Yet Another Face of the Creation Operator

Abstract

If one defines the creation operator as an abstract weighted shift of weights $\mathop {\{ \sqrt n + 1\} }\nolimits_{n = 0}^\infty$, then the operator

$$\frac{1}{{\sqrt 2 }}(x - \frac{d}{{dx}})$$

appears as a face (read: unitary image) of it while the operator of multiplication by the independent variable in the Segal-Bargmann space does as another. In [11] a finite difference operator

$$\sqrt x f(x - 1) - \sqrt {af} (x),a > 0$$

was recorded. It may be considered as yet another face of the creation one. Our intension here is to invite attention to this operator by considering the case in detail.

Wlodzimierz Mlak in memoriam