  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.
