The Commutant of Some Shift Operators
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Complex Analysis and Operator Theory
The Commutant of Some Shift Operators Ali Abkar1 · Guangfu Cao2 · Kehe Zhu3,4 Received: 7 March 2020 / Accepted: 11 July 2020 © Springer Nature Switzerland AG 2020
Abstract We obtain a characterization for the commutant of certain weighted shift operators of higher multiplicity. The characterization is based on the realization of the operator as multiplication by monomials on weighted Bergman spaces of the unit disk. Keywords Weighted shift · Multiplication operator · Commutant · Weighted Bergman space Mathematics Subject Classification 47B37 · 47B38 · 46E22 · 30H10
Communicated by Raymond Mortini. Cao was supported by NNSF of China (Grant No. 11671152). Zhu was supported by NNSF of China (Grant No. 11720101003), STU Scientific Research Foundation for Talents (Grant No. NTF17009), Guangdong Basic and Applied Basic Research Foundation (No. 2019A1515110178), and Key Projects of Fundamental Research in Universities of Guangdong Province (Grant No. 2018KZDXM034).
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Kehe Zhu [email protected] Ali Abkar [email protected] Guangfu Cao [email protected]
1
Department of Pure Mathematics, Faculty of Science, Imam Khomeini International University, Qazvin 34149, Iran
2
Department of Mathematics, South China Agricultural University, Guangzhou 510640, Guangdong, China
3
Department of Mathematics and Statistics, State University of New York, Albany, NY 12222, USA
4
Department of Mathematics, Shantou University, Shantou 515063, Guangdong, China 0123456789().: V,-vol
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A. Abkar et al.
1 Introduction Let S be a bounded linear operator on a Hilbert space H . The commutant of S, denoted (S) , is the algebra of all bounded linear operators T on H such that T S = ST . There are very few examples of operators whose commutant algebra can be explicitly described. One such example is the classical unilateral shift S on l 2 , which maps any sequence (a0 , a1 , a2 , . . .) to (0, a0 , a1 , . . .). If S is realized as the operator of multiplication by z on the Hardy space H 2 of the unit disk D, then the commutant of S consists exactly of multiplication operators Mϕ : H 2 → H 2 , where ϕ ∈ H ∞ , the algebra of all bounded analytic functions on D. Thus the commutant of the unilateral shift can be identified with H ∞ . The operator Mz : H 2 → H 2 is the simplest example of shift operators and the simplest example of Toeplitz operators. The commutant of more general shift operators and more general Toeplitz operators has been studied in the literature by many authors, see [1,2,5,12–14,16,17,20,21,24]. As can be seen from these papers, even the description of the commutant of the Toeplitz operator M B on H 2 induced by a Blaschke product with two zeros is highly non-trivial. One of the motivations for the study of the commutant of an operator S is the attempt to identify all reducing subspaces of S, which is equivalent to the identification of all orthogonal projections in (S) . Once again, there are not many examples for which this can be achieved. However, there has been so
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