Abstract
We study Toeplitz operators with radial and separately radial symbols on the weighted Bergman spaces on the unit ball. The unitary equivalence of such operators with multiplication operators on \(\ell ^2\) spaces was previously obtained by analytic methods in Grudsky et al. (J Oper Theory 49(2):325–346, 2003) and Quiroga-Barranco and Vasilevski (Integral Equ Oper Theory 59(1):67–98, 2007), respectively. We prove that the same constructions can be performed with a purely representation theoretic approach to obtain the same conclusions and formulas. However, our method is shorter, more elementary and more elucidating.
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Dawson, M., Ólafsson, G., Quiroga-Barranco, R.: Commuting Toeplitz operators on bounded symmetric domains and multiplicity-free restrictions of holomorphic discrete series. J. Funct. Anal. 268(7), 1711–1732 (2015)
Goodman, R., Wallach, N.R.: Symmetry, representations, and invariants. In: Graduate Texts in Mathematics, vol. 255. Springer. Dordrecht (2009)
Grudsky, S., Karapetyants, A., Vasilevski, N.: Toeplitz operators on the unit ball in \({\mathbb{C}}^{n}\) with radial symbols. J. Oper. Theory 49(2), 325–346 (2003)
Grudsky, S., Quiroga-Barranco, R., Vasilevski, N.: Commutative \(C^*\)-algebras of Toeplitz operators and quantization on the unit disk. J. Funct. Anal. 234(1), 1–44 (2006)
Quiroga-Barranco, R., Vasilevski, N.: Commutative algebras of Toeplitz operators on the Reinhardt domains. Integr. Equ. Oper. Theory 59(1), 67–98 (2007)
Quiroga-Barranco, R., Vasilevski, N.: Commutative \(C^{*}\)-algebras of Toeplitz operators on the unit ball. I. Bargmann-type transforms and spectral representations of Toeplitz operators. Integr. Equ. Oper. Theory 59(3), 379–419 (2007)
Quiroga-Barranco, R., Vasilevski, N.: Commutative C-algebras of Toeplitz operators on the unit ball. II. Geometry of the level sets of symbols. Integr. Equ. Oper. Theory 60(1), 89–132 (2008)
Wallach, N.R.: The analytic continuation of the discrete series I. Trans. Am. Math. Soc. 251, 1–17 (1979)
Zhu, K.: Spaces of holomorphic functions in the unit ball. In: Graduate Texts in Mathematics, vol. 226. Springer, New York (2005)
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Research supported by SNI and a Conacyt Grant.
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To Sergei Grudsky on the occasion of his 60th birthday.
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Quiroga-Barranco, R. Separately radial and radial Toeplitz operators on the unit ball and representation theory. Bol. Soc. Mat. Mex. 22, 605–623 (2016). https://doi.org/10.1007/s40590-016-0111-0
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DOI: https://doi.org/10.1007/s40590-016-0111-0