Functional calculus of quantum channels for the holomorphic discrete series of SU(1,1)

The tensor product of two holomorphic discrete series representations of $SU(1,1)$ can be decomposed as a direct multiplicity-free sum of infinitely many holomorphic discrete series representations. I shall introduce equivariant quantum channels for each component of the direct sum by mapping the tensor product of an operator and the identity onto the projection onto one of the irreducible components, generalizing the construction of pure equivariant quantum channels for compact groups. Then I calculate the functional calculus of this operator for polynomials and prove a limit formula for the trace of the functional calculus for any differentiable function. The methods I used are the theory of reproducing kernel Hilbert spaces and a Plancherel theorem for the disk $D=SU(1,1)/U\left(1\right)$, together with exact constants for the eigenvalues of the Berezin transform. I prove that the limit of the trace of the functional calculus can be expressed using generalized Husimi functions or using Berezin transforms.