N. Vasilevski - Academia.edu (original) (raw)

Papers by N. Vasilevski

Journal of Mathematical Sciences

Our purpose is to characterize the so-called horizontal Fock-Carleson type measures for derivativ... more Our purpose is to characterize the so-called horizontal Fock-Carleson type measures for derivatives of order k (we write it k-hFC for short) for the Fock space as well as the Toeplitz operators generated by sesquilinear forms given by them. We introduce real coderivatives of k-hFC type measures and show that the C*-algebra generated by Toeplitz operators with the corresponding class of symbols is commutative and isometrically isomorphic to certain C*-subalgebra of L ∞ (R n). The above results are extended to measures that are invariant under translations along Lagrangian planes.

We study the interrelations and differences between compactness properties of commutators and sem... more We study the interrelations and differences between compactness properties of commutators and semi-commutators of Toeplitz operators for different classes (algebras) of their defining symbols. The importance and interest to this question is caused by the constitutive influence of these properties on the structure of corresponding Toeplitz and related operator algebras.

ABSTRACT We exhibit a surprising but natural connection among the Bergman space structure, commut... more ABSTRACT We exhibit a surprising but natural connection among the Bergman space structure, commutative algebras of Toeplitz operators and pencils of hyperbolic straight lines. The commutative C*-algebras of Toeplitz operators on the unit disk can be classified as follows. Each pencil of hyperbolic straight lines determines the set of symbols consisting of functions which are constant on corresponding cycles, the orthogonal trajectories to lines forming a pencil. It turns out that the C*-algebra generated by Toeplitz operators with this class of symbols is commutative.

Journal of Functional Analysis, 2014

The classical theory of Toeplitz operators in spaces of analytic functions deals usually with sym... more The classical theory of Toeplitz operators in spaces of analytic functions deals usually with symbols that are bounded measurable functions on the domain in question. A further extension of the theory was made for symbols being unbounded functions, measures, and compactly supported distributions, all of them subject to some restrictions. In the context of a reproducing kernel Hilbert space we propose a certain framework for a 'maximally possible' extension of the notion of Toeplitz operators for a 'maximally wide' class of 'highly singular' symbols. Using the language of sesquilinear forms we describe a certain common pattern for a variety of analytically defined forms which, besides covering all previously considered cases, permits us to introduce a further substantial extension of a class of admissible symbols that generate bounded Toeplitz operators. Although our approach is unified for all reproducing kernel Hilbert spaces, for concrete operator consideration in this paper we restrict ourselves to Toeplitz operators acting on the standard Fock (or Segal-Bargmann) space.

International Society for Analysis, Applications and Computation, 2000

Let V be a pointed convex cone in IR n having vertex at the origin. Denote by T v = IR n + iV the... more Let V be a pointed convex cone in IR n having vertex at the origin. Denote by T v = IR n + iV the tube domain over V, and denote by A ∈(T v ) the Bergman space on T v , i.e., the subspace of. L 2 (T v ) consisting of all functions analytic in T V . Studying the structure of the Bergman space, we obtain several connections between the Bergman and Hardy spaces, as well as between the corresponding Bergman and Szego projections.

International Society for Analysis, Applications and Computation, 1999

Differential Operators and Related Topics, 2000

Consider the space L2(@, dp,), where dp, is the Gaussian measure, and its Fock subspace F 2 (@) c... more Consider the space L2(@, dp,), where dp, is the Gaussian measure, and its Fock subspace F 2 (@) consisting of all analytic (entire) functions in @. We introduce the so-called true-poly-Fock spaces, and prove that L2(@, dp,) is the direct sum of the Fock and all true-poly-Fock spaces. The structure of these spaces and connections between them are described. The orthogonal (Bargmann type) projections onto true-poly-Fock spaces are given.

The paper is devoted to the study of Toeplitz operators with piecewise continuous symbols. We cla... more The paper is devoted to the study of Toeplitz operators with piecewise continuous symbols. We clarify the geometric regularities of the behaviour 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.

