Lumpability of linear evolution equations in Banach spaces
Let \(X\) and \(Y\) be Banach spaces and \(M:X\to Y \) be a surjective bounded operator. In \(X\) consider the abstract Cauchy problem \(x'(t)=Ax(t)\), \(x(0)=x_0\), where \(A\) is a generator of \(C_0\)-semigroup. Let \(A\in \mathcal B(X)\). It is proved that the following statements are equivalent: {\parindent=0.7cm\begin{itemize}\item[(1)] there exists \(\hat{A}\in \mathcal B(Y)\) such that \(MA=\hat{A}M\), \item[(2)] \(\ker(M)\subseteq\ker(MA)\), \item[(3)] \(\mathrm{ran}(M^*)\) is invariant under \(A^*\). \end{itemize}} Let \(T(t)\) be a semigroup generated by \(A\). It is proved that the following statements are equivalent: {\parindent=0.7cm\begin{itemize}\item[(4)] \(\ker(M)\) is invariant under \(T(t)\) for all \(t>0\), \item[(5)] there exists a linear operator \(\hat{A}\) on \(M(\mathrm{dom}(A))\) such that \(\hat{A}\) is a generate of \(C_0\)-semigroup and \(MA=\hat{A}M\). \item[(6)] \(\mathrm{ran}(M^*)\) is invariant under \(T^*(t)\). \end{itemize}} It is proved that there exists (unbounded) operator \(\hat{A}\) on \(Y\) such that \(MA=\hat{A}M\) if and only if the following holds: {\parindent=0.7cm\begin{itemize}\item[(7)] \(A(\ker(M)\cap\mathrm{dom}(A))\subset\ker(M)\), \item[(8)] there exists \(\lambda>0\) such that \((\lambda I-A)\) is surjective from \(\ker(M)\cap \mathrm{dom}(A)\) to \(\ker(M)\). \end{itemize}}
scientific article; zbMATH DE number 6672803