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We study Hankel operators on the harmonic Bergman space b2(B), where B is the open unit ball in R ~, n > 2. We show that if f is in C(/)) then the Hankel operator with symbol f is compact. For the proof we have to extend the definition of Hankel operators to the spaces b~(B), 1 < p < 0% and use an interpolation theorem. We also use the explicit formula for the orthogonal projection of L2(B, dV) onto b2(B). This result implies that the commutator and semi-commutator of Toeplitz operators with symbols in C(/)) are compact.

Let Ω be an open simply connected proper subset of the complex plane. We identify, up to isomorphism, which groups are possible for the group of unitary composition operators of a Hardy-Smirnov space defined on Ω. We also study the relationship between the geometry of Ω and the corresponding group.

Abstract. The operator that takes the function f to ψf • φ is a weighted composition operator. We study numerical ranges of some classes of weighted composition operators on H 2 , the Hardy-Hilbert space of the unit disc. We consider the case where φ is a rotation of the unit disc and identify a class of convexoid operators. In the case of isometric weighted composition operators we give a complete classification of their numerical ranges. We also consider the inclusion of zero in the interior of the numerical range.