Analytic tridiagonal reproducing kernels

The paper characterizes the reproducing kernel Hilbert spaces with orthonormal bases of the form {(a(n,0) + a(n,1)z +... + a(n,J)z(J))z(n), n greater than or equal to 0}. The primary focus is on the tridiagonal case where J = 1, and on how it compares with the diagonal case where J = 0. The question of when multiplication by z is a bounded operator is investigated, and aspects of this operator are discussed. In the diagonal case, M-z is a weighted unilateral shift. It is shown that in the tridiagonal case, this need not be so. and an example is given in which the commutant of M-z on a tridiagonal space is strikingly different from that on any diagonal space.