Mathematische Nachrichten, 1993

Advances in Analysis - Proceedings of the 4th International ISAAC Congress, 2005

Clifford Algebras and their Applications in Mathematical Physics, 1992

Mathematische Nachrichten, 2009

ABSTRACT Toeplitz operators with discontinuous presymbols on the Fock space are studied. These pr... more ABSTRACT Toeplitz operators with discontinuous presymbols on the Fock space are studied. These presymbols can have two limit values at the points of some subset in ℂn, generally speaking of nonzero measure. Nevertheless the Toeplitz operator algebra still admits a commutative symbolic calculus.

Journal of Functional Analysis, 2006

A family of recently discovered commutative C * -algebras of Toeplitz operators on the unit disk ... more A family of recently discovered commutative C * -algebras of Toeplitz operators on the unit disk can be classified as follows. Each pencil of hyperbolic straight lines determines a set of symbols consisting of functions which are constant on the corresponding cycles, the orthogonal trajectories to lines forming a pencil. The C * -algebra generated by Toeplitz operators with such symbols turns out to be commutative. We show that these cases are the only possible ones which generate the commutative C * -algebras of Toeplitz operators on each weighted Bergman space.  2005 Elsevier Inc. All rights reserved.

Integral Equations and Operator Theory, 2003

Let G ⊂ C be a domain with smooth boundary and let α be a C 2diffeomorphism on G satisfying the C... more Let G ⊂ C be a domain with smooth boundary and let α be a C 2diffeomorphism on G satisfying the Carleman condition α•α = id G . We denote by R the C * -algebra generated by the Bergman projection of G, all multiplication operators aI (a ∈ C(G)) and the operator W ϕ = | det Jα| ϕ • α, where det Jα is the Jacobian of α. A symbol algebra of R is determined and Fredholm conditions are given. We prove that the C * -algebra generated by the Bergman projection of the upper half-plane and the operator (W ϕ)(z) = ϕ(−z) is isomorphic and isometric to C 2 × M2(C).

Integral Equations and Operator Theory, 2004

Integral Equations and Operator Theory, 2002

The paper is devoted to the study of specific pr0p~e~ies of Toeplitz operators with (unbounded, i... more The paper is devoted to the study of specific pr0p~e~ies of Toeplitz operators with (unbounded, in general) radial symbols a = a(r). Boundedness and compactness conditions, as well as examples, are given. It turns out that there exist non-zero symbols which generate zero Toeplitz operators. We characterize such symbols, as well as the class of symbols for which Ta = 0 implies a(r) = 0 a.e. For each compact set M there exists a Toeplitz operator T~ such that sp T~ --ess-sp T~ --M. We show that the set of symbols which generate bounded Toeplitz operators no longer forms an algebra under pointwise multiplication. Besides the algebra of Toeplitz operators we consider the algebra of Weyl pseudodifferential operators obtained from Toeplitz ones by means of the Bargmann transform. Rewriting our Toeplitz and Weyl pseudodifferential operators in terms of the Wick symbols we come to their spectral decompositions.

Integral Equations and Operator Theory, 1996

The structure of the C*-algebra with identity generated by two orthogonal projections is well und... more The structure of the C*-algebra with identity generated by two orthogonal projections is well understood. All irreducible representations of this algebra are either two-dimensional or one--dimensional. The situation becomes unpredictable in the case of the C*-algebra generated by three orthogonal projections. Even in the more specific case when two of the projections commute, the algebra under consideration may have infinite dimensional irreducible representations.

Integral Equations and Operator Theory, 1994

Right, left and two-sided group convolution operators for a class of step two nilpotent Lie group... more Right, left and two-sided group convolution operators for a class of step two nilpotent Lie groups are considered. To understand the structure of these operators the direct integral construction and symplectic changes of variables are used. As the application to the complex analysis ...