The Ascent And Descent Of Semigroups Of Operators

The definitions of ascent and descent of linear operators are given by A.E.Taylor ([1],1966) and D.C.Lay([2],1970) in the first time. They present some results about spectral analysis of linear operators by using ascent and descent. These contents can be seen in [3] which is witten by A.E.Taylor and D.C.Lay. In [3], the definitions of ascent and descent of linear operators are given. In 1991, S.M.Verduyn([4]) proposed the definition of ascent of semigroups of linear operators and discussed the completeness of root subspaces of an infinitesimal generator of semigroups of linear operators by using it. But it is unusual for systematic study to ascent and descent of semigroups of linear operators.In this paper,we will study the ascent and descent of semigroups of linear operators systematically. We mainly solve the following problems:Note: Here α(T(t)) and δ(T(t)) denote the ascent and descent of semigroups of linear operators T(t) respectively.(I) Semigroups of operators with finite ascent and descent have the property which decomposes the space into the direct sum .Theorem 2.5. If α(T{t)) = p < ∞ and δ(T{t)) = q < ∞, then α(T(t)) = δ{T(t))=p and(II) The conditions on which ascent and descent of semigroups of linear operators are finite are presented.Theorem 2.6. Let linear operator T(t)(t ≥ 0) be a C₀ semigroup defined on a Banach space X, α(T(t)) = p and δ(T(t)) = q. If there exists m > 0 such that X = R(T(m))N(T{m)), then α(T(t)) = δ(T(t))≤ m, moreover, X = R(T(p)) N(T(q)).Theorem 2.7. Let A be an infinitesimal generator of a linear operator C₀ semigroup T(t)(t ≥ 0) defined on a complex Banach space X. If the resolvent R(z, A) is a meromorphic function with finite exponential type, then the ascent of {T(t)} is finite.Theorem 2.8. Let A be a discrete operator and an infinitesimal generator of a linear operator C₀ semigroup T(t)(t ≥ 0) defined on a separable complex Banach space X. If there is a sequence of circular arc contours C_l(l = 1,2, ...) such that(i)Cl is contained in the left plane Re(z) < 0, and constitutes a closed contour, where αl is a positive real number.