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      Aspects of multivariable operator theory on weighted symmetric Fock spaces

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      Author
      Kaptanoğlu, H. T.
      Date
      2014
      Source Title
      Communications in Contemporary Mathematics
      Print ISSN
      0219-1997
      Electronic ISSN
      1793-6683
      Publisher
      World Scientific Publishing
      Volume
      16
      Issue
      5
      Pages
      1350034-1 - 1350034-49
      Language
      English
      Type
      Article
      Item Usage Stats
      129
      views
      101
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      Abstract
      We obtain all Dirichlet spaces Fq, q ∈ ℝ, of holomorphic functions on the unit ball of ℂN as weighted symmetric Fock spaces over ℂN. We develop the basics of operator theory on these spaces related to shift operators. We do a complete analysis of the effect of q ∈ ℝ in the topics we touch upon. Our approach is concrete and explicit. We use more function theory and reduce many proofs to checking results on diagonal operators on the Fq. We pick out the analytic Hilbert modules from among the Fq. We obtain von Neumann inequalities for row contractions on a Hilbert space with respect to each Fq. We determine the commutants and investigate the almost normality of the shift operators. We prove that the C∗-algebras generated by the shift operators on the Fq fit in exact sequences that are in the same Ext class. We identify the groups K0 and K1 of the Toeplitz algebras on the Fq arising in K-theory. Radial differential operators are prominent throughout. Some of our results, especially those pertaining to lower negative values of q, are new even for N = 1. Many of our results are valid in the more general weighted symmetric Fock spaces Fb that depend on a weight sequence b. © World Scientific Publishing Company.
      Keywords
      Analytic Hilbert module
      Bergman
      Busby invariant
      Commutant
      C∗-algebra
      Dirichlet
      Drury-Arveson
      Extension
      Fock
      Fredholm
      Hardy
      hyponormal
      K-groups
      multiplier
      Radial differential operator
      Reproducing kernel Hilbert space
      Row contraction
      Shift
      Short exact sequence
      Spectrum
      Subnormal
      Toeplitz
      Virtual point
      Vvon Neumann inequality
      47A13
      47B32
      19K33
      32A36
      32A37
      46E20
      46E22
      46L08
      47A20
      47A30
      47A53
      47B35
      47B37
      47B38
      47C15
      Permalink
      http://hdl.handle.net/11693/26637
      Published Version (Please cite this version)
      http://dx.doi.org/10.1142/S021919971350034X
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      • Department of Mathematics 635
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