Pick interpolation in several variables

Hamilton, Ryan

doi:10.1090/S0002-9939-2013-11571-6

Pick interpolation in several variables
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by Ryan Hamilton PDF

Proc. Amer. Math. Soc. 141 (2013), 2097-2103 Request permission

Abstract:

We investigate the Pick problem for the polydisk and unit ball using dual algebra techniques. Some factorization results for Bergman spaces are used to describe a Pick theorem for any bounded region in $\mathbb {C}^d$.

References

M. B. Abrahamse, The Pick interpolation theorem for finitely connected domains, Michigan Math. J. 26 (1979), no. 2, 195–203. MR 532320, DOI 10.1307/mmj/1029002212
Jim Agler and John E. McCarthy, Nevanlinna-Pick interpolation on the bidisk, J. Reine Angew. Math. 506 (1999), 191–204. MR 1665697, DOI 10.1515/crll.1999.004
Jim Agler and John E. McCarthy, Pick interpolation and Hilbert function spaces, Graduate Studies in Mathematics, vol. 44, American Mathematical Society, Providence, RI, 2002. MR 1882259, DOI 10.1090/gsm/044
Hari Bercovici, The algebra of multiplication operators on Bergman spaces, Arch. Math. (Basel) 48 (1987), no. 2, 165–174. MR 878428, DOI 10.1007/BF01189287
Hari Bercovici, Ciprian Foias, and Carl Pearcy, Dual algebras with applications to invariant subspaces and dilation theory, CBMS Regional Conference Series in Mathematics, vol. 56, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1985. MR 787041, DOI 10.1090/cbms/056
Hari Bercovici and Derek Westwood, The factorization of functions in the polydisc, Houston J. Math. 18 (1992), no. 1, 1–6. MR 1159434
Brian Cole, Keith Lewis, and John Wermer, Pick conditions on a uniform algebra and von Neumann inequalities, J. Funct. Anal. 107 (1992), no. 2, 235–254. MR 1172022, DOI 10.1016/0022-1236(92)90105-R
Kenneth R. Davidson and Ryan Hamilton, Nevanlinna-Pick interpolation and factorization of linear functionals, Integral Equations Operator Theory 70 (2011), no. 1, 125–149. MR 2786738, DOI 10.1007/s00020-011-1862-7
Ronald G. Douglas and Jaydeb Sarkar, Some remarks on the Toeplitz corona problem, Hilbert spaces of analytic functions, CRM Proc. Lecture Notes, vol. 51, Amer. Math. Soc., Providence, RI, 2010, pp. 81–89. MR 2648868, DOI 10.1090/crmp/051/05
Steven G. Krantz, Function theory of several complex variables, 2nd ed., The Wadsworth & Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1992. MR 1162310
Scott McCullough, Nevanlinna-Pick type interpolation in a dual algebra, J. Funct. Anal. 135 (1996), no. 1, 93–131. MR 1367626, DOI 10.1006/jfan.1996.0005
B. Prunaru, A factorization theorem for multiplier algebras of reproducing kernel Hilbert spaces, Canadian Math. Bull., to appear.