Tridiagonal Reproducing Kernels and Subnormality

We consider analytic reproducing kernel Hilbert spaces H with orthonormal bases of the formf(an + bnz)z n : n 0g. If bn = 0 for all n, then H is a diagonal space and multiplication by z, Mz, is a weighted shift. Our focus is on providing extensive classes of examples for which Mz is a bounded subnormal operator on a tridiagonal spaceH where bn6 0. The Aronszajn sum of H and (1 z)H where H is either the Hardy space or the Bergman space on the disk are two such examples.