Joel Rosenfeld, Benjamin Russo, and Warren E. Dixon

Published in Journal of Mathematical Analysis and Applications, Volume 463, Issue 2, 15 July 2018, Pages 576-592

This paper investigates the function theoretic properties of two reproducing kernel functions based on the Mittag-Leffler function that are related through a composition. Both spaces provide one parameter generalizations of the traditional Bargmann-Fock space. In particular, the Mittag-Leffler space of entire functions yields many similar properties to the Bargmann-Fock space, and several results are demonstrated involving zero sets and growth rates. The second generalization, the Mittag-Leffler space of the slitted plane, is a reproducing kernel Hilbert space (RKHS) of functions for which Caputo fractional differentiation and multiplication by $z^q$ (for $q>0$) are densely defined adjoints of one another.

Published in Operators and Matrices, 2017, Volume 11, Number 2, pp 397–433

An operator $T$ is called a $3$-isometry if there exist operators $B_1(T^*,T)$ and $B_2(T^*,T)$ such that
$$Q(n) = T^{*n}T^{n} = 1 + nB_1(T^*,T)+n^2B_2(T^*,T)$$
for all natural numbers $n$. An operator $J$ is a Jordan operator of order $2$ if $J=U+N$ where $U$ is unitary, $N$ is nilpotent order $2$, and $U$ and $N$ commute.
An easy computation shows that $J$ is a $3$-isometry and that the restrictions of $J$ to an invariant subspace is also a $3$-isometry.
Those $3$-isometries which are the restriction of a Jordan operator to an invariant subspace can be identified, using the theory of completely positive maps, in terms of a positivity condition on the operator pencil $Q(s)$.
In this article, we establish the analogous result in the multi-variable setting and show, by modifying an example of Choi, that an additional hypothesis is necessary.
Lastly we discuss the joint spectrum of sub-Jordan tuples and derive the results of $3$-symmetric operators as a corrolary.

Published in Integral Equations and Operator Theory, January 2016, Volume 84, Issue 1, pp 69–87

An operator $T$ on Hilbert space is a $3$-isometry if there exist operators $B_1$ and $B_2$ such that $$T^{*n}T^n = I+nB_1 +n^2 B_2.$$
An operator $J$ is a Jordan operator if it the sum of a unitary $U$ and nilpotent $N$ of order two which commute.
If $T$ is a $3$-isometry and $c>0$, then $I-c^{-2} B_2 + sB_1 + s^2B_2$ is positive semidefinite for all real $s$ if and only if it is the restriction of a Jordan operator $J = U + N$
with the norm of $N$ at most $c$. As a corollary, an analogous result for $3$-symmetric operators, due to
Helton and Agler, is recovered.