Unbounded Toeplitz Operators Research Articles

Unbounded Toeplitz Operators with Rational Symbols

Unbounded (and bounded) Toeplitz operators (TO) with rational symbols are analysed in detail showing that they are densely defined closed and have finite dimensional kernels and deficiency spaces. The latter spaces as well as the domains, ranges, spectral and Fredholm points are determined. In particular, in the symmetric case, i.e., for a real rational symbol the deficiency spaces and indices are explicitly available. The concluding section gives a brief overview on the research on unbounded TO in order to locate the present contribution. Regarding properties of unbounded TO in general, it furnishes some new results recalling the close relationship to Wiener-Hopf operators and, in case of semiboundedness, to singular operators of Hilbert transformation type. Specific symbols considered in the literature admit further analysis. Some conclusions are drawn for semibounded integrable and real square-integrable symbols. There is an approach to semibounded TO, which starts from closable semibounded forms related to a Toeplitz matrix. The Friedrichs extension of the TO associated with such a form is studied. Finally, analytic TO and Toeplitz-like operators are briefly examined, which in general differ from the TO treated here.

Kernels of Unbounded Toeplitz Operators and Factorization of Symbols

We consider kernels of unbounded Toeplitz operators in H^p({mathbb {C}}^{+}) in terms of a factorization of their symbols. We study the existence of a minimal Toeplitz kernel containing a given function in H^p({mathbb {C}}^{+}), we describe the kernels of Toeplitz operators whose symbol possesses a certain factorization involving two different Hardy spaces and we establish relations between the kernels of two operators whose symbols differ by a factor which corresponds, in the unit circle, to a non-integer power of z. We apply the results to describe the kernels of Toeplitz operators with non-vanishing piecewise continuous symbols.

Unbounded Toeplitz Operators

This partly expository article develops the basic theory of unbounded Toeplitz operators on the Hardy space H2, with emphasis on operators whose symbols are not square integrable. Unbounded truncated Toeplitz operators on coinvariant subspaces of H2 are also studied.

Sharp two-weight inequalities for singular integrals, with applications to the Hilbert transform and the Sarason conjecture

We prove two-weight norm inequalities for Calderón–Zygmund singular integrals that are sharp for the Hilbert transform and for the Riesz transforms. In addition, we give results for the dyadic square function and for commutators of singular integrals. As an application we give new results for the Sarason conjecture on the product of unbounded Toeplitz operators on Hardy spaces.

Toeplitz operators in Segal-Bargmann spaces of vector-valued functions vector-valued functions

We discuss new results concerning unbounded Toeplitz operators defined in Segal-Bargmann spaces of (vector-valued) functions, i.e. the space of all entire functions which are square summable with respect to the Gaussian measure in $\mathrm{C}^n$. The problem of finding adjoints of analytic Toeplitz operators is solved in some cases. Closedness of the range of analytic Toeplitz operators is studied. We indicate an example of an entire function inducing a Toeplitz operator, for which the space of polynomials is not a core though it is contained in its domain.

Unbounded Toeplitz operators in the Segal-Bargmann space, III

The paper is a continuation of our previous study of unbounded Toeplitz operators in the Segal-Bargmann space; [6], [7]. This work is divided into separate sections, devoted to different topics described below. In Section 2 some necessary preliminaries are given. Section 3 is devoted to the problem of computing the adjoint of Toeplitz operators. A class of entire symbols ' is found for which T ' has been computed. Section 4 deals with Toeplitz operators as generators of contraction semigroups. In turn Section 5 describes Fischer pairs of polynomials (the notion introduced by H. Shapiro in [11]), from the point of view of perturbation theory. The last Section 6 ends up with remarks and open problems. We thank anonymous referee for his helpful and careful remarks.

The invertibility of the product of unbounded Toeplitz operators

In this note I give necessary and sufficient conditions on outer functionsf andg for the operator $$T_f T_{\bar g} $$ to be bounded and invertible on H2. I also discuss the relationship of this question to two open questions in operator theory and weighted norm inequalities.

On the theory of unbounded Toeplitz operators

On the theory of unbounded Toeplitz operators