Finite Rank Perturbations of Toeplitz Operators

We study finite rank perturbations of the Brown-Halmos type results involving products of Toeplitz operators acting on the Bergman space.      

Commutants of Analytic Toeplitz Operators on the Harmonic Bergman Space

We study the commuting problem for Toeplitz operators on the harmonic Bergman space of the unit disk. We show that an analytic Toeplitz operator and a co-analytic Toeplitz operator with certain noncyclicity hypothesis can commute only when one of their symbols is constant. We also obtain analogous results for semi-commutants.     

Products of Toeplitz Operators on the Bergman Space

In 1962 Brown and Halmos gave simple conditions for the product of two Toeplitz operators on Hardy space to be equal to a Toeplitz operator. Recently, Ahern and Cucković showed that a similar result holds for Toeplitz operators with bounded harmonic symbols on Bergman space. For general symbols, the situation is much more complicated. We give necessary and sufficient conditions for the product to be a Toeplitz operator (Theorem 6.1), an explicit formula for the symbol of the product in certain cases (Theorem 6.4), and then show that almost anything can happen (Theorem 6.7).     

Norms of Moore-Penrose Inverses of Fredholm Toeplitz Operators

In this paper we estimate the norm of the Moore-Penrose inverse T(a)⁺ of a Fredholm Toeplitz operator T(a) with a matrix-valued symbol a∈L_{N × N}^∞ defined on the complex unit circle. In particular, we show that in the ”generic case” the strict inequality ||T(a)⁺|| > ||a⁻¹||_∞ holds. Moreover, we discuss the asymptotic behavior of ||T(t^ra)⁺|| for . The results are illustrated by numerical experiments.     

Herz Classes and Toeplitz Operators in the Disk

In this paper we decompose into diadic annuli and consider the class S_p,q of Toeplitz operators T_φ for which the sequence of Schatten norms belongs to ℓ^q, where φ_n = φχ A_n. We study the boundedness and compactness of the operators in S_p,q and we describe the operators T_φ , φ ≥ 0 in these spaces in terms of weighted Herz norms of the averaging operator of the symbols φ.     

A Note on the Range of Toeplitz Operators

A well known lemma attributed to Coburn states that a Toeplitz operator with non-trivial kernel acting on the Hardy space must have dense range. We show that the range of a non-zero Toeplitz operator with non-trivial kernel must contain all polynomials and state this in a precise form.     

Toeplitz Operators on Analytic Besov Spaces

We characterize complex measures μ on the unit disk for which the Toeplitz operator T _μ is bounded or compact on the analytic Besov spaces B _p with 1 ≤ p < ∞. Research supported in part by NSF grant, DMS 0200587 (first author); and by a NSERC grant (third author).     

On hyponormal Toeplitz operators with polynomial and circulant-type symbols

This paper characterises those hyponormal Toeplitz operators on the Hardy space of the unit circle among all Toeplitz operators that have polynomial symbols with circulant-type sets of coefficients.Supported in part by The Natural Sciences and Engineering Research Council of CanadaSupported in part by BSRI-96-1420 and KOSEF 94-0701-02-01-3     

Dynamics of Properties of Toeplitz Operators with Radial Symbols

In the case of radial symbols we study the behavior of different properties (boundedness, compactness, spectral properties, etc.) of Toeplitz operators T_a⁽⁾ acting on weighted Bergman spaces over the unit disk , in dependence on , and compare their limit behavior under with corresponding properties of the initial symbol a.     

On Toeplitz operators with quasihomogeneous symbols

In this paper, we give some basic results concerning Toeplitz operators whose symbol is of the form e^{i p θ}ϕ, where ϕ is a radial function, then use these results to characterize all Toeplitz operators which commute with them.Received: 12 June 2004; revised: 27 January 2005     

Semi-commutators of Toeplitz operators on the Bergman space

In this paper several necessary and sufficient conditions are obtained for the semi-commutator of Toeplitz operators andT _g with bounded pluriharmonic symbols on the unit ball to be compact on the Bergman space. Using -harmonic function theory on the unit ball we show that with bounded pluriharmonic symbolsf andg is zero on the Bergman space of the unit ball or the Hardy space of the unit sphere if and only if eitherf org is holomorphic.The author was supported in part by the National Science Foundation.     

