Completeness of the system of eigenvectors of off-diagonal operator matrices and its applications in elasticity theory

Abstract

Huang Jun-Jie(黄俊杰) a) , Alatancang(阿拉坦仓) a) , and Wang Hua(王 华) a)b) †

Completeness of system of root vectors of upper triangular infinite-dimensional Hamiltonian operators appearing in elasticity theory

“…Since the operator P Q possesses the countable eigenvalues, by (ii) of Lemma 1 in [14], the off-diagonal operator A = 0 P Q 0 also possesses the countable…”

“…Therefore, (14) is valid. Moreover, we assume that there is another constant sequence {d k , d −k , d k , d −k | k ∈ Λ} such that the expansion (14) is valid.…”

“…Therefore, the system {U k , V k | k = ±1, ±2, · · · } of eigenvectors of the Hamiltonian operator H B is complete in X 4 . Necessity Similar to the proof of Theorem 1 in [14], we omit it here. Therefore, the proof is completed.…”

“…The method breaks the limitation of self-adjointness, extends the Hilbert-Schmidt theorem, and brings many important problems to be solved [6][7][8][9][10] . However, its mathematical foundation, i.e., the completeness of the system of eigenvectors or root vectors, has not been systemically established [11][12][13][14] . Moreover, the geometric multiplicity and algebraic multiplicity of the eigenvalue should be first discussed.…”

On the completeness of eigen and root vector systems for fourth-order operator matrices and their applications

“…20, No. 10 (2011) 100202 vector system of the operator matrix M is complete in the sense of Cauchy principal value in the space X 4 if and only if the vector systems { x k } k∈Λ and { y 0 , P * x k /λ k } k∈Λ are both orthogonal bases in X, where λ k = √ ν k .…”