We study generalized inner functions on a large family of Reproducing Kernel Hilbert Spaces. We show that the only inner functions which are entire are the normalized monomials.
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A. Aleman, M. Hartz, J. E. McCarthy, and D. Richter, The Smirnov class for spaces with complete Pick property, J. London Math. Soc, 96 (2017), 228–242.
C. Bénéteau, M. Fleeman, D. Khavinson, D. Seco, and A. A. Sola, Remarks on inner functions and optimal approximants, Canad. Math. Bull., 61 (2018) 704–716.
P. L. Duren and A. Schuster, Bergman Spaces, Amer. Math. Soc. (Providence, RI, 2004).
O. El Fallah, K. Kellay, J. Mashreghi, and T. Ransford, A Primer on the Dirichlet Space, Cambridge University Press (2014).
E. Fricain, J. Mashreghi, and D. Seco, Cyclicity in reproducing kernel Hilbert spaces of analytic functions, Comput. Methods Funct. Theory, 14 (2014) 665–680.
J. B. Garnett, Bounded Analytic Functions, Academic Press Inc. (1981).
M. T. Nowak, R. Rososzczuk and M. Woloszkiewicz-Cyll, Extremal functions in weighted Bergman spaces, Complex Var. Elliptic Equ., 62 (2017), 98–109.
H. N. Salas, A note on strictly cyclic weighted shifts, Proc. Amer. Math. Soc, 83 (1981), 555–556.
D. Seco, Some problems on optimal approximants, in: Recent Progress on Operator Theory and Approximation in Spaces of Analytic Functions, Contemp. Math., vol. 679, Amer. Math. Soc. (Providence, RI, 2017) pp. 193–206.
H. Shapiro and A. L. Shields, On the zeros of functions with finite Dirichlet integral and some related function spaces, Math. Z., 80 (1962), 217–299.
A. L. Shields, Weighted shift operators and analytic function theory, in: Topics in Operator Theory, Math. Surveys, vol. 13, Amer. Math. Soc. (Providence, RI, 1974), pp. 49–128.
The authors thank I. Efraimidis, M. Hartz and the anonymous referee for careful reading and valuable comments.
The authors acknowledge the financial support by the Severo Ochoa Programme for Centers of Excellence in R&D (SEV-2015-0554) at ICMAT.
The second author is also grateful for support from the Spanish Ministry of Economy and Competitiveness, through grant MTM2016-77710-P.
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Cobos, A., Seco, D. No Entire Inner Functions. Anal Math 46, 39–45 (2020). https://doi.org/10.1007/s10476-020-0015-0
Key words and phrases
- Reproducing kernel Hilbert space
- inner function
- Dirichlet space
Mathematics Subject Classification
- primary 30J05
- secondary 30H05