Commutative algebras generated by Toeplitz operators - Gelfand theory and spectral properties
Final Report Abstract
In this project we were analyzing Toeplitz operators acting on the standard weighted Bergman space over the complex unit ball Bn in Cn. Moreover, we studied the structure of C∗- or Banach algebras generated by such operators. In most cases we assumed the invariance of the operator symbols under certain group actions. Typically this induces an additional structure of the algebra (e.g. commutativity). Part A of the project deals with commutative Banach-Toeplitz algebras subordinate to the quasi-elliptic automorphism subgroup of Aut(Bn) and their Gelfand theory. As a main result we explicitly describe the maximal ideal space and the Gelfand transform on a dense subalgebra. A series of applications is given, such as the proof of the inverse closedness (in some cases) or a description of the radical. Part B is concerned with Toeplitz operators having radial symbols and algebras generated by them. All elements of the algebra act as diagonal operators with respect to the standard monomial basis of the Bergman space. We characterize the eigenvalue sequences of radial Toeplitz operators by solving a weighted Hausdorff moment problem and we characterize the C∗-algebra generated by (radial) Toeplitz operators. As a result this algebra is isomorphic isometric to a C∗-algebra of slowly oscillating sequences (in the sense of Schmidt). The analysis in B required a series of approximation results of bounded operators by Toeplitz operators which are based on the notion of the (m, λ)-Berezin transform. Part C studies C∗-algebras generated by Toeplitz operators on the weighted Bergman space over B2. The operator symbols are assumed to be invariant under a torus action. Each element of the algebra leaves invariant the summands in a suitable decomposition of the Bergman space over B2. We consider different symbol classes and the corresponding operator algebras. In a noncommutative setting (corresponding to the classical Toeplitz algebra) we give a complete list of the irreducible representations. As a surprise to us an interesting series of representations appears that is induced by a kind of ”quantization effect”.
Publications
- On the structure of commutative Banach algebras generated by Toeplitz operators on the unit ball. Quasi-elliptic case. I: Generating subalgebras, J. Funct. Anal. 265 (2013), 2956-2990
W. Bauer, N. Vasilevski
(See online at https://doi.org/10.1016/j.jfa.2013.08.006) - (m, λ)-Berezin transform and approximation of operators on weighted Bergman spaces over the unit ball, Operator Theory: Advances and Appliations 240 (2014), 45-68
W. Bauer, N. Vasilevski, C. H. Yanez
(See online at https://doi.org/10.1007/978-3-319-06266-2_3) - Eigenvalue characterization of radial operators on weighted Bergman spaces over the unit ball, Integr. Equ. Oper. Theory 78 (2014) (2), 271-300
W. Bauer, N. Vasilevski, C. H. Yanez
(See online at https://doi.org/10.1007/s00020-013-2101-1) - On the structure of commutative Banach algebras generated by Toeplitz operators on the unit ball. Quasi-elliptic case. II: Gelfand Theory, Complex Anal. Oper. Theory 9 (2015), 593-630
W. Bauer, N. Vasilevski
(See online at https://doi.org/10.1007/s11785-014-0385-z) - On algebras generated by Toeplitz operators and their representations, J. Funct. Anal. 272(2) (2017), 705-737
W. Bauer, N. Vasilevski
(See online at https://doi.org/10.1016/j.jfa.2016.09.013)
