Schatten class Toeplitz operators on the Bergman space

Abstract: The study of Toeplitz operator was initiated in a paper of Otto Toeplitz in 1911. The objective of this paper is to discuss some algebraic properties of Toeplitz operators in Bergman Space. These properties include some general properties together with compactness and boundedness properties of operator have been studied. Key words: Operator matrix, Bergman space, Analytic. Reproducing Kernel, Berzin Transform

2016

The definition of Toeplitz operators in the Bergman space of square integrable analytic functions in the unit disk in the complex plane is extended in such a way that it covers many cases where the traditional definition does not work. This includes, in particular, highly singular symbols such as measures, distributions, and certain hyperfunctions. Bibliography: 22 titles.

Houston Journal of Mathematics, 1986

Let G be a countable union of annuli centered at 0 and contained in the unit disc D; let 1G be the characteristic function of G. Let T G be the Toeplitz operator on the Bergman space A 2 with symbol 1G. We show that the essential spectrum of T G is connected, and we give upper and lower bounds on the spectrum and essential spectrum in terms of the radii of the annuli. 1. Introduction. Let C be the complex plane and let D be the open unit disc. Let dA be normalized (Lebesgue) area measure on D; define Løø(D) as the complex-valued functions defined on D which are measurable and essentially bounded with respect to dA. Define L2(D) to be the complex-valued functions defined on D which are measurable and square-integrable with respect to dA. L2(D) is a Hilbert _ space with inner product (f,g)L2 = fD f fg dA. Define the Bergman space A 2 to be the analytic functions in L2(D). It is well known that A 2 is a closed subspace of L2(D) and is thus a Hilbert space (see [4]). Let P be the projection from L2(D) onto A 2. Let fC Løø(D); the Toeplitz operator on A 2 with symbol f, denoted Tf, is defined by Tf(g) = P(fg), g C A 2. We shall denote the spectrum of Tf by o(Tf). Let B(A 2) be the Banach algebra of bounded linear operators on A2; let •c(A 2) denote the ideal of compact operators on A 2. The quotient algebra B(A2)/•c(A 2) is referred to as the Calkin algebra, and the quotient map rr: B(A 2)-• B(A2)fic(A 2) is referred to as the Calkin map. The essential spectrum of Tf, denoted oe(Tf), is the spectrum of rr(Tf) in the Calkin algebra. In this work we shall consider Toeplitz operators on A 2 for which the symbol is the characteristic (indicator) function of a measurable subset G of D. An important 397 398 JAMES W. LARK, III application of results concerning these operators is in the work of Voas [8]. Let 1G denote the characteristic function of G; i.e., 1G(Z) = 1 if z E G, 1G(Z) = 0 if z • G. We shall use the symbol T G to represent the operator T1G on A 2. It is easy to see that T G is self-adjoint, and that o(T G) _C [0,1]. Since T G is self-adjoint (and thus rr(TG)is self-adjoint), then sup o(T G) = sup{X: 3, E O(TG)} = I[TGII and sup oe(T G) = sup{X: 3, G oe(TG)} = I[TGlle (the essential norm of TG).

Journal für die reine und angewandte Mathematik (Crelles Journal), 2000

It is well-known that Toeplitz operators on the Hardy space of the unit disc are characterized by the equality S * 1 T S 1 = T , where S 1 is the Hardy shift operator. In this paper we give a generalized equality of this type which characterizes Toeplitz operators with harmonic symbols in a class of standard weighted Bergman spaces of the unit disc containing the Hardy space and the unweighted Bergman space. The operators satisfying this equality are also naturally described using a slightly extended form of the Sz.-Nagy-Foias functional calculus for contractions. This leads us to consider Toeplitz operators as integrals of naturally associated positive operator measures in order to take properties of balayage into account.

Studia Mathematica, 2015

A full description of the membership in the Schatten ideal S p (A 2 ω) for 0 < p < ∞ of the Toeplitz operator acting on large weighted Bergman spaces is obtained.

Rocky Mountain Journal of Mathematics, 2014

Afrika Matematika, 2019

An operator T on a Hilbert space is hyponormal if T*T-TT* is positive. In this work we consider hyponormality of Toeplitz operators on the Bergman space with a logarithmic weight. Under a smoothness assumption we give a necessary condition when the symbol is of the form f + g with f , g analytic on the unit disk. We also find a sufficient condition when f is a monomial and g a polynomial.

2016

Abstract. In this paper we give necessary and sufficient conditions for the hyponormality of the Toeplitz operator Tϕ on the weighted Bergman space A 2 α(D) for ϕ(z) = ā−3 z̄ 3 + ā−1z̄+ a1z+ a3z 3 and ϕ(z) = ā−3z̄ 3 + ā−2z̄ 2 + a2z 2 + a3z 3. 1.

2021

As an extension to the study of Toeplitz operators on the Bergman space, the notion of H-Toeplitz operators Bφ is introduced and studied. Necessary and sufficient conditions under which H-Toeplitz operators become co-isometry and partial isometry are obtained. Some of the invariant subspaces and kernels of H-Toeplitz operators are studied. We have obtained the conditions for the compactness and Fredholmness for H-Toeplitz operators. In particular, it has been shown that a non-zero HToeplitz operator can not be a Fredholm operator on the Bergman space. Moreover, we have also discussed the necessary and sufficient conditions for commutativity of H-Toeplitz operators.

2007