斜对角无穷维Hamilton算子的数值域和二次数值域

Numerical Range and Quadratic Numerical Range of Off-Diagonal Infinite Dimensional Hamiltonian Operators

摘要: 本文研究了斜对角无穷维Hamilton算子的数值域和二次数值域，利用数值域和二次数值域的定义以及酉算子的不变性得到了其数值域和二次数值域关于虚轴、实轴、原点对称的充分条件，即D(B)=D(C)。其次给出了斜对角无穷维Hamilton算子数值域和二次数值域的闭包包含谱集的结论。

Abstract: In this paper, the numerical range and quadratic numerical range of diagonal infinite dimensional Hamiltonian operators are studied. By using the definitions of numerical range and quadratic numerical range and the invariance of unitary operator, we obtain the sufficient conditions for the numerical range and quadratic numerical range are symmetric with respect to the imaginary axis, the real axis and the origin, D(B)=D(C). Secondly, we give the conclusion that the closure of the numerical range and quadratic numerical range of off-diagonal infinite dimensional Hamiltonian operators contains spectral sets.