Pick reproducing kernel Hilbert spaces. In particular, we exhibit uncountably many discs in the ball of \(\ell ^2\) which are multiplier biholomorphic but have non-isomorphic multiplier algebras. We also show that there are closed discs in the ball of \(\ell ^2\) which are varieties, and examine their multiplier algebras. In finite balls, we provide a counterpoint to a result of Alpay, Putinar and Vinnikov by providing a proper rational biholomorphism of the disc onto a variety \(V\) in \({\mathbb {B}}_2\) such that the multiplier algebra is not all of \(H^\infty (V)\) . We also show that the transversality property, which is one of their hypotheses, is a consequence of the smoothness that they require." />
Multipliers of Embedded Discs
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  • 作者：Kenneth R. Davidson ; Michael Hartz…
  • 关键词：Non ; selfadjoint operator algebras ; Reproducing kernel Hilbert spaces ; Multiplier algebra ; Isomorphism problem ; Embedded discs ; 47L30 ; 47A13 ; 46E22
  • 刊名：Complex Analysis and Operator Theory
  • 出版年：2015
  • 出版时间：February 2015
  • 年：2015
  • 卷：9
  • 期：2
  • 页码：287-321
  • 全文大小：366 KB
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    3. Alpay, D., Putinar, M., Vinnikov, V.: A Hilbert space approach to bounded analytic extension in the ball. Commun. Pure Appl. Anal. 2, 139-45 (2003) CrossRef
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    5. Cohn, H.: Conformal mappings on Riemann surfaces. McGraw-Hill Book Company, New York (1967)
    6. Arveson, W.: Subalgebras of \(C^*\) -algebras III: multivariable operator theory. Acta Math. 181, 159-28 (1998) CrossRef
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    8. Cowen, C., MacCluer, B.: Some problems on composition operators. Contemp. Math. 213, 17-5 (1998) CrossRef
    9. Davidson, K., Pitts, D.: Nevanlinna–Pick interpolation for non-commutative analytic Toeplitz algebras. Integral Equ. Oper. Theory 31, 321-37 (1998) CrossRef
    10. Davidson, K., Ramsey, C., Shalit, O.: The isomorphism problem for some universal operator algebras. Adv. Math. 228, 167-18 (2011) CrossRef
    11. Davidson, K., Ramsey, C., Shalit, O.: Operator algebras for analytic varieties. Trans. Am. Math. Soc. (to appear). arXiv:1201. 4072v4 [math.OA]
    12. Duren, P., Schuster, A., Vukoti?, D.: On uniformly discrete sequences in the disk, Quadrature domains and their applications. In: Oper. Theory Adv. Appl., vol. 156, pp. 131-50. Birkh?user, Basel (2005)
    13. Feller, W.: An introduction to probability theory and its applications, vol. I, 3rd ed. Wiley, New York (1968)
    14. Garnett, J.: Two remarks on interpolation by bounded analytic functions, Banach spaces of analytic functions (Proc. Pelczynski Conf., Kent State Univ., Kent, Ohio, 1976). In: Lecture Notes in Math., vol. 604, pp. 32-0. Springer, Berlin (1977)
    15. Garnett, J.: Bounded Analytic Functions. Academic Press, Dublin (1981)
    16. Gilbarg, D., Trudinger, N.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (2001)
    17. Han, Q.: A basic course in partial differential equations. In: Graduate Studies in Mathematics, vol. 120. American Mathematical Society, USA (2011)
    18. Hartz, M.: Topological isomorphisms for some universal operator algebras. J. Funct. Anal. 263, 3564-587 (2012) CrossRef
    19. Hoffman, K.: Banach spaces of analytic functions. Prentice Hall, Englewood Cliffs (1962)
    20. Kerr, M., McCarthy, J., Shalit, O.: On the isomorphism question for complete multiplier algebras. Integral Eqs. Oper. Theory 76, 39-3 (2013) CrossRef
    21. Rudin, W.: Function Theory in the Unit Ball of \({\mathbb{C}}^n\) . Springer, Berlin (1980) CrossRef
    22. Whittaker, E.T., Watson, G.N.: A course of modern analysis. In: An Introduction to the General Theory of Infinite Processes and of Analytic Functions; With an Account of the Principal Transcendent
  • 刊物类别：Mathematics and Statistics
  • 刊物主题：Mathematics
    Mathematics
    Operator Theory
    Analysis
  • 出版者：Birkh盲user Basel
  • ISSN：1661-8262
文摘
We consider a number of examples of multiplier algebras on Hilbert spaces associated to discs embedded into a complex ball in order to examine the isomorphism problem for multiplier algebras on complete Nevanlinna-a href='/search?dc.title=Pick&facet-content-type=ReferenceWorkEntry&sortOrder=relevance' class='reference-link webtrekk-track' gaCategory="Internal link" gaLabel="Pick" gaAction="reference keyword">Pick reproducing kernel Hilbert spaces. In particular, we exhibit uncountably many discs in the ball of \(\ell ^2\) which are multiplier biholomorphic but have non-isomorphic multiplier algebras. We also show that there are closed discs in the ball of \(\ell ^2\) which are varieties, and examine their multiplier algebras. In finite balls, we provide a counterpoint to a result of Alpay, Putinar and Vinnikov by providing a proper rational biholomorphism of the disc onto a variety \(V\) in \({\mathbb {B}}_2\) such that the multiplier algebra is not all of \(H^\infty (V)\) . We also show that the transversality property, which is one of their hypotheses, is a consequence of the smoothness that they require.

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