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# On the range and kernel of Toeplitz and little Hankel operators

### Abstract

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.

Key words: Toeplitz operators, little Hankel operators, Bergman space, inner functions, range and kernel of operators.

### Article Information

 Title On the range and kernel of Toeplitz and little Hankel operators Source Methods Funct. Anal. Topology, Vol. 19 (2013), no. 1, 55-67 MathSciNet MR3088318 zbMATH 1289.47058 Milestones Received 23/02/2012 Copyright The Author(s) 2013 (CC BY-SA)

### Authors Information

Namita Das
P. G. Department of Mathematics, Utkal University, Vani Vihar, Bhubaneswar, 751004, Orissa, India

Pabitra Kumar Jena
P. G. Department of Mathematics, Utkal University, Vani Vihar, Bhubaneswar, 751004, Orissa, India

### Citation Example

Namita Das and Pabitra Kumar Jena, On the range and kernel of Toeplitz and little Hankel operators, Methods Funct. Anal. Topology 19 (2013), no. 1, 55-67.

### BibTex

@article {MFAT641,
AUTHOR = {Das, Namita and Jena, Pabitra Kumar},
TITLE = {On the range and kernel of Toeplitz and little Hankel operators},
JOURNAL = {Methods Funct. Anal. Topology},
FJOURNAL = {Methods of Functional Analysis and Topology},
VOLUME = {19},
YEAR = {2013},
NUMBER = {1},
PAGES = {55-67},
ISSN = {1029-3531},
MRNUMBER = {MR3088318},
ZBLNUMBER = {1289.47058},
URL = {http://mfat.imath.kiev.ua/article/?id=641},
}