On a class of Toeplitz and little Hankel operators on L2a (U+)

Abstract

In this paper we establish certain algebraic properties of Toeplitz operators and a class of little Hankel operators defined on the Bergman space of the upper half plane. We show that if K is a compact operator on L2a (U+),M(s) = i?s/i+s , ?a(s) = (c?1)+sd/(1+c)s?d where a = c + id ? D, s ? U+ and J f(s) = f (?s) then lim |a|?1? ||K ? TJ(M??a)KT+ M??a || = 0 and for ?,? ? h?(D), if ??s(??M)T??M ? T??M??s(??M) is compact, then lim w=x+iy y?0 ||c([??s(??M)dw] ? [?* ??Mdw]) + c([?J(??M)dw] ? [?* ?s(??M)dw])|| = 0, where dw(s) = 1/?? w+i/w?i (?2i)Imw/(s+w)2 ,w ? U+, ?? is the little Hankel operator on L2a (U+) with symbol ? and ?s is a function defined onU+ with |?s| = 1, for all s ? U+. Applications of these results are also obtained.