Π_(k)空间上算子的运算及谱性质

The Operation and Spectral Properties of Operators onΠ_(k)Space

考虑Pontrjagin空间上具有零性不变子空间算子在正规分解下三角表示的运算问题,给出算子的基本运算公式。基于这些运算公式,得到正规算子、酉算子的充分条件和自伴条件的充分必要条件,以及一个一般算子点谱与近似点谱的性质。这些性质也反映出三角算子矩阵的谱分割性质,并证明自伴算子的非实谱是共轭成对出现的。

Under normal decomposition,we consider the operation of the triangle representation for operators which have a neutral invariant subspace on Pontrjagin space,and establish the basic operating formula of the operators.Based on these formulas,the conditions of the formality and unitarity of the operators are given respectively as well as the sufficient and necessary conditions of the self-adjointness of the operators.At the same time,the properties of point spectrum and approximate point spectrum of a general operator are obtained.These properties reflect the spectral partition property of triangular operator matrix,and prove the fact that the non-real spectrum of self-adjoint operator appears in conjugate pairs.