No abstract

Abstract.The symbols of hyponormal Toeplitz operators are completely described and those are also studied, being related with the extremal problems of Hardy spaces. Moreover, we discuss Halmos's question about a subnormal Toeplitz operator when the self-commutator is finite rank.

No abstract

Let (X 9 J^9m) be a probability measure space and A a subalgebra of L°°(m), containing the constant functions. Srinivasan and Wang defined A to be a weak-*Dirichlet algebra if A + A (the complex conjugate) is weak-*dense in L°°(m) and the integral is multiplicative on A, \fgdm = \fdm \gβm for /, ge A. In this paper the notion of extended weak-*Dirichlet algebra is introduced; A is an extended weak-*Dirichlet algebra if A + A is weak-*dense in L'im) and if the conditional expectation E^ to some sub σ-algebra & is multiplicative on A. Then most of important theorems proved for weak-*Dirichlet algebras are generalized in the context of extended weak-*Dirichlet algebras, for instance, Szegδ's theorem and Beuring's theorem. Besides, our approach will yield several theorems which were not known even for weak-*Dirichlet algebras.

Two generalizations of the classical Hankel operators are defined on an abstract Hardy space that is associated with a uniform algebra. In this paper the norms of Hankel operators are studied. This has applications to weighted norm inequalities for conjugation operators, and invertible Topelitz operators. The results in this paper have applications to concrete uniform algebras, for example, a polydisc algebra and a uniform algebra which consists of rational functions.