The quaternionic evolution operator

Abstract

In the recent years, the notion of slice regular functions has allowed the introduction of a quaternionic functional calculus. In this paper, motivated also by the applications in quaternionic quantum mechanics, see Adler (1995) [1], we study the quaternionic semigroups and groups generated by a quaternionic (bounded or unbounded) linear operator T = T 0 + i T 1 + j T 2 + k T 3 . It is crucial to note that we consider operators with components T ℓ ( ℓ = 0 , 1 , 2 , 3 ) that do not necessarily commute. Among other results, we prove the quaternionic version of the classical Hille–Phillips–Yosida theorem. This result is based on the fact that the Laplace transform of the quaternionic semigroup e t T is the S-resolvent operator ( T 2 − 2 Re [ s ] T + | s | 2 I ) − 1 ( s ¯ I − T ) , the quaternionic analogue of the classical resolvent operator. The noncommutative setting entails that the results we obtain are somewhat different from their analogues in the complex setting. In particular, we have four possible formulations according to the use of left or right slice regular functions for left or right linear operators.