Aspects of multivariable operator theory on weighted symmetric Fock spaces
Author
Kaptanoğlu, H. T.
Date
2014Source 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
ArticleItem Usage Stats
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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 moduleBergman
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
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46E20
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47A20
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47B35
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47C15