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On the Commutativity of a Certain Class of Toeplitz Operators

100%

Louhichi I. , Randriamahaleo F. , Zakariasy L.

nr 1

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.

Product of weighted Hankel and weighted Toeplitz operators

60%

Datt G. , Porwal D. K.

tom Vol. 46, nr 3

571--583

In this paper, we discuss some properties of the weighted Hankel operator H(...) and describe the conditions on which the weighted Hankel operator H(...) and weighted Toeplitz operator T(...), with (…) on the space H(...) being a sequence of positive numbers with (…), commute. It is also proved that if a non-zero weighted Hankel operator H(...) commutes with T(...), which is not a multiple of the identity, then H(...), for some (…).

Shifted model spaces and their orthogonal decompositions

60%

Câmara M. C. , Kliś-Garlicka K. , Ptak M.

tom Vol. 44, no. 3

341--357

We generalize certain well known orthogonal decompositions of model spaces and obtain similar decompositions for the wider class of shifted model spaces, allowing us to establish conditions for near invariance of the latter with respect to certain operators which include, as a particular case, the backward shift S*. In doing so, we illustrate the usefulness of obtaining appropriate decompositions and, in connection with this, we prove some results on model spaces which are of independent interest. We show moreover how the invariance properties of the kernel of an operator T, with respect to another operator, follow from certain commutation relations between the two operators involved.