The Mittag Leffler Reproducing Kernel Hilbert Spaces of Entire and Analytic Functions

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.

Lifting Commuting 3-Isometric Tuples

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.

A Lifting Theorem for 3-Isometries

Scott McCullough and Ben Russo

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.

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