Commuting Toeplitz operators with harmonic symbols

Mellin Transform, Monomial Symbols, and Commuting Toeplitz Operators

This paper studies algebras of functions on the unit disk generated byH∞(D) and bounded harmonic functions. Using these algebras, we characterize compact semicommutators and commutators of Toeplitz operators with harmonic symbols on the Bergman space.

We study the commutator of the multiplication and harmonic Bergman projection, Hankel and Toeplitz operators on the harmonic Bergman spaces. The same type operators have been well studied on the analytic Bergman spaces. The main difficulty of this study is that the bounded harmonic function space is not an algebra! In this paper, we characterize theLp boundedness and compactness of these operators with harmonic symbols. Results about operators in Schatten classes, the cut-off phenomenon and general symbols are also included.

In this paper we completely characterize when the commutator and semi-commutator of two monomial-type Toeplitz operators on the Fock space over ℂn have finite rank. In sharp contrast to Bergman space over the unit ball case, it turns out that there are many other cases on the Fock space for (semi-)commuting Toeplitz operators.

We study the commuting problem for Toeplitz operators on the Hardy space of the polydisk. We consider two symbols where one is an arbitrarily bounded symbol and the other is a bounded symbol which has the form of a sum of an analytic function and a co-analytic function. We then give a necessary and sufficient condition on the symbols for which the corresponding Toeplitz operators commute each other.

Zero products of Toeplitz operators with harmonic symbols

<abstract><p>We first describe commuting Toeplitz operators with harmonic symbols on weighted harmonic Bergman spaces. Then, a sufficient condition for hyponormality on weighted Bergman spaces of the punctured unit disk, when the analytic part of the symbol is a monomial, is shown.</p></abstract>

This chapter takes up several independent topics in Euclidean Fourier analysis, all having some bearing on the subject of partial differential equations. Section 1 elaborates on the relationship between the Fourier transform and the Schwartz space, the subspace of $L^{1}(\mathbb{R}^{N})$ consisting of smooth functions with the property that the product of any iterated partial derivative of the function with any polynomial is bounded. It is possible to make the Schwartz space into a metric space, and then one can consider the space of continuous linear functionals; these continuous linear functionals are called “tempered distributions.” The Fourier transform carries the space of tempered distributions in one-one fashion onto itself. Section 2 concerns weak derivatives, and the main result is Sobolev’s Theorem, which tells how to recover information about ordinary derivatives from information about weak derivatives. Weak derivatives are easy to manipulate, and Sobolev’s Theorem is therefore a helpful tool for handling derivatives without continually having to check the validity of interchanges of limits. Sections 3–4 concern harmonic functions, those functions on open sets in Euclidean space that are annihilated by the Laplacian. The main results of Section 3 are a characterization of harmonic functions in terms of a mean-value property, a reflection principle that allows the extension to all of Euclidean space of any harmonic function in a half space that vanishes at the boundary, and a result of Liouville that the only bounded harmonic functions in all of Euclidean space are the constants. The main result of Section 4 is a converse to properties of Poisson integrals for half spaces, showing that harmonic functions in a half space are given as Poisson integrals of functions or of finite complex measures if their $L^{p}$ norms over translates of the bounding Euclidean space are bounded. Sections 5–6 concern the Calderon–Zygmund Theorem, a far-reaching generalization of the theorem concerning the boundedness of the Hilbert transform. Section 5 gives the statement and proof, and two applications are the subject of Section 6. One of the applications is to Riesz transforms, and the other is to the Beltrami equation, whose solutions are “quasiconformal mappings.” Sections 7–8 concern multiple Fourier series for smooth periodic functions. The theory is established in Section 7, and an application to traces of integral operators is given in Section 8.

In this paper it is shown that Toeplitz operators on Bergman space form a dense subset of the space of all bounded linear operators, in the strong operator topology, and that their norm closure contains all compact operators. Further, the C<sup>*</sup>-algebra generated by them does not contain all bounded operators, since all Toeplitz operators belong to the essential commutant of certain shift. The result holds in Bergman spaces A<sup>2</sup>(Ω) for a wide class of plane domains Ω⊂C, and in Fock spaces A<sup>2</sup>(C<sup>N</sup>), N≧1.

The paper is devoted to the study of Toeplitz operators with piecewise continuous symbols. We clarify the geometric regularities of the behavior of the essential spectrum of Toeplitz operators in dependence on their crucial data: the angles between jump curves of symbols at a boundary point of discontinuity and on the limit values reached by a symbol at that boundary point. We show then that the curves supporting the symbol discontinuities, as well as the number of such curves meeting at a boundary point of discontinuity, do not play any essential role for the Toeplitz operator algebra studied. Thus we exclude the curves of symbol discontinuity from the symbol class definition leaving only the set of boundary points (where symbols may have discontinuity) and the type of the expected discontinuity. Finally we describe the C*-algebra generated by Toeplitz operators with such symbols.

One of the major goals in the theory of Toeplitz operators on the Bergman space over the unit disk D in the complex place C is to completely describe the commutant of a given Toeplitz operator, that is, the set of all Toeplitz operators that commute with it. Here we shall study the commutants of a certain class of quasihomogeneous Toeplitz operators defined on the harmonic Bergman space.

An equation with a Hardy space Toeplitz operator can be solved by Wiener–Hopf factorization. However, Wiener–Hopf factorization does not work for Bergman space Toeplitz operators. The only way we see to tackle equations with a Toeplitz operator on the Bergman space is to have recourse to approximation methods. The paper is intended as a review of and an illustrated tour through several such methods and thus through some beautiful topics in the intersection of operator theory, complex analysis, differential geometry, and numerical mathematics.

Products of Hankel and Toeplitz Operators on the Bergman Space

Let \Omega be a subdomain of \mathbb{C} and let \mu be a positive Borel measure on \Omega . In this paper, we study the asymptotic behavior of the eigenvalues of compact Toeplitz operators T_\mu acting on Bergman spaces on \Omega . Let (\lambda _n(T_\mu)) be the decreasing sequence of the eigenvalues of T_\mu , and let \rho be an increasing function such that \rho (n)/n^A is decreasing for some A>0 . We give an explicit necessary and sufficient geometric condition on \mu in order to have \lambda _n(T_\mu)\asymp 1/\rho (n) . As applications, we consider composition operators C_\varphi , acting on some standard analytic spaces on the unit disc \mathbb{D} . First, we give a general criterion ensuring that the singular values of C_\varphi satisfy s_n(C_\varphi ) \asymp 1/\rho(n) . Next, we focus our attention on composition operators with univalent symbols, where we express our general criterion in terms of the harmonic measure of \varphi (\mathbb{D}) . We finally study the case where \partial \varphi (\mathbb{D}) meets the unit circle in one point and give several concrete examples. Our method is based on upper and lower estimates of the trace of h(T_\mu) , where h is a suitable concave or convex function.

Preliminary Text. Do not use. 15 years ago the function theory and operator theory connected with the Hardy spaces was well understood (zeros; factorization; interpolation; invariant subspaces; Toeplitz and Hankel operators, etc.). None of the techniques that led to all the information about Hardy spaces worked on their close relatives the Bergman spaces. Most mathematicians who worked in the intersection of function theory and operator theory thought that progress on the Bergman spaces was unlikely. Now the situation has completely changed. Today there are rich theories describing the Bergman spaces and their operators. Research interest and research activity in the area has been high for several years. A book is badly needed on Bergman spaces and the three authors are the right people to write it.