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Abstract. An analogue of the Paley-Wiener theorem is developed for weighted Bergman spaces of analytic functions in the upper half-plane. The result is applied to show that the invariant subspaces of the shift operator on the standard Bergman space of the unit disk can be identified with those of a convolution Volterra operator on the space L 2 (R + , (1/t)dt).

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Adjoints of certain operators of composition type are calculated. Specifically, on the classical Hardy space H 2 (D) of the open unit disk D operators of the form C B (f) = f • B are considered, where B is a finite Blaschke product. C * B is obtained as a finite linear combination of operators of the form Tg A B T h , where g and h are rational functions, Tg , T h are associated Toeplitz operators and A B is defined by A B (f)(z) = 1 n B(ξ)=z f (ξ).

Abstract. Boundedness of weighted composition operators Wu,ϕ acting on the classical Dirichlet space D as W h,ϕ f = h (f •ϕ) is studied in terms of the multiplier space associated to the symbol ϕ, i.e., M(ϕ) = {u ∈ D : Wu,ϕ is bounded on D}. A prominent role is played by the multipliers of the Dirichlet space. As a consequence, the spectrum of Wu,ϕ in D whenever ϕ is an automorphism of the unit disc is studied, extending a recent work of Hyvärinen, Lindström, Nieminen and Saukko [13] to the context of the Dirichlet space.