Linear-Fractional Composition Operators in Several Variables

We investigate properties of linear-fractional composition operators C_φ on Hardy and Bergman spaces of the ball in that are motivated by a formula for the self-commutator [C_φ^* ,C_φ]. In particular, we characterize when certain commutators [C_φ, C_σ] are compact, and give conditions under which is compact, where is multiplication by the monomial z^β. Our results allow us to determine when C_φ is essentially normal, for φ belonging to a large class of linear-fractional symbols.     

Commutative Algebras of Toeplitz Operators on the Reinhardt Domains

Let D be a bounded logarithmically convex complete Reinhardt domain in centered at the origin. Generalizing a result for the one-dimensional case of the unit disk, we prove that the C ^*-algebra generated by Toeplitz operators with bounded measurable separately radial symbols (i.e., symbols depending only on is commutative. We show that the natural action of the n-dimensional torus defines (on a certain open full measure subset of D) a foliation which carries a transverse Riemannian structure having distinguished geometric features. Its leaves are equidistant with respect to the Bergman metric, and the orthogonal complement to the tangent bundle of such leaves is integrable to a totally geodesic foliation. Furthermore, these two foliations are proved to be Lagrangian. We specify then the obtained results for the unit ball.     

Commuting of Toeplitz Operators in Several Complex Variables

1 IntroductionLet be the unit ball Bn or the polydisk Dn in Cn, and dA denote the Lebesgue measure on , normalized so that the measure of equals 1. L2/a() is The Bergman space. For z, the Bergman reproducing kernel is a function Kz L2/a). The normalized Bergman reproducing kernel k is the function Kz Kz2. Here, as elsewhere in this paper, the norm 2 and the inner product are taken in the space L2(,dA).Let P denote the orthogonal projection of L2(, dA) onto L2/a(). For a function f L …     

Hypercyclic Pairs of Coanalytic Toeplitz Operators

A pair of commuting operators, (A,B), on a Hilbert space is said to be hypercyclic if there exists a vector such that {A ⁿ B ^k x : n, k ≥ 0} is dense in . If f, g ∈H ^∞(G) where G is an open set with finitely many components in the complex plane, then we show that the pair (M ^* _f , M ^* _g ) of adjoints of multiplcation operators on a Hilbert space of analytic functions on G is hypercyclic if and only if the semigroup they generate contains a hypercyclic operator. However, if G has infinitely many components, then we show that there exists f, g ∈H ^∞(G) such that the pair (M ^* _f , M ^* _g ) is hypercyclic but the semigroup they generate does not contain a hypercyclic operator. We also consider hypercyclic n-tuples.     

Hyponormality of Toeplitz operators with rational symbols

In this article we introduce a notion of `division' for rational functions and then give a criterion for hyponormality of (f, g are rational functions) in the cases where g divides f. Furthermore we show that we may assume, without loss of generality, that g divides f when we consider the hyponormality of . Supported in part by a grant from Faculty Research Fund, Sungkyunkwan University, 2004. Supported in part by a grant (R14-2003-006-01000-0) from the Korea Research Foundation.     

Commutative <Emphasis Type="Italic">C</Emphasis>*-Algebras of Toeplitz Operators on the Unit Ball,I. Bargmann-Type Transforms and Spectral Representations of Toeplitz Operators

Extending known results for the unit disk, we prove that for the unit ball there exist n+2 different cases of commutative C*-algebras generated by Toeplitz operators, acting on weighted Bergman spaces. In all cases the bounded measurable symbols of Toeplitz operators are invariant under the action of certain commutative subgroups of biholomorphisms of the unit ball. This work was partially supported by CONACYT Projects 46936 and 44620, México.     

Finite-Rank Products of Toeplitz Operators in Several Complex Variables

For any α > −1, let A²_α be the weighted Bergman space on the unit ball corresponding to the weight (1 – |z|²)^α. We show that if all except possibly one of the Toeplitz operators are diagonal with respect to the standard orthonormal basis of A²_α and has finite rank, then one of the functions must be the zero function.      

Essential Norms of Composition Operators

We obtain simple estimates for the essential norm of a composition operator acting from the Hardy space H ^p to H ^q , p > q, in one or several variables. When p = and q = 2 our results give an exact formula for the essential norm.