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Vietnam Journal of Mathematics 38:4 (2010) 381-393  

 

On Asymptotic Property of Toeplitz Operators

Namita Das and Madhusmita Sahoo

 

Abstract. In this paper we derive certain asymptotic properties of Toeplitz operators on Hardy and Bergman spaces. More precisely, we have shown that if (T)) and converges to an operator L in the strong operator topology for all inner functions (T) then , a Toeplitz oper-ator on (T) and if T is an operator in the Hankel algebra, the norm closed algebra generated by all Toeplitz and all Hankel operators together on (T), then the sequence converges to a Toeplitz operator on (T) in the strong operator topology for all inner functions (T). As an application, we have shown that if (T)) and is of fi-nite rank for all (T) then , where (T)and F is a finite rank operator. This is an extension of the work done for the scalar valued case in [2] and [12]. Asymptotic properties of Toeplitz operators and Hankel operators defined on the Berg-man space were also analysed.

2000 Mathematics Subject Classification: 47B35.

Keywords: Hardy space, Bergman space, Toeplitz operators, Hankel operators, Inner functions.

 

 

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