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Invariant quadratic forms on spaces of holomorphic functions

Existence and uniqueness of invariant quadratic forms on spaces of holomorphic functions is studied. Invariance is with respect to the action of the automorphism group of the underlying domain given by , where ϕ is an automorphism and J ϕ is its Jacobian determinant. Quadratic forms invariant under this action are called q-invariant forms. For q > 0, it is shown, under mild assumptions on the Bergman kernal, that there is a unique q-invariant form, which is positive definite and determines a unique functional Hilbert space. In the case of the ball, there is a unique q-invariant form for all real q, which is positive definite for q > 0, and positive semi-definite for q = 0 and a discrete set of negative q. In those cases when the form is positive semi-definite, integral formulas are obtained for the associated (semi) norms.