Diagonalization of non-selfadjoint analytic semigroups and application to the shape memory alloys operator (Q5937223)

Here, the abstract elliptic problem on \(\mathbb{R}^+\times H\) (\(H\) is a Hilbert space) of Dirichlet type \(u'(t)= Au(t)\), \(u(0)=\varphi\), \(t>0\), is discussed. The operator \(A\) is densely defined and not necessarily selfadjoint on \(H\). Under the assumption that the resolvent of \(A\) belongs to the Carleman class \(C_p\) for \(p\in (0,1/2)\), it is shown that the solution \(u(t)= e^{-t\sqrt A}\varphi\) can be expanded into a series of generalized eigenvalues of the operator \(A\). A similar result is also given in the case \(p\geq 1/2\).