Analytical Theory of Semi-groups

Abstract

The analytical theory of semi-groups of bounded linear operators in a B-space deals with the exponential functions in infinite dimensional function spaces. It is concerned with the problem of determining the most general bounded linear operator valued function T(t), t ≧ 0, which satisfies the equations

$$(T(t + s) = T(t).T(s),T(0) = I$$

The problem was investigated by E. Hille [2] and K. Yosida [5] independently of each other around 1948. They introduced the notion of the infinitesimal generator A of T(t) defined by

$${\rm A} = s - \mathop {\lim {t^{ - 1}}}\limits_{t \downarrow 0} (T(t) - 1)$$

, and discussed the generation of T(t) in terms of A and obtained a characterization of the infinitesimal generator A in terms of the spectral property of A.