Study On The Two Classes Of Subspaces On Banach Space

In this paper,we mainly study the two class important invariant subspaces of linearly bounded operator on complex Banach spaces which is the analytical core and the quasi nilpotent space, we mainly discuss this two class important invariant subspaces by means of the local spectral theory, leading to some new results.Fristly,we discuss the hereditary invariability of the closed property of the an-alytical core to restriction on its invariant subspace(see Lemma 2.2.2 to Theorem 2.2.7); then we discuss the relationship between the analytical core of TS-λI and ST -λI(see Theorem 2.3.1 to Cor.2.3.2); we discuss the relationship between the analytical core K(TS) of the opetator TS and the analytical core K(T),K(S) when TS=ST (see Theorem 2.4.8); we introduce the analytical core's spectrum and the (K) property using the analytical core and discuss the relationship between the (K) property and SEVP.Secondly, we discuss the quasi nilpotent space. We discuss the ralationship between the closed property of TS and ST(see Theroem 3.2.2) and prove that TS∈HP if and only if ST∈HP;we give the equivalent description for the quasi-nilpotent part of a semi-B-Fredholm opetator to be closed by means of the SEVP (see Theroem 3.3.2). Moreover, we prove the stability that quasi nilpotent space of a opetator which have generalized Kato decomposition is always closed(see Theroem 3.3.3).