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

To a densely defined, but not necessarily selfadjoint, operator A on a Hilbert space H we consider on H the following abstract ‘‘elliptic’’ problem of Dirichlet type: ½u Žt . AuŽt ., t 0 uŽ0. . Then, in this paper, we establish that for every t 0, the solution uŽt. e t ʹA can be expanded into a series of generalized eigenvectors of the operator A provided that its resolvent belongs to Carleman class C for some p 0, 1 . A p 2 similar result holds for t large enough if the inverse A 1 belongs to Carleman class C for every p 1 . ŽSee Theorem 3.1 and Theorem 3.2.. Furthermore, we p 2 apply these obtained results to the shape memory alloys non-selfadjoint operator Ž 0 I . n n n C D4 D2 , 0, 0, and D x 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 C for every p 1 , so that we get in this case p 2 more regularity in the sense that the operatorial series Ý k 1etCPk converges strongly in E to the analytic semigroup etC for every t 0 Žthe P are the k projectors into the root subspaces of C.. A similar result holds for e t ʹC provided that t is large enough. ŽSee Theorem 4.1 and Theorem 4.2 in Section 4 for the precise result..