Nonautonomous exponential dichotomy (Q2480710)

The notion of \textit{bievolution family} \(U\subset \mathcal L(X)\) over a B-space \(X\) is introduced in analogy with the treatment of bisemigroups related to exponentially dichotomous operators in [\textit{H.\,Bart, I.\,Goldberg} and \textit{M.\,A.\thinspace Kaashoek} J.~Funct.\ Anal.\ 68, 1--42 (1986; Zbl 0606.47021); see also the author's forthcoming monograph ``Exponentially dichotomous operators and applications'' Oper.\ Theory:\ Adv.\ Appl.\ 182; Basel:\ Birkhäuser (2008; Zbl 1158.47001)]. It is proved that by way of \([E_U(t)f](\tau) = U(\tau,\tau-t)f(\tau-t)\) for \((\tau,\tau-t)\in \Delta_{+}\cup \Delta_{-}\) (a disjoint union of the half-planes \(\Delta_{\pm} = \{t,s \in \mathbb R^2: {\pm}(t-s)\geq 0\})\), each bievolution family \(U\) determines a strongly continuous ``evolutionary'' bisemigroup \(E_U\) on \(L^p(\mathbb R,X)\), \(1\leq p <\infty\). The same is true of \(E_U\) on \(C_0(\mathbb R,X)\) if and only if \(U\) is a bievolution family.