 and 49(6), 1990, 1307–1310.
A. Kheifets Scattering matrices and Parseval Equality in Abstract Interpolation Problem, Ph.D. Thesis, 1990, Kharkov State University.
A. Kheifets, The abstract interpolation problem and applications, in: Holomorphic spaces (Ed. D. Sarason, S. Axler, J. McCarthy), pp. 351–379, Cambridge Univ. Press, Cambridge, 1998.
A. Kheifets and P. Yuditskii, An analysis and extension of V.P. Potapov’s approach to interpolation problems with applications to the generalized bitangential Schur-Nevanlinna-Pick problem andj-inner-outer factorization, in: Matrix and Operator Valued Functions (Ed. I. Gohberg and L.A. Sakhnovich), Oper. Theory Adv. Appl. 72, Birkhäuser, Basel, 1994, pp. 133–161.
R.B. Leech, Factorization of analytic functions and operator inequalities, Integral Equations and Operator Theory 78 (2014) no. 1, 71–73.
S. McCullough and T.T. Trent, Invariant subspaces and Nevanlinna-Pick kernels, J. Funct. Anal. 178 (2000), no. 1, 226–249.
G. Popescu, Characteristic functions for infinite sequences of noncommuting operators, J. Operator Theory 22 (1989), no. 1, 51–71.
G. Popescu, On intertwining dilations for sequences of noncommuting operators, J. Math. Anal. Appl. 167 (1992) no. 2, 382–402.
G. Popescu, Multivariable Nehari problem and interpolation, J. Funct. Anal. 200 (2003), no. 2, 536–581.
G. Popescu, Entropy and multivariable interpolation, Mem. Amer. Math. Soc. 184 (2006), no. 868.
G. Popescu, Free Holomorphic functions on the unit ball of \(B({\mathcal {H}})^n\), J. Funct. Anal. 241 (2006), 268–333.
G. Popescu, Free Holomorphic functions on the unit ball of \(B({\mathcal {H}})^n\)II, J. Funct. Anal. 258 (2010), 1513–1578.
V. P. Potapov, The multiplicative structure ofJ-contractive matrix functions, Amer. Math. Soc. Transl. 15 (1960) 131–243.
G. Salomon, O.M. Shalit, and E. Shamovich, Algebras of bounded noncommutative analytic functions on subvarieties of the noncommutative unit ball, Trans. Amer. Math. Soc. 370 no. 12 (2018), 8639–8690.
D. Sarason, Generalized interpolation inH ∞, Trans. Amer. Math. Soc. 127 (1967), 179–203.
Acknowledgements
The work of the second author was partially supported by Simons Foundation Grant 524539.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Ball, J.A., Bolotnikov, V. (2020). Interpolation by Contractive Multipliers Between Fock Spaces. In: Alpay, D., Fritzsche, B., Kirstein, B. (eds) Complex Function Theory, Operator Theory, Schur Analysis and Systems Theory. Operator Theory: Advances and Applications(), vol 280. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-44819-6_7
Download citation
DOI: https://doi.org/10.1007/978-3-030-44819-6_7
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-030-44818-9
Online ISBN: 978-3-030-44819-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)