Domain Regularity and Semigroup Commutation Relations

Abstract

In the present Chapter, we obtain several generalizations of a classical commutation relation from the theory of matrix Lie algebras and Lie groups, often called the adjoint representation identity. Specifically, if A and B are operators on a finite-dimensional space E, this relation takes the form

$$ \begin{gathered} \exp (tA)B \exp ( - tA) = \exp (t adA)(B) \hfill \\ = \sum {\left\{ {{t^{k}}/k!{{(ad A)}^{k}}(B):O \leqslant k < \infty } \right\}} \hfill \\ \end{gathered} $$

(1)

where ad A(C) = AC - CA. Here, we shall primarily be concerned with extensions of (l) to cases where A and B are linear endomorphisms of infinite-dimensional spaces E_∞ or D for which the exponentials in (l) can still be interpreted reasonably in terms of other endomorphisms of these spaces. As is well-known (and essentially recapitulated in Chapter 2), the standard matrix arguments using rearrangements of power series apply equally well to bounded Banach space operators A, B.