A mean value inequality with applications to Bergman space operators

A Generalized Mean Value Inequality for Subharmonic Functions and Applications

If u ≥ 0 is subharmonic on a domain Ω in Rn and p > 0, then it is well-known that there is a constant C(n, p) ≥ 1 such that u(x)p ≤C(n, p)M V (up,B(x,r)) for each ball B(x,r) ⊂ Ω. We recently showed that a similar result holds more generally for functions of the form ψ◦ u where ψ : R+ → R+ may be any surjective, concave function whose inverse ψ−1 satisfies the ∆2-condition. Now we point out tha...

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_...

Positive Toeplitz Operators on the Bergman Space

In this paper we find conditions on the existence of bounded linear operators A on the Bergman space La(D) such that ATφA ≥ Sψ and ATφA ≥ Tφ where Tφ is a positive Toeplitz operator on L 2 a(D) and Sψ is a self-adjoint little Hankel operator on La(D) with symbols φ, ψ ∈ L∞(D) respectively. Also we show that if Tφ is a non-negative Toeplitz operator then there exists a rank one operator R1 on L ...

Intermediate Hankel Operators on the Bergman Space

Products of Toeplitz Operators on a Vector Valued Bergman Space

We give a necessary and a sufficient condition for the boundedness of the Toeplitz product TF TG∗ on the vector valued Bergman space L 2 a(C ), where F and G are matrix symbols with scalar valued Bergman space entries. The results generalize those in the scalar valued Bergman space case [4]. We also characterize boundedness and invertibility of Toeplitz products TF TG∗ in terms of the Berezin t...