Conditionally positive definite unilateral weighted shifts

Abstract

In a recent paper [15], Hilbert space operators T with the property that each sequence of the form {‖Tnh‖2}n=0∞ 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 unilateral 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 unilateral 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 emphases on unexpected differences from the analogous problem for subnormal unilateral weighted shifts.