Let H 2 (D 2 ) be the Hardy space over the bidisk. For sequences of Blaschke products {ϕ n (z): −∞ < n < ∞} and {ψ n (w): −∞ < n < ∞} satisfying some additional conditions, we may define a Rudin type invariant subspace M. We shall determine the rank of H 2 (D 2 ) M for the pair of operators T * z and T * w .

Let H 2 ðD 2 Þ be the Hardy space over the bidisk. Let fj n ðzÞg nf0 be a sequence of one variable inner functions such that j n ðzÞ=j nþ1 ðzÞ is a nonconstant inner function for every n f 0. Associated with them, we have an invariant subspace M of H 2 ðD 2 Þ. When j 0 ðzÞ is a Blaschke product, it is determined rankðM m wMÞ for the fringe operator F z on M m wM and rank M as an invariant subspace of H 2 ðD 2 Þ.

An elementary proof of the Aleman, Richter and Sundberg theorem concerning the invariant subspaces of the Bergman space is given. (2000). Primary 47A15, 32A35; Secondary 47B35. Mathematics Subject Classification

We determine the norm and the essential norm of the difference of weighted composition operators on the space of bounded harmonic functions on the open unit disk. The argument is done on the boundary. Mathematics Subject Classification (2010). Primary 47B38; Secondary 30H10.Keywords. Weighted composition operator, the space of bounded harmonic functions, essential norm.

We consider the component problem on the sets of weighted composition operators on the spaces of bounded harmonic and analytic functions on the open unit disk with the operator norms, respectively. Especially, we shall determine path connected components in the sets of noncompact weighted composition operators. u(z) = ∂D u(e iθ)P z (e iθ) dm(e iθ) for z ∈ D, where P z is the Poisson kernel for the point z ∈ D and m is the normalized Lebesgue measure on ∂D. Then u ∈ h ∞. Besides, for f ∈ h ∞ , f * = f on D. Let S(D) be the set of analytic self-maps of D. For each ϕ ∈ S(D), we may define the composition operator C ϕ by C ϕ f = f •ϕ, where f is analytic on D. Composition operators on various analytic function spaces have been studied extensively during the past few decades. See [2, 14] for an overview of these results. Presently, some of the open questions in this field are related to the topological structure of the set of composition operators. For even f ∈ h ∞ , C ϕ f = f • ϕ is also harmonic on D. So the question of properties of C ϕ on h ∞ naturally arises. In [1], Choa and the first and third authors first considered properties of composition operators on h ∞ and, although h ∞ is not algebra, introduced the concept of weighted composition operators M u C ϕ on h ∞ defined by (M u C ϕ f)(z) = ∂D u(e iθ)(f • ϕ) * (e iθ)P z (e iθ) dm(e iθ)