Wandering Montel theorems for Hilbert space valued holomorphic functions
HTML articles powered by AMS MathViewer
- by Jim Agler and John E. McCarthy
- Proc. Amer. Math. Soc. 146 (2018), 4353-4367
- DOI: https://doi.org/10.1090/proc/14086
- Published electronically: May 24, 2018
- PDF | Request permission
Abstract:
We prove that if $\{ u^k \}$ is a sequence of holomorphic functions that takes values in an infinite dimensional Hilbert space $\mathcal {H}$, there are unitaries $\{ U^k \}$ on $\mathcal {H}$ so that $U^k u^k$ has a subsequence that converges locally uniformly. We also prove a non-commutative version of this result.References
- J. Agler and J. E. M\raise.45exc Carthy. Global holomorphic functions in several non-commuting variables II, Canadian Math. Bull., to appear, arXiv:1706.09973.
- Jim Agler and John E. McCarthy, Global holomorphic functions in several noncommuting variables, Canad. J. Math. 67 (2015), no. 2, 241–285. MR 3314834, DOI 10.4153/CJM-2014-024-1
- Jim Agler and John E. McCarthy, The implicit function theorem and free algebraic sets, Trans. Amer. Math. Soc. 368 (2016), no. 5, 3157–3175. MR 3451873, DOI 10.1090/tran/6546
- D. Alpay and D. S. Kalyuzhnyĭ-Verbovetzkiĭ, Matrix-$J$-unitary non-commutative rational formal power series, The state space method generalizations and applications, Oper. Theory Adv. Appl., vol. 161, Birkhäuser, Basel, 2006, pp. 49–113. MR 2187742, DOI 10.1007/3-7643-7431-4_{2}
- W. Arendt and N. Nikolski, Vector-valued holomorphic functions revisited, Math. Z. 234 (2000), no. 4, 777–805. MR 1778409, DOI 10.1007/s002090050008
- Sriram Balasubramanian, Toeplitz corona and the Douglas property for free functions, J. Math. Anal. Appl. 428 (2015), no. 1, 1–11. MR 3326973, DOI 10.1016/j.jmaa.2015.03.005
- J. A. Ball, G. Marx, and V. Vinnikov, Interpolation and transfer function realization for the non-commutative Schur-Agler class, to appear, arXiv:1602.00762.
- Joseph A. Ball, Gilbert Groenewald, and Tanit Malakorn, Conservative structured noncommutative multidimensional linear systems, The state space method generalizations and applications, Oper. Theory Adv. Appl., vol. 161, Birkhäuser, Basel, 2006, pp. 179–223. MR 2187744, DOI 10.1007/3-7643-7431-4_{4}
- Joseph A. Ball, Gregory Marx, and Victor Vinnikov, Noncommutative reproducing kernel Hilbert spaces, J. Funct. Anal. 271 (2016), no. 7, 1844–1920. MR 3535321, DOI 10.1016/j.jfa.2016.06.010
- J. William Helton, Igor Klep, and Scott McCullough, Proper analytic free maps, J. Funct. Anal. 260 (2011), no. 5, 1476–1490. MR 2749435, DOI 10.1016/j.jfa.2010.11.007
- J. William Helton and Scott McCullough, Every convex free basic semi-algebraic set has an LMI representation, Ann. of Math. (2) 176 (2012), no. 2, 979–1013. MR 2950768, DOI 10.4007/annals.2012.176.2.6
- J. William Helton, J. E. Pascoe, Ryan Tully-Doyle, and Victor Vinnikov, Convex entire noncommutative functions are polynomials of degree two or less, Integral Equations Operator Theory 86 (2016), no. 2, 151–163. MR 3568011, DOI 10.1007/s00020-016-2317-y
- Dmitry S. Kaliuzhnyi-Verbovetskyi and Victor Vinnikov, Foundations of free noncommutative function theory, Mathematical Surveys and Monographs, vol. 199, American Mathematical Society, Providence, RI, 2014. MR 3244229, DOI 10.1090/surv/199
- Kang-Tae Kim and Steven G. Krantz, Normal families of holomorphic functions and mappings on a Banach space, Expo. Math. 21 (2003), no. 3, 193–218. MR 2006009, DOI 10.1016/S0723-0869(03)80001-4
- J. E. Pascoe, The inverse function theorem and the Jacobian conjecture for free analysis, Math. Z. 278 (2014), no. 3-4, 987–994. MR 3278901, DOI 10.1007/s00209-014-1342-2
- J. E. Pascoe and R. Tully-Doyle, Free Pick functions: representations, asymptotic behavior and matrix monotonicity in several noncommuting variables, J. Funct. Anal. 273 (2017), no. 1, 283–328. MR 3646301, DOI 10.1016/j.jfa.2017.04.001
- Gelu Popescu, Free holomorphic functions on the unit ball of $B(\scr H)^n$, J. Funct. Anal. 241 (2006), no. 1, 268–333. MR 2264252, DOI 10.1016/j.jfa.2006.07.004
Bibliographic Information
- Jim Agler
- Affiliation: Department of Mathematics, University of California San Diego, La Jolla, California 92093
- MR Author ID: 216240
- John E. McCarthy
- Affiliation: Department of Mathematics, Washington University, St. Louis, Missouri 63130
- MR Author ID: 271733
- ORCID: 0000-0003-0036-7606
- Received by editor(s): September 8, 2017
- Received by editor(s) in revised form: January 5, 2018
- Published electronically: May 24, 2018
- Additional Notes: The first author was partially supported by National Science Foundation Grant DMS 1665260
The second author was partially supported by National Science Foundation Grant DMS 1565243 - Communicated by: Stephan Ramon Garcia
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 4353-4367
- MSC (2010): Primary 32A19, 47L25
- DOI: https://doi.org/10.1090/proc/14086
- MathSciNet review: 3834664