Toeplitz Operators on the Weighted Bergman Space over the Two-Dimensional Unit Ball (original) (raw)
Related papers
Integral Equations and Operator Theory, 2012
Let A 2 λ (B n) denote the standard weighted Bergman space over the unit ball B n in C n. New classes of commutative Banach algebras T (λ) which are generated by Toeplitz operators on A 2 λ (B n) have been recently discovered in Vasilevski (Integr Equ Oper Theory 66(1):141-152, 2010). These algebras are induced by the action of the quasi-elliptic group of biholomorphisms of B n. In the present paper we analyze in detail the internal structure of such an algebra in the lowest dimensional case n = 2. We explicitly describe the maximal ideal space and the Gelfand map of T (λ). Since T (λ) is not invariant under the *-operation of L(A 2 λ (B n)) its inverse closedness is not obvious and is proved. We remark that the algebra T (λ) is not semi-simple and we derive its radical. Several applications of our results are given and, in particular, we conclude that the essential spectrum of elements in T (λ) is always connected.
Algebras of Toeplitz Operators on the n-Dimensional Unit Ball
Complex Analysis and Operator Theory
We study C *-algebras generated by Toeplitz operators acting on the standard weighted Bergman space A 2 λ (B n) over the unit ball B n in C n. The symbols f ac of generating operators are assumed to be of a certain product type, see (1.1). By choosing a and c in different function algebras S a and S c over lower dimensional unit balls B ℓ and B n−ℓ , respectively, and by assuming the invariance of a ∈ S a under some torus action we obtain C *-algebras T λ (S a , S c) whose structural properties can be described. In the case of k-quasi-radial functions S a and bounded uniformly continuous or vanishing oscillation symbols S c we describe the structure of elements from the algebra * This work was partially supported by CONACYT Project 238630, México and by DFG (Deutsche Forschungsgemeinschaft), Project BA 3793/4-1. T λ (S a , S c), derive a list of irreducible representations of T λ (S a , S c), and prove completeness of this list in some cases. Some of these representations originate from a "quantization effect", induced by the representation of A 2 λ (B n) as the direct sum of Bergman spaces over a lower dimensional unit ball with growing weight parameter. As an application we derive the essential spectrum and index formulas for matrix-valued operators.
Quasi-radial operators on the weighted Bergman space over the unit ball
Communications in Mathematical Analysis, 2014
We study the so-called quasi-radial operators, i.e., the operators that are invariant under the subgroup of the unitary group U(n) formed by the block-diagonal matrices with unitary blocks of fixed dimensions. The quasi-radial Toeplitz operators appear naturally and play a crucial role under the study of the commutative Banach (not C *) algebras of Toeplitz operators [1, 8]. They form an intermediate class of operators between the Toeplitz operators with radial a = a(r), r = √ |z 1 | 2 +. .. + |z n | 2 , and separately-radial a = a(|z 1 |,. .. , |z n |) symbols.
Complex Analysis and Operator Theory, 2014
Extending our results in Bauer and Vasilevski (J Funct Anal 265(11):2956-2990, 2013) the present paper gives a detailed structural analysis of a class of commutative Banach algebras B k (h) generated by Toeplitz operators on the standard weighted Bergman spaces A 2 λ (B n) over the complex unit ball B n in C n. In the most general situation we explicitly determine the set of maximal ideals of B k (h) and we describe the Gelfand transform on a dense subalgebra. As an application to the spectral theory we prove the inverse closedness of algebras B k (h) in the full algebra of bounded operators on A 2 λ (B n) for certain choices of h. Moreover, it is remarked that B k (h) is not semi-simple. In the case of k = (n) we explicitly describe the radical Rad B n (h) of the algebra B n (h). This result generalizes and simplifies the characterization of Rad B 2 (1), which was given in Bauer and Vasilevski (Integr Equ Oper Theory 74:199-231, 2012). Keywords Weighted Bergman space • Gelfand theory • Commutative Toeplitz algebra • Generalized Berezin transform Communicated by Vladimir Bolotnikov. W. Bauer has been supported by an "Emmy-Noether scholarship" of DFG (Deutsche Forschungsgemeinschaft). N. Vasilevski has been partially supported by CONACYT Project 102800, México.
