In this paper we decompose D into diadic annuli {An : n ∈ N} and consider the class Sp,q of Toeplitz operators Tϕ for which the sequence of Schatten norms Tϕ n p n∈N belongs to q , where ϕn = ϕχA n . We study the boundedness and compactness of the operators in Sp,q and we describe the operators Tϕ, ϕ ≥ 0 in these spaces in terms of weighted Herz norms of the averaging operator of the symbols ϕ. (2000). Primary 47B35, Secondary 46E30. Mathematics Subject Classification

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Let D be the unit disk. For A ⊂ L∞(D) containing piecewise continuous functions, we study the C * − algebras RA generated by the Bergman projection for D and operators of multiplication by functions of A. These algebras are related to the algebra generated by more than two projections depending on how many limits a function has at a boundary point. We find the description of the symbol algebra of RA, denoted here by RA. Interesting facts about representations of RA are found and we construct a special family of coefficients such that the algebra RA has irreducible representations of predefined dimensions. Mathematics Subject Classification (2000). 47B34, 47L25, 44A15.

The classical result by Brown and Halmos (J Reine Angew Math 213:8-102, 1964) implies that there is no nontrivial commutative C * -algebra generated by Toeplitz 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 there are many nontrivial commutative C * and Banach algebras generated by Toeplitz operators acting on the Bergman spaces. In the paper we show that the situation on the multidimensional Hardy space H 2 (S 2n−1 ) is drastically different from the one on H 2 (S 1 ). We represent the Hardy space H 2 (S 2n−1 ) as a direct sum of weighted Bergman spaces over B n−1 , and use the already known results for the Bergman space operators to describe a variety of nontrivial commutative C * and Banach algebras generated by Toeplitz operators acting on the multidimensional Hardy space H 2 (S 2n−1 ).

We study Hilbert spaces of super-holomorphic functions (including anti-commuting Grassmann variables) in the setting of bounded symmetric domains, more precisely for the matrix ball of arbitrary size. Our main results concern the classification of irreducible representations of the associated Toeplitz C * -algebra and an explicit decomposition of the super-Bergman space as a direct sum of vector-valued (ordinary) Bergman spaces.