The interest of the Hilbert spaces L λ 2 , of Chapter V is due to the fact that the group G of holomorphic automorphisms of D acts on them by (irreducible) unitary representations. This statement must be made a little more precise. A unitary (resp. bounded) representation is properly speaking a strongly continuous homomorphism g → T (g)into the unitary (resp. bounded) transformations of a Hilbert space, such that T (g) T (h) = T (gh) for all g,h∈G. A slightly more general notion is that of a (unitary, or bounded) projective representation,where one requires only T (g)T (h) = c (g, h)T (gh) with some c (g, h) ∈ ℂ.
KeywordsHolomorphic Function Analytic Continuation Unitary Representation Toeplitz Operator Bergman Space
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