self-commutators of composition operators with monomial symbols on the bergman space

Self-commutators of composition operators with monomial symbols on the Bergman space

Let $varphi(z)=z^m, z in mathbb{U}$, for some positive integer $m$, and $C_varphi$ be the composition operator on the Bergman space $mathcal{A}^2$ induced by $varphi$. In this article, we completely determine the point spectrum, spectrum, essential spectrum, and essential norm of the operators $C^*_varphi C_varphi, C_varphi C^*_varphi$ as well as self-commutator and anti-self-commutators of $C_...

On the Toeplitz operators with piecewise continuous symbols on the Bergman space

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 t...

On a Class of Composition Operators on Bergman Space

Let D= {z ∈ C : |z| < 1} be the open unit disk in the complex plane C. Let A2(D) be the space of analytic functions on D square integrable with respect to the measure dA(z) = (1/π)dx dy. Given a ∈D and f any measurable function on D, we define the function Ca f by Ca f (z) = f (φa(z)), where φa ∈ Aut(D). The map Ca is a composition operator on L2(D,dA) and A2(D) for all a ∈D. Let (A2(D)) be the...

Toeplitz Operators on the Bergman Space of Planar Domains with Essentially Radial Symbols

Let Ω be a bounded multiply-connected domain in the complex plane C, whose boundary ∂Ω consists of finitely many simple closed smooth analytic curves γj j 1, 2, . . . , n where γj are positively oriented with respect to Ω and γj ∩ γi ∅ if i / j. We also assume that γ1 is the boundary of the unbounded component of C\Ω. LetΩ1 be the bounded component of C\γ1, and Ωj j 2, . . . , n the unbounded c...

Toeplitz operators with distributional symbols on Bergman spaces

On Commutators of Operators on Hilbert Space

1. In this note we first generalize a result of P. R. Halmos [3] concerning commutators of (bounded ) operators on Hubert space. Then we obtain some partial results on a problem of commutators in von Neumann algebras which is closely related to another problem raised by Halmos in [4]. Let 3C be any (infinite-dimensional) Hubert space, and let £(3C) denote the algebra of all bounded operators on...