Matrices of operators and regularized semigroups (Q1319312)

Let \({\mathcal A}\) be the matrix of operators \((p_{i,j} (A_ 1,\dots,A_ n))_{i,j=1}^ m\), on \(X^ n\), where \(X\) is a Banach space, \(p_{i,j}\), \(1\leq i,j\leq m\) are polynomials and \(iA_ 1,\dots, iA_ n\) are commuting generators of bounded linear strongly continuous groups. Let \(| A|^ 2 \equiv \sum_{k=1}^ n A_ k^ 2\). We show that when \((p_{i,j})\) is Petrowsky-correct, that is, the spectrum of \((p_{i,j}(x))\) is contained in a fixed left half-plane, for all \(x\) in \(\mathbb{R}^ n\), then \(\exists\) a constant \(r\), depending on \(m\), \(n\) and the maximum degree of \(p_{i,j}\), such that \({\mathcal A}\) generates an exponentially bounded \((1+| A|^ 2)^{-r}\)-regularized semigroup. When the numerical range of \((p_{i,j}(x))\) is contained in a fixed left half-plane, for all \(x\) in \(\mathbb{R}^ n\), then the dependence on \(m\) disappears. We show that our choice of \(r\) is best possible. We also show that, for any \((p_{i,j})\), there exists \(C\), with dense range, such that \({\mathcal A}\) generates a \(C\)-regularized semigroup.