一类无穷维Hamilton算子的谱刻画及其可逆性

SPECTRUM AND INVERTIBILITY OF A CLASS OF INFINITE DIMENSIONAL HAMILTONIAN OPERATORS

研究了定义域为对角形的上三角无穷维Hamilton算子:H=的谱刻画及其可逆性;当A的剩余谱不含关于虚轴对称的点时,H的谱等于A的谱与A的谱关于虚轴对称分支的并集,并得到了H的预解集为空及其可逆的充要条件;作为结论的应用,得到当A为无穷维Hamilton算子时,H的点谱、剩余谱、连续谱和谱分别等于A的点谱、剩余谱、连续谱和谱.

This paper is concerned with the spectrum and invertibility of a class of infinite dimensional Hamiltonian operators described by H =[],where D（H）= D（A）×D（—A~＊） and A is a linear operator in a Hilbert space.If residual spectrum of A excludes the symmetric point about imaginary axis,then the spectrum of H is composed of the spectrum of A and its symmetric points about imaginary axis.The sufficient and necessary conditions for the resolvent set of H to be empty and for H to be invertible are obtained. In particular,if A itself is a Hamiltonian operator,the point spectrum,residual spectrum, continuous spectrum of H is exactly same as those of A.