Conditionally positive definite unilateral weighted shifts

TL;DR

This paper investigates conditionally positive definite unilateral weighted shifts, characterizing them via moment sequences, solving backward extension problems, and exploring flatness issues with notable differences from subnormal shifts.

Contribution

It provides a complete characterization of conditionally positive definite unilateral weighted shifts and addresses the backward extension and flatness problems in this context.

Findings

Characterization via formal moment sequences

Solution to backward extension problem

Distinct flatness properties compared to subnormal shifts

Abstract

In a recent paper [15], Hilbert space operators $T$ with the property that each sequence of the form $\{\|T^n h\|^2\}_{n=0}^{\infty}$ is conditionally positive definite in a semigroup sense were introduced. In the present paper, this line of research is continued in depth in the case of unilateral weighted shifts. The conditional positive definiteness of weighted shifts is characterized in terms of formal moment sequences. The description of the representing triplet, the main object canonically associated with such operators, is provided. The backward extension problem for conditionally positive definite weighted shifts is solved, revealing a new feature that does not appear in the case of other operator classes. Finally, the flatness problem in this context is discussed, with an emphases on unexpected differences from the corresponding problem for subnormal weighted shifts.