Journal of Operator Theory

Volume 70, Issue 2, Autumn 2013 pp. 477-494.

Tridiagonal reproducing kernels and subnormality

Authors: Gregory T. Adams (1), Nathan S. Feldman (2), and Paul J. McGuire (3)
Author institution: (1) Mathematics Department, Bucknell University, Lewisburg, PA 17837, U.S.A.
(2) Mathematics Department, Washington and Lee University, Lexington, VA 24450, U.S.A.
(3) Mathematics Department, Bucknell University, Lewisburg, PA 17837, U.S.A.

Summary: We consider analytic reproducing kernel Hilbert spaces $\mathcal{H}$ with orthonormal bases of the form $\{(a_n + b_n z) z^n : n \geqslant 0 \}$. If $b_n = 0$ for all $n$, then $H$ is a diagonal space and multiplication by $z$, $M_z$, is a weighted shift. Our focus is on providing extensive classes of examples for which $M_z$ is a bounded subnormal operator on a tridiagonal space $\mathcal{H}$ where $b_n\neq 0$. 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.

DOI: http://dx.doi.org/10.7900/jot.2011sep12.1942
Keywords: analytic reproducing kernel, subnormal operator, tridiagonal kernel

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