Spectral Analysis of the Bounded Linear Operator in the Reproducing Kernel Space W 2 m(D)

We first introduce some related definitions of the bounded linear operator L in the reproducing kernel space W 2 m(D). Then we show spectral analysis of L and derive several property theorems.


Introduction
It is well known that spectral analysis of linear operators [1] is an important topic in functional analysis. For example, the matrix eigenvalue in linear algebra and eigenvalue problems for differential equation have been discussed emphatically. Two major reasons are as follows. Firstly, spectral analysis arises from vibration frequency problems and the stability theory of system. Secondly, spectral analysis comes from the need of discussing the structure of the operator and solving the corresponding equation by using the eigenvalue and spectral theorem. Also, spectral analysis can be used to study the structure of the solution for homogeneous or nonhomogeneous differential system and the normalized form of matrix which can be obtained clearly by matrix eigenvalues.
So far, the spectral decomposition method [2] has become a central topic in the theory of spectral analysis of linear operators. This method has been successfully applied in Hilbert space and perfect spectral decomposition theorem [3]. In recent years, the spectral decomposition method has been developed into spectral theorems of spectral operators and decomposable operators in Banach space [4,5].
The paper is organized as follows. In Section 2, we introduce some related definitions for the eigenvalue of the bounded linear operator in the reproducing kernel space 2 ( ). In Section 3, the regular point and the spectral point of bounded linear operator in 2 ( ) are given. In Section 4, we show spectral analysis of the bounded linear operator and also establish several theorems. Section 5 ends this paper with a brief conclusion.

Related Definitions
Definition 1. Let be an abstract set, 2 ( ) the reproducing kernel space, and [ 2 ( ) → 2 ( )] the bounded linear operator space. ∀ ∈ [ 2 ( ) → 2 ( )] with , ∈ N, if there exists nonvanishing vector ∈ 2 ( ), such that Then is called an eigenvalue of and is called the eigenvector of according to , where denotes the identity operator.
and all eigenvectors and zero vector of compose the eigenvector space which is denoted by .
then is the eigenvalue of if and only if the following integral equation has nonzero solution: is a linear independence vector system, then (3) can be converted into the equivalent equation Thus we have the following results.

Regular Point and Spectral Point
In Definitions 1-3, we introduce the eigenvalue, eigenvector, eigenvector space, and dim of for the homogeneous equation (1). However, for many problems in mathematics and physics, we just need to solve the following nonhomogeneous equation: where is a given operator, is a given vector, and is an unknown vector. In order to discuss this problem, we need to introduce the following definitions and theorems.
Particularly, when D( ) = 2 ( ) and is a bounded linear operator, we can derive the following results.
if is a regular operator, then * is also a regular operator and by taking conjugate on both sides of the above formulas, we obtain * ( −1 ) In view of Theorem 6, we can see that * is a regular operator and ( * ) −1 = ( −1 ) * . The proof is complete.
(1) − is a regular operator, that is, − is a one-toone linear operator from D( ) to 2 ( ). In addition, the inverse operator ( − ) −1 is a linear bounded operator. Then is called a regular point of . All regular points compose the regular set of , which is denoted by ( ).
(2) If is not a regular point, then is called a spectral point of . All spectral points compose the spectral set of , which is denoted by ( ).
In view of Definition 8, we have ( ) ⋃ ( ) = C. Then we have the following property results. Proof. ⇒ Since R( − ) = 2 ( ), then ∀ ∈ 2 ( ), ∃ ∈ 2 ( ) such that ( − ) = . In addition, in view of the boundedness of ( − ) −1 and the Cauchy-Schwartz inequality, we have that is, Hence, ( − ) −1 exists and is a bounded linear operator. Summing up the above, − is a regular operator, where is a regular point of .
Lemma 9 shows that ∀ ∈ 2 ( ); when is a continuous linear operator and is the regular point of , has a unique solution . Furthermore, the continuity of depends on the right term. In other words, if { } =1 are column vectors and → , then → .

Lemma 10. Let be a bounded linear operator in the repro-
Proof. Otherwise, the invertible operator can convert the nonvanishing vector to nonvanishing vector. Hence, there exists ̸ = 0 such that ( − ) = 0. That is, is not the eigenvalue of . When 2 ( ) is a finite dimension space and is not the eigenvalue of , we can derive that = − is an invertible mapping. Obviously, R( ) = 2 ( ). In fact, let { } =1 be the basis of 2 ( ); then {( − ) } =1 is a linear independent system in 2 ( ) and also a basis of 2 ( ). Therefore, R( ) = 2 ( ). In view of the inverse operator Theorem, ( − ) −1 is bounded. It follows that ∈ ( ).
So, the proof of the theorem is complete.
Lemma 10 shows that regular point and spectral point are absolutely opposite for finite dimension normed spaces. That is, spectral point of can only be an eigenvalue in finite dimension normed space. This is entirely consistent with the conclusion of the theory of linear algebra. But if 2 ( ) is an infinite-dimensional space and is not the eigenvalue of , then may not be a regular point of , so far as − is not a map from 2 ( ) to 2 ( ).
For example, let ∀ ∈ C, ∫ ( ) = ( ) has only zero solutions. Hence, has not eigenvalue. That is, zero is not the eigenvalue. However, the range of values is all functions of the from ∫ ( ) for (0 − ). This shows that the spectral point is complex in infinite-dimensional space for the operator . Now, we will classify the spectral set by three situations.
(a) If − is not one-to-one, then is called point spectral of ; the set of point spectral is denoted by ( ).

(b) If
− is one-to-one and R( − ) is dense in 2 ( ), then is called continuous spectral of ; the set of continuous spectral is denoted by ( ).
(1) Consider 1 ∈ ( ). namely, Furthermore, we have Then we obtain It follows that is a regular point of and In addition, when | | > ‖ ‖, we have ‖ / ‖ < 1. In view of (2) of Theorem 11, we have The proof is complete.
The proof is complete.
From the purpose of solving equations, spectral radius has the following meanings.
(1) For | | > ( ), due to the fact that is a regular point of , then for any ∈ 2 ( ), ( − ) = has a unique solution .
(2) For | | ≤ ( ), it cannot guarantee this equation has a solution for any ∈ 2 ( ). In many practical problems, in order to calculate the spectral range, one needs to estimate the spectral radius. In terms of (4) of Theorem 13, we can get ( ) ≤ ‖ ‖. In practical terms, this estimate is convenient, but it is imprecise. Proof. In terms of (4) of Theorem 13, ( ) ≤ lim → ∞ √‖ ‖.
The proof is complete.
Definition 17 is the finite-dimensional space concept nilpotent operator in the infinite-dimensional space to promote. In the spectral theory of operators, generalized nilpotent operator is a kind of important operator.
In terms of Theorem 16 and the spectral radius theorem, one can obtain that the generalized nilpotent operator has only a spectral point 0.
In terms of the property of , one obtains Note that This shows that is a generalized nilpotent operator; spectral point = 0 is not the eigenvalue of .

Conclusions
This paper first introduces the eigenvalue, eigenvector, eigenvector space, and dim of the bounded linear operator in the reproducing kernel space 2 ( ). Then we show some definitions and properties of the regular operator. The regular set and spectral set of bounded linear operator are also introduced. From the solvability of the equation, we show the spectral classification and give three conditions. Finally, we introduce the spectral analysis of the bounded linear operator . It includes the definitions of spectral radius, nilpotent operator, approximate spectral point, and remainder spectral point. We also establish some property theorems of the bounded linear operator in the reproducing kernel space