Toeplitz Type Operators Research Articles

Hankel- and Toeplitz-Type Operators on the Unit Ball

Let Bm be the unit ball in the m-dimensional complex plane Cm with the weighted measuredμαz=α+1α+2···α+mπm1−|z|2αdmzα>−1.From the viewpoint of the Cauchy–Riemann operator we give an orthogonal direct sum decomposition for L2(Bm,dμα(z)), i.e., L2(Bm,dμα(z))=⊕n∈Z+,σ∈ΔAσn, where the components A(+,+,…,+)0 and A(−,−,…,−)0 are just the weighted Bergman and conjugate Bergman spaces, respectively. Using the simplex polynomials from T. H. Koornwinder and A. L. Schwartz (1997, Constr. Approx13, 537–567), we obtain an orthogonal basis for every subspace. As an application of the orthogonal decomposition, we define the Hankel- and Toeplitz-type operators and discuss Sp-criteria for these kinds of operators.

Pluriharmonic Symbols of Commuting Toeplitz Type Operators on the Weighted Bergman Spaces

AbstractA class of Toeplitz type operators acting on the weighted Bergman spaces of the unit ball in the n-dimensional complex space is considered and two pluriharmonic symbols of commuting Toeplitz type operators are completely characterized.

Pluriharmonic symbols of commuting Toeplitz type operators

Certain Toeplitz type operators acting on the Bergman space A1 of the unit ball are considered and pluriharmonic symbols of commuting Toeplitz type operators are characterised by using M-harmonic function theory.

Toeplitz-Hankel type operators on Dirichlet spaces

LetLα,2 be the function space defined by the (weighted) Dirichlet energy integrals on the unit disk of complex plane. By constructing new orthogonal polynomials we give an orthogonal decomposition\(L^{\alpha ,2} = \oplus _{k = 0}^\infty (A_k \oplus \bar A_k )\) such thatA0α is just the (weighted) Dirichlet space. We define three kinds of Toeplitz and Hankel type operators, develop their boundedness andSp-criteria, and reveal cut-off phenomena and Wu' phenomena.

Krein space approach to H/sup ∞/ mixed sensitivity minimization for a class of infinite dimensional systems

Motivated by the computation of the H/sup /spl infin// optimal performance and optimal mixed sensitivity for a class of infinite dimensional systems, the author considers the explicit computation of the norm and minimal symbols of a class of Hankel plus Toeplitz type operators with irrational symbols using the Krein space approach. The author shows how to explicitly compute the norm and parameterize all the minimal symbols. The results are used to obtain a complete solution to the H/sup /spl infin// optimal mixed sensitivity synthesis problem for a class of infinite dimensional systems: an explicit formula for computing the H/sup /spl infin// optimal performance and explicit parameterization of all optimal and suboptimal mixed sensitivities. An illustrative example is given. >

Toeplitz and Hankel type operators on an annulus

Let Ω be a regular domain in the extended complex plane, i.e., it is a bounded domain and its boundary consists of a finite number of disjoint analytic simple closed curves. Let dm(z) be the Lebesgue area measure on Ω and let ds = dm(z)/ω(z) be the Poincare metric on Ω, a Riemannian metric of negative constant curvature. It may be proved that Ω(z) ≈ Euclidean distance from z to the boundary of Ω (see [8]).