1,287 research outputs found
On the Structure of Commutative Banach Algebras Generated by Toeplitz Operators on the Unit Ball. Quasi-Elliptic Case. II: Gelfand Theory
No full textExtending 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 generated by Toeplitz operators on the standard weighted Bergman spaces over the complex unit ball in . In the most general situation we explicitly determine the set of maximal ideals of and we describe the Gelfand transform on a dense subalgebra. As an application to the spectral theory we prove the inverse closedness of algebras in the full algebra of bounded operators on for certain choices of . Moreover, it is remarked that is not semi-simple. In the case of we explicitly describe the radical of the algebra . This result generalizes and simplifies the characterization of , which was given in Bauer and Vasilevski (Integr Equ Oper Theory 74:199-231, 2012)
Spectral zeta function of a sub-Laplacian on product sub-Riemannian manifolds and zeta-regularized determinant
No full textWe analyze the spectral zeta function for sub-Laplace operators on product manifolds M x N. Starting from suitable conditions on the zeta functions on each factor, the existence of a meromorphic extension to the complex plane and real analyticity in a zero neighbourhood is proved. In the special case of N = S(1) and using the Poisson summation formula, we obtain expressions for the zeta-regularized determinant. Moreover, we can calculate limit cases of such determinants by inserting a parameter into our formulas. This is a generalization of results in Furutani and de Gosson (2003) [1] and in particular it applies to an intrinsic sub-Laplacian on U(2) congruent to S(3) x S(1) induced by a sum of squares of canonical vector fields on S(3); cf. Bauer and Furutani (2008) [2]. Finally, the spectral zeta function of a sub-Laplace operator on Heisenberg manifolds is calculated by using an explicit expression of the heat kernel for the corresponding sub-Laplace operator on the Heisenberg group; cf. Beals et al. (2000) [18] and Hulanicki (1976) [19]. (C) 2010 Elsevier B.V. All rights reserved
Algebraic properties and the finite rank problem for Toeplitz operators on the Segal–Bargmann space
No full textAbstractWe study three different problems in the area of Toeplitz operators on the Segal–Bargmann space in Cn. Extending results obtained previously by the first author and Y.L. Lee, and by the second author, we first determine the commutant of a given Toeplitz operator with a radial symbol belonging to the class Sym>0(Cn) of symbols having certain growth at infinity. We then provide explicit examples of zero-products of non-trivial Toeplitz operators. These examples show the essential difference between Toeplitz operators on the Segal–Bargmann space and on the Bergman space over the unit ball. Finally, we discuss the “finite rank problem”. We show that there are no non-trivial rank one Toeplitz operators Tf for f∈Sym>0(Cn). In all these problems, the growth at infinity of the symbols plays a crucial role
Das Studium der Fachdidaktik der Geographie in der 1. und 2. Phase der Ausbildung der Geographielehrer
No full textDas Studium der Fachdidaktik der Geographie in der 1. und 2. Phase der Ausbildung der Geographielehrer / Ludwig Bauer ; Wolfram Hausmann. - In: Geographie / hrsg. von Ludwig Bauer ... - München : Oldenbourg, 1976. - S. 155- 174
Commuting Toeplitz operators with quasi-homogeneous symbols on the Segal–Bargmann space
No full textAbstractLet T=Tzlz¯k with l,k∈N0 be a Toeplitz operator with monomial symbol acting on the Segal–Bargmann space over the complex plane. We determine the symbols Ψ of polynomial growth at infinity such that TΨ and Tzlz¯k commute on the space of all holomorphic polynomials. By using polar coordinates we represent Ψ as an infinite sum Ψ(reiθ)=∑j=−∞∞Ψj(r)eijθ. Then we are able to reduce the above problem to the case of quasi-homogeneous symbols Ψ=Ψjeijθ. We obtain the radial part Ψj(r) in terms of the inverse Mellin transform of an expression which is a product of Gamma functions and a trigonometric polynomial. If we allow operator symbols of higher growth at infinity, we point out that in some of the cases more than one Toeplitz operator TΨjeijθ exists commuting with T
Commutative Toeplitz Banach Algebras on the Ball and Quasi-Nilpotent Group Action