On algebras generated by Toeplitz operators and their representations
Journal of Functional Analysis, 2017
We study Banach and C *-algebras generated by Toeplitz operators acting on weighted Bergman spaces A 2 λ (B 2) over the complex unit ball B 2 ⊂ C 2. Our key point is an orthogonal decomposition of A 2 λ (B 2) into a countable sum of infinite dimensional spaces, each one of which can be identified with a differently weighted Bergman space A 2 μ (D) over the complex unit disk D. Moreover, all elements of the above algebras leave each of the summands in the above decomposition invariant and their restriction to each level acts as a compact perturbation of a Toeplitz operator on A 2 μ (D). The symbols of the generating Toeplitz operators are chosen to be suitable extensions to B 2 of families S of bounded functions on D. Symbol classes S that generate important classical commutative and non-commutative Toeplitz algebras in L(A 2 μ (D)) are of particular interest. In this paper we discuss various examples. In the case of S = C(D) and S = C(D) ⊗ L ∞ (0, 1) we characterize all irreducible representations of the resulting Toeplitz operator C *-algebras. Their Calkin algebras are described and * This work was partially supported by CONACYT Project 238630, México and through the Deutsche Forschungsgemeinschaft, DFG Sachmittelbeihilfe BA 3793/4-1. 1 index formulas are provided.
2016
We consider the set of all Toeplitz operators acting on the weighted Bergman space over the upper half-plane whose L ∞-symbols depend only on the argument of the polar coordinates. The main result states that the uniform closure of this set coincides with the C *-algebra generated by the above Toeplitz operators and is isometrically isomorphic to the C *-algebra of bounded functions that are very slowly oscillating on the real line in the sense that they are uniformly continuous with respect to the arcsinh-metric on the real line.
Commutative Toeplitz Algebras and Their Gelfand Theory: Old and New Results
Complex Analysis and Operator Theory
We present a survey and new results on the construction and Gelfand theory of commutative Toeplitz algebras over the standard weighted Bergman and Hardy spaces over the unit ball in \mathbb {C}^n$$ C n . As an application we discuss semi-simplicity and the spectral invariance of these algebras. The different function Hilbert spaces are dealt with in parallel in successive chapters so that a direct comparison of the results is possible. As a new aspect of the theory we define commutative Toeplitz algebras over spaces of functions in infinitely many variables and present some structural results. The paper concludes with a short list of open problems in this area of research.
Parabolic Quasi-radial Quasi-homogeneous Symbols and Commutative Algebras of Toeplitz Operators
Topics in Operator Theory, 2010
We describe new Banach (not C* !) algebras generated by Toeplitz operators which are commutative on each weighted Bergman space over the unit ball Bn, where n > 2. For n = 2 all these algebras collapse t o t h e single C*-algebra generated by Toeplitz operators with quasi-parabolic symbols. As a by-product, we describe t h e situations when t h e product of mutually commuting Toeplitz operators is a Toeplitz operator itself. . Primary 47B35; Secondary 47L80, 32A36.
Journal of Functional Analysis, 2013
Extending our results in Bauer and Vasilevski (J Funct Anal 265(11):2956-2990, 2013) the present paper gives a detailed structural analysis of a class of commutative Banach algebras B k (h) generated by Toeplitz operators on the standard weighted Bergman spaces A 2 λ (B n) over the complex unit ball B n in C n. In the most general situation we explicitly determine the set of maximal ideals of B k (h) and we describe the Gelfand transform on a dense subalgebra. As an application to the spectral theory we prove the inverse closedness of algebras B k (h) in the full algebra of bounded operators on A 2 λ (B n) for certain choices of h. Moreover, it is remarked that B k (h) is not semi-simple. In the case of k = (n) we explicitly describe the radical Rad B n (h) of the algebra B n (h). This result generalizes and simplifies the characterization of Rad B 2 (1), which was given in Bauer and Vasilevski (Integr Equ Oper Theory 74:199-231, 2012). Keywords Weighted Bergman space • Gelfand theory • Commutative Toeplitz algebra • Generalized Berezin transform Communicated by Vladimir Bolotnikov. W. Bauer has been supported by an "Emmy-Noether scholarship" of DFG (Deutsche Forschungsgemeinschaft). N. Vasilevski has been partially supported by CONACYT Project 102800, México.