Diagonalization of Non-selfadjoint Analytic Semigroups and Application to the Shape Memory Alloys Operator

AbstractTo a densely defined, but not necessarily selfadjoint, operator A on a Hilbert space H we consider on R+×H the following abstract “elliptic” problem of Dirichlet type:[formula] Then, in this paper, we establish that for every t>0, the solution [formula] can be expanded into a series of generalized eigenvectors of the operator A provided that its resolvent belongs to Carleman class Cp for some p∈]0,12[. A similar result holds for t large enough if the inverse A−1 belongs to Carleman class Cp for every p>12. (See Theorem 3.1 and Theorem 3.2.) Furthermore, we apply these obtained results to the shape memory alloys non-selfadjoint operator [formula] and Dn=∂n/∂xn when acting on an appropriate Hilbert space E of functions on the interval [0,1], by establishing that the inverse C−1 belongs to the Carleman class Cp for every p>12, so that we get in this case more regularity in the sense that the operatorial series ∑∞k=1etCPk converges strongly in E to the analytic semigroup etC for every t>0 (the Pk are the projectors into the root subspaces of C). A similar result holds for [formula] provided that t is large enough. (See Theorem 4.1 and Theorem 4.2 in Section 4 for the precise result.