Common properties of bounded linear operators AC and BA: Local spectral theory

Let X, Y   be Banach spaces, A:X→Y$A:X\to Y$ and B,C:Y→X$B,C:Y\to X$ be bounded linear operators satisfying operator equation ABA=ACA$ABA=ACA$. In this paper, we show that AC and BA share the local spectral properties such as Bishop's property (β), subscalarity and Dunford's property (C). Also, the quasi-nilpotent part and the analytic core for AC and BA are studied.     

Common properties of the operator products in local spectral theory

Let X, Y be Banach spaces, A, D : X → Y and B, C : Y → X be the bounded linear operators satisfying operator equation set AC D = DBD,DBA = AC A.In this paper, we show that AC and BD share some basic operator properties such as the injectivity and the invertibility. Moreover, we show that AC and BD share many common local spectral properties including SVEP, Bishop property(β) and Dunford property(C).     

Common properties of operatorsRS andSR andp-hyponormal operators

LetR andS be bounded linear operators on a Bananch space. We discuss the spectral and subdecomposable properties and properties concerning invariant subspaces common toRS andSR. We prove that, by these properties,p-hyponormal and log-hyponormal operators and their generalized Aluthge transformations are all subdecomposable operators;T andT(r, 1–r)(0<r<1) have same spectral structure and equal spectral parts ifT denotesp-hyponormal or dominant operator; for everyT L(H), 0<r<1,T has nontrivial (hyper-)invariant subspace ifT(r, 1–r) does.This research was supported by the National Natural Science Foundation of China.     

Spectral theory for general nonautonomous/random dispersal evolution operators

We investigate the spectral theory of the following general nonautonomous evolution equation

     

Spectral theory of difference operators with almost constant coefficients II

We show that the operators whose coefficients are approximately constant in a general sense have an absolutely continuous spectrum which is equal to that of the corresponding constant coefficient operator. For such operators, the absolutely continuous spectrum can be read off from the associated characteristic polynomial. This generalizes the classical results on second-order operators and extends those of higher order differential operators to the difference setting. Our approach relies on an analysis of the associated difference equation with the help of uniform asymptotic summation techniques.     

Generalization of the determinants for trace-potent linear operators

SupposeA is a bounded linear operator on a separable Hilbert space withA ^m of trace class for some positive integerm. A generalized determinant for the operatorI–A is defined, its properties studied and this determinant is then used to exhibit an inversion formula forI–A.     

Spectral properties of a compactly perturbed linear span of projections

Spectral properties of the sum of a linear span of projections and a compact nonnegative operator are considered. In particular, we prove partial completeness results for certain parts of the system of eigenfunctions. The main tool is a transformation the original spectral problem to that of a monic weakly hyperbolic pencil.     

Perturbation of the Drazin inverse for closed linear operators

We investigate the perturbation of the Drazin inverse of a closed linear operator recently introduced by second author and Tran, and derive explicit bounds for the perturbations under certain restrictions on the perturbing operators. We give applications to the solution of perturbed linear equations, to the asymptotic behaviour ofC ₀-semigroups of linear operators, and to perturbed differential equations. As a special case of our results we recover recent perturbation theorems of Wei and Wang.     

On semibounded restrictions of self-adjoint operators

After the von Neumann's remark [10] about pathologies of unbounded symmetric operators and an abstract theorem about stability domain [9], we develope here a general theory allowing to construct semibounded restrictions of selfadjoint operators in explicit form. We apply this theory to quantum-mechanical momentum (position) operator to describe corresponding stability domains. Generalization to the case of measurable functions of these operators is considered. In conclusion we discuss spectral properties of self-adjoint extensions of constructed self-adjoint restrictions.