Algebras Generated by Toeplitz Operators on the Hardy Space over the Siegel Domain
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Complex Analysis and Operator Theory
Algebras Generated by Toeplitz Operators on the Hardy Space over the Siegel Domain Armando Sánchez-Nungaray1
· Nikolai Vasilevski2
Received: 17 February 2020 / Accepted: 25 September 2020 © Springer Nature Switzerland AG 2020
Abstract Following the approach of Loaiza and Vasilevski (Equ Oper Theory 92(3): 33, 2020, https://doi.org/10.1007/s00020-020-02580-x; “Operator Theory, Functional Analysis and Applications”, Oper Theory Adv Appl 28(2), 2020, to appear), we give two different representations of the Hardy space on the Siegel domain in terms of the Bergman space and terms of the direct integral of the weighted Fock spaces. Based on these representations we describe then various commutative and non-commutative C ∗ -algebras, which are generated by Toeplitz operators acting on the Hardy space over the Siegel domain. Keywords Toeplitz operator · C*-algebra · Bergman space · Siegel domain Mathematics Subject Classification Primary 47B35; Secondary 47L80 · 32A36
1 Introduction In the paper, we study several algebras, C ∗ and Banach, commutative and noncommutative, which are generated by Toeplitz operators, with specific classes of
Communicated by H. Turgay Kaptanoglu. This work was partially supported by CONACYT Project 238630, México. This article is part of the topical collection “Higher Dimensional Geometric Function Theory and Hypercomplex Analysis” edited by Irene Sabadini, Michael Shapiro and Daniele Struppa.
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Nikolai Vasilevski [email protected] Armando Sánchez-Nungaray [email protected]
1
Facultad de Matemáticas, Universidad Veracruzana, Lomas del Estadio S/N, Zona Universitaria, 91000 Xalapa, Ver., México
2
Departamento de Matemáticas, CINVESTAV, Apartado Postal 14-740, 07000 México, D.F., México 0123456789().: V,-vol
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symbols, that act on the multidimensional Hardy space. The interest in this problem was primarily caused by the following observation. The classical result by Brown and Halmos [8] implies that there is no nontrivial commutative C ∗ -algebra generated by operators acting on the Hardy space H 2 (S 1 ), while there are only two commutative Banach algebras. One of them is generated by Toeplitz operators with analytic symbols, and the other one is generated by Toeplitz operators with conjugate analytic symbols. At the same time, as it was quite unexpectedly observed recently, there are many nontrivial commutative C ∗ -algebras generated by Toeplitz operators acting on the weighted Bergman spaces on the unit disk, as well as, many nontrivial commutative both C ∗ and Banach algebras generated by Toeplitz operators acting on the weighted Bergman spaces on the multidimensional unit ball. Thus a natural question emerges: what is the situation with commutative C ∗ and Banach algebras generated by Toeplitz operators on the multidimensional Hardy space? The first step in this direction was made in [1], where Z. Akkar, following all the reasonings in the characterization of commutative C ∗ -algeb
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