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Volume 51
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Volume 49
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Volume
48
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Vietnam
Journal of Mathematics 38:4 (2010) 381-393
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On Asymptotic Property of Toeplitz Operators
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Namita Das and
Madhusmita Sahoo
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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.
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2000 Mathematics Subject Classification: 47B35.
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Keywords: Hardy space, Bergman space, Toeplitz
operators, Hankel operators, Inner functions.
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