Title: Conditional Positive Definiteness and Complete Monotonicity in Subnormal Operator Theory

Abstract: On a Hilbert space H, a subnormal operator S is the restriction, to an invariant subspace, of a normal operator N (i.e., N^∗N = NN^∗). Born in the 1950’s, subnormal operator theory has developed into one of the central areas of modern operator theory, sitting at the intersection of complex function theory and functional analysis. Powered by S. Brown’s 1978 discovery of nontrivial invariant subspaces for subnormal operators, and J. Thomson’s proof that bounded point evaluations always exist (which ultimately led to R. Olin and J. Thomson’s proof that subnormal operators are reflexive), subnormal operator theory rapidly moved to center stage, with vibrant connections to rational approximation theory, complex geometry, infinite moment divisibility, and the theory of Bernstein functions.

We will first describe the various presentations of a separable infinite dimensional Hilbert space H, and some of the operators acting on H. Next, we will briefly focus on (single and bivariate) weighted shifts. For sequences  of positive real numbers, called weights, we will describe the weighted shift operators having the property of moment in finite divisibility (MID); that is, for any p > 0, the Schur power is subnormal. We will show that is MID if and only if certain infinite matrices log (0) and log (1) are conditionally positive definite (CPD). (Here is the sequence of moments associated with , log (0) ; log (1) are the canonical Hankel matrices whose positive semi-definiteness determines the subnormality of , and log is calculated entry-wise, i.e., in the sense of Schur or Hadamard.)

All Agler shifts are MID; in turn, Agler shifts belong to a larger class HWS of homographic shifts, that is, shifts whose weights-squared sequences are of the form

α_n :=an+b/cn+d, where a, b, c, d are real numbers and ad − bc nonzero . The entire class consists of subnormal operators, a property inherited by regular subshifts; that is, shifts whose weights are of the form β_n := α_pn+r, for p, r in N₀ and 0 ≤ r < p.

Next, we will use conditional positive definiteness to establish a new bridge between k–hyponormality and n–contractivity, which sheds significant new light on how the two well known staircases from hyponormality to subnormality interact. As a consequence, we prove that a contractive weighted shift Wα is MID if and only if for all p > 0and and are CPD.

Motivated by the family HWS, we then consider the geometrically regular shifts (GRWS), whose weights-squared sequences are of the form α ◦ ϕ, where ϕ : N₀ → N₀ is defined recursively by ϕ(0) := q and ϕ(n) := ϕ(n − 1) + pr+n, with p, q, r ∈N₀ and p ≥ 2. Rather surprisingly, the GRWS class offers a wide variety of structural behavior, encompassing complete hyperexpansivity, k-hyponormality, subnormality, moment infinite divisibility, and even weights-squared sequences interpolated by Bernstein functions.

During the lecture, we will discuss some basic aspects of the GRWS theory in the concrete case of weights-squared of the form for

and−1 ≤ N, D ≤ 1. In this case, the structural properties can be summarized, on a first pass,

in an intriguing radially-invariant diagram. One particular sector in this diagram directly relates to Bernstein functions, a close relative of completely monotone functions, which are Laplace transforms of positive Borel measures on [0, +∞).

We will end with a recent discovery of previously uncatalogued Bernstein functions naturally arising from GRWS and expressible as rational functions with interlacing zeros and poles.

The talk is based on joint work over several years with C. Benhida, G. Exner, L.A. Fialkow,

I.S. Hwang, S. Kim, S.H. Lee, W.Y. Lee, H.M. M¨oller, T. Prasad, J. Yoon, S. Yoo, and

E.H. Zerouali.

Short Bio: Professor Raúl E. Curto is a Professor of Mathematics working on Analysis and Operator Theory and Analysis. He is our Collegiate Fellow and an AMS fellow.

He got his PhD degree from SUNY at Stony Brook in 1987 and joined the department in 1981.

Special Note: This talk is a continuation of the April 2 presentation, which was interrupted due to weather. Professor Raul will briefly review the material covered on April 2.
Everyone is welcome to attend this session.