Inversions of Hermite semigroup

Abstract

Let { e − c H | c ⩾ 0 } \{ {e^{ - cH}}|c \geqslant 0\} be the Hermite semigroup on the real line R \mathbb {R} . Then a representation is constructed for inversions of the semigroup, and it gives a representation of e − c H {e^{ - cH}} for c > 0 c > 0 . Moreover, some characterizations of the domain in which, for c > 0 , e − c H c > 0,\;{e^{ - cH}} is well defined are examined.