No full textStudying commutative C -algebras generated by Toeplitz operators on the unit ball it was proved that, given a maximal commutative subgroup of biholomorphisms of the unit ball, the C -algebra generated by Toeplitz operators, whose symbols are invariant under the action of this subgroup, is commutative on each standard weighted Bergman space. There are five different pairwise non-conjugate model classes of such subgroups: quasi-elliptic, quasi-parabolic, quasi-hyperbolic, nilpotent and quasi-nilpotent. Recently it was observed in Vasilevski (Integr Equ Oper Theory. 66:141-152, 2010) that there are many other, not geometrically defined, classes of symbols which generate commutative Toeplitz operator algebras on each weighted Bergman space. These classes of symbols were subordinated to the quasi-elliptic group, the corresponding commutative operator algebras were Banach, and being extended to C -algebras they became non-commutative. These results were extended then to the classes of symbols, subordinated to the quasi-hyperbolic and quasi-parabolic groups. In this paper we prove the analogous commutativity result for Toeplitz operators whose symbols are subordinated to the quasi-nilpotent group. At the same time we conjecture that apart from the known C -algebra cases there are no more new Banach algebras generated by Toeplitz operators whose symbols are subordinated to the nilpotent group and which are commutative on each weighted Bergman space.DFG (Deutsche Forschungsgemeinschaft); CONACYT, Mexico [102800
Compactness characterization of operators in the Toeplitz algebra of the Fock space Fαp
No full textAbstractFor 1<p<∞ let Tpα be the norm closure of the algebra generated by Toeplitz operators with bounded symbols acting on the standard weighted Fock space Fαp. In this paper, we will show that an operator A is compact on Fαp if and only if A∈Tpα and the Berezin transform Bα(A) of A vanishes at infinity
On the structure of commutative Banach algebras generated by Toeplitz operators on the unit ball. Quasi-elliptic case. I: Generating subalgebras
No full textExtending recent results in [3] to the higher dimensional setting n >= 3 we provide a further step in the structural analysis of a class of commutative Banach algebras generated by Toeplitz operators on the standard weighted Bergman space over the n-dimensional complex unit ball. The algebras B-k (h) under study are subordinated to the quasi-elliptic group of automorphisms of B-n and in terms of their generators they were described in [23]. We show that B-k(h) is generated in fact by an essentially smaller set of operators, i.e., the Toeplitz operators with k-quasi-radial symbols and a finite set of Toeplitz operators with "elementary" k-quasi-homogeneous symbols. Then we analyze the structure of the commutative subalgebras corresponding to these two types of generating symbols. In particular, we describe spectra, joint spectra, maximal ideal spaces and the Gelfand transform. (C) 2013 Elsevier Inc. All rights reserved.DFG (Deutsche Forschungsgemeinschaft); CONACYT, Mexico [102800
On The Structure of A Commutative Banach Algebra Generated By Toeplitz Operators With Quasi-Radial Quasi-Homogeneous Symbols
No full textLet denote the standard weighted Bergman space over the unit ball in . New classes of commutative Banach algebras which are generated by Toeplitz operators on 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 . 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 . Since is not invariant under the -operation of its inverse closedness is not obvious and is proved. We remark that the algebra 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 is always connected
Commuting Toeplitz operators on the Segal–Bargmann space
No full textAbstractConsider two Toeplitz operators Tg, Tf on the Segal–Bargmann space over the complex plane. Let us assume that g is a radial function and both operators commute. Under certain growth condition at infinity of f and g we show that f must be radial, as well. We give a counterexample of this fact in case of bounded Toeplitz operators but a fast growing radial symbol g. In this case the vanishing commutator [Tg,Tf]=0 does not imply the radial dependence of f. Finally, we consider Toeplitz operators on the Segal–Bargmann space over Cn and n>1, where the commuting property of Toeplitz operators can be realized more easily
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