Abstract. This 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 ≥ 0}.The primary focus is on the tridiagonal case where J = 1 and how it compares to 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 Mz 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 Mz on a tridiagonal space is strikingly different from that on any diagonal space.
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