2 × