P. K. Jena

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Articles: 1

On the range and kernel of Toeplitz and little Hankel operators

Namita Das, Pabitra Kumar Jena

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 19 (2013), no. 1, 55-67

In this paper we study the interplay between the range and kernel of Toeplitz and little Hankel operators on the Bergman space. Let $T_\phi$ denote the Toeplitz operator on $L^2_a(\mathbb{D})$ with symbol $\phi \in L^\infty(\mathbb{D})$ and $S_\psi$ denote the little Hankel operator with symbol $\psi \in L^\infty(\mathbb{D}).$ We have shown that if ${\operatorname{Ran}} (T_\phi) \subseteq {\operatorname{Ran}} (S_\psi)$ then $\phi \equiv 0$ and find necessary and sufficient conditions for the existence of a positive bounded linear operator $X$ defined on the Bergman space such that $T_\phi X=S_\psi$ and ${\operatorname{Ran}} (S_\psi) \subseteq {\operatorname{{\operatorname{Ran}}}} (T_\phi).$ We also obtain necessary and sufficient conditions on $\psi \in L^\infty(\mathbb{D})$ such that ${\operatorname{Ran}} (T_\psi)$ is closed.


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