Semigroup Generation Properties Of Two Classes Of Off-diagonal Hamiltonian Operator Matrices

Since the Stone Theorem was given in 1932 by M.H.Stone,the theory of linear operator semigroups in Banach spaces has been gradually developed and increasingly approached completeness and,the operator semigroup method has become one of the important methods in mathematical physics.A Hamiltonian operator matrix is a class of non-self-adjoint operator in product space,which appears naturally in a Hamiltoni-an system.In this paper,we study the semigroup generation properties for two classes of off-diagonal Hamiltonian operator matrices H=(?)under the appropriate space frame.Firstly,when B is positive definite and C is negative definite,D((-C)1/2)×D(B1/2)is defined as a Hilbert space,in which,the problems for H to generate a contractive and an analytic semigroup are studied,and it is obtained that the spectrum of H has a Hamiltonian structure.Secondly,for the non-negative off-diagonal Hamiltonian operator matrix,when B and C are both positive definite,the relationship between the point spec-trum of H and that of product operators BC and CB are discussed in D(C1/2×D(B1/2).Furthermore,the sufficient and necessary conditions for the operator matrix to generate a contractive and an analytic semigroup are obtained.