Functional Calculus And Perturbation Theory Of Self-adjoint Operators

H is a Hilbert space.V is a dense subspace of H.A(t)is a family operators of self-adjoint operator which has the same domain.A(t)has compact resolvent.A(t)has some properties of continuity,is their eigenvalue can be parameterized continuous,or even real analytic,twice differentiable and C1 function?Rellich showed that when A(t)is real analytic,its eigenvalues can be parameterized by real analytic function.Kato proved that when A(t)is smooth,its eigenvalues can be parameterized by twice differentiable functions.Kriegl and Michor prove that when A(t)is C1,?,then its eigenvalues can be parameterized by the C1 function.In this paper,we arrange the conclusions of Kriegl and Michor.If A(t)is Cn,n?3 and the multiplicity of those eigenvalues is not exceeded n/3,then the eigenvalues near it can be parameterized by twice differentiable functions.In the segment of the sectorial operator calculus,we apply a holomorphic function f to the sectorial operator B to obtain the real analytic map B?f(B).Finally,the perturbation of the self-adjoint operator and the calculus of the sectorial operator are used to study the self-adjoint operators with compact resolvent on mapping space.In the first chapter,firstly,I introduce some concepts and conclusions about Banach spaces that are often used in this paper.Secondly,some conclusions about functional calculus in banach Spaces are introduced.Finally,I will talk about some conclusions about the calculus of sectorial operators.In the second chapter,the first block is the perturbation theory of the self-adjoint operator,and the second block is some functional calculus results of the sectorial operator.In the third chapter,I mainly introduce the mapping space on Riemannian geometry,completing the mapping space into Sobolev space according specific norm on the mapping space of section.In the first order natural bundle,I talk about the concept of fiber metric,covariant derivative and Laplacian.Discuss real analytic map from g??Ho(S2+T*M)to fiber metrics,covariant derivatives and Laplace operators.In last chapter,we study the operator 1+?9 by using the functional calculus of the sectorial operator,then I use the perturbation theory of the self-adjoint operator to parameterize its eigenvalues.