Algebra generated by Toeplitz operators with $${\mathbb {T}}$$ T -invariant symbols
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ORIGINAL ARTICLE
Algebra generated by Toeplitz operators with T-invariant symbols Nikolai Vasilevski1 Received: 23 April 2020 / Accepted: 10 August 2020 Sociedad Matemática Mexicana 2020
Abstract We study the structure of the C -algebras generated by Toeplitz operators acting on the weighted Bergman space A2k ðB2 Þ on the two-dimensional unit ball, whose symbols are invariant under the action of the group T. We consider three principally different basic cases of its action t : ðz1 ; z2 Þ7!ðtz1 ; tk2 z2 Þ, with k2 ¼ 1; 0; 1. The properties of the corresponding Toeplitz operators as well as the structure of the C algebra generated by them turn out to be drastically different for these three cases. Keywords Toeplitz operators T-invariant symbol Bergman space
Mathematics Subject Classification Primary 47B35 Secondary 32A36
1 Introduction The standard definition of the Toeplitz operators fits, as a particular case, to the following general operator theory setting. Let H be a Hilbert space, H0 be its closed subspace, and let P be the orthogonal projection of H onto H0 . Given a bounded linear operator A, acting on H, introduce its compression TA :¼ PAjH0 onto the subspace H0 . The main question of interest is how the properties of the operator A (or of an algebra generated by such operators) determine the properties of its compression (or the algebra generated by the corresponding compressions). In the context of Toeplitz operators on the Bergman space, H is a weighted space L2 ðD; dvk Þ on some domain of holomorphy D, H0 is its subspace A2k ðDÞ consisting of all analytic in D functions, called the (weighted) Bergman space. The orthogonal & Nikolai Vasilevski [email protected] 1
Departamento de Matema´ticas, CINVESTAV, Apartado Postal 14-740, 07000 Mexico, D.F., Mexico
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projection Bk of L2 ðD; dvk Þ onto the Bergman space A2k ðDÞ is called the Bergman projection. The operator A is just the multiplication operator aI, by a 2 L1 ðDÞ. Note that, although the properties of the multiplication operator aI, as well as, the properties of the (commutative !) C -algebra generated by them are very simple and completely understood, the properties of its compression Toeplitz operator Ta ¼ Bk aIjA2 ðDÞ , with generic L1 -symbol a, still do not fully understood and cannot k be characterized in detail. At the same time, many deep partial results have already been obtained in this direction. One of the common strategies here is to select a subclass S L1 ðDÞ of symbols, obeying some specific properties, and study the Toeplitz operators, with symbols from S, as well as the algebras generated by them. Of course, one of the best possible options would be to find a symbol class S, so that the corresponding Toeplitz operators also generate a commutative C -algebra. Fortunately, as it was quite unexpectedly observed recently, such classes S do exist. Recall, in this connection, the case of the unit ball Bn , the domain treated in the paper. The result (see [9] for details) reads as follows. Gi
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