Generalized 𝑞-Fock spaces and structural identities

Alpay, Daniel; Cerejeiras, Paula; Kaehler, Uwe; Schneider, Baruch

doi:10.1090/proc/16720

Generalized $q$-Fock spaces and structural identities
HTML articles powered by AMS MathViewer

by Daniel Alpay, Paula Cerejeiras, Uwe Kaehler and Baruch Schneider

Proc. Amer. Math. Soc.

DOI: https://doi.org/10.1090/proc/16720

Published electronically: April 15, 2024

HTML | PDF | Request permission

Abstract:

Using $q$-calculus we study a family of reproducing kernel Hilbert spaces which interpolate between the Hardy space and the Fock space. We give characterizations of these spaces in terms of classical operators such as integration and backward-shift operators, and their $q$-calculus counterparts. Furthermore, these new spaces allow us to study intertwining operators between classic backward-shift operators and the q-Jackson derivative.

References

Daniel Alpay, The Schur algorithm, reproducing kernel spaces and system theory, SMF/AMS Texts and Monographs, vol. 5, American Mathematical Society, Providence, RI; Société Mathématique de France, Paris, 2001. Translated from the 1998 French original by Stephen S. Wilson. MR 1839648
Daniel Alpay, A complex analysis problem book, Birkhäuser/Springer, Cham, 2016. Second edition [of MR2848640]. MR 3560222, DOI 10.1007/978-3-319-42181-0
Daniel Alpay, Paula Cerejeiras, Uwe Kähler, and Trevor Kling, Commutators on Fock spaces, J. Math. Phys. 64 (2023), no. 4, Paper No. 042102, 21. MR 4569624, DOI 10.1063/5.0080723
Daniel Alpay, Fabrizio Colombo, and Irene Sabadini, The Fock space as a De Branges-Rovnyak space, Integral Equations Operator Theory 91 (2019), no. 6, Paper No. 51, 12. MR 4032209, DOI 10.1007/s00020-019-2550-2
Daniel Alpay, Aad Dijksma, James Rovnyak, and Hendrik de Snoo, Schur functions, operator colligations, and reproducing kernel Pontryagin spaces, Operator Theory: Advances and Applications, vol. 96, Birkhäuser Verlag, Basel, 1997. MR 1465432, DOI 10.1007/978-3-0348-8908-7
Daniel Alpay, Palle Jorgensen, Ron Seager, and Dan Volok, On discrete analytic functions: products, rational functions and reproducing kernels, J. Appl. Math. Comput. 41 (2013), no. 1-2, 393–426. MR 3017129, DOI 10.1007/s12190-012-0608-2
Daniel Alpay and Motke Porat, Generalized Fock spaces and the Stirling numbers, J. Math. Phys. 59 (2018), no. 6, 063509, 12. MR 3816055, DOI 10.1063/1.5035352
N. Alpay, A new characterization of the Hardy space and of other spaces of analytic functions, Istanbul J. Math. 1 (2023), no. 1, 1–11, DOI 10.26650/ijmath.2023.00005.
N. Aronszajn, La théorie des noyaux reproduisants et ses applications. I, Proc. Cambridge Philos. Soc. 39 (1943), 133–153 (French). MR 8639, DOI 10.1017/S0305004100017813
V. Bargmann, On a Hilbert space of analytic functions and an associated integral transform, Comm. Pure Appl. Math. 14 (1961), 187–214. MR 157250, DOI 10.1002/cpa.3160140303
V. Bargmann, Remarks on a Hilbert space of analytic functions, Proc. Nat. Acad. Sci. U.S.A. 48 (1962), 199–204. MR 133009, DOI 10.1073/pnas.48.2.199
Yue Cai, Richard Ehrenborg, and Margaret Readdy, $q$-Stirling identities revisited, Electron. J. Combin. 25 (2018), no. 1, Paper No. 1.37, 18. MR 3785016, DOI 10.37236/6829
Yue Cai and Margaret A. Readdy, $q$-Stirling numbers: a new view, Adv. in Appl. Math. 86 (2017), 50–80. MR 3614191, DOI 10.1016/j.aam.2016.11.007
Peter L. Duren, Theory of $H^{p}$ spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR 268655
Kenneth Hoffman, Banach spaces of analytic functions, Dover Publications, Inc., New York, 1988. Reprint of the 1962 original. MR 1102893
F. H. Jackson, On $q$-definite integrals, Quart. J. 41 (1910), 193–203.
Victor Kac and Pokman Cheung, Quantum calculus, Universitext, Springer-Verlag, New York, 2002. MR 1865777, DOI 10.1007/978-1-4613-0071-7
Toufik Mansour and Matthias Schork, Commutation relations, normal ordering, and Stirling numbers, Discrete Mathematics and its Applications (Boca Raton), CRC Press, Boca Raton, FL, 2016. MR 3408225
Stephen C. Milne, A $q$-analog of restricted growth functions, Dobinski’s equality, and Charlier polynomials, Trans. Amer. Math. Soc. 245 (1978), 89–118. MR 511401, DOI 10.1090/S0002-9947-1978-0511401-8
Walter Rudin, Real and complex analysis, 3rd ed., McGraw-Hill Book Co., New York, 1987. MR 924157
Hans van Leeuwen and Hans Maassen, A $q$ deformation of the Gauss distribution, J. Math. Phys. 36 (1995), no. 9, 4743–4756. MR 1347109, DOI 10.1063/1.530917
Kehe Zhu, Analysis on Fock spaces, Graduate Texts in Mathematics, vol. 263, Springer, New York, 2012. MR 2934601, DOI 10.1007/978-1-4419-8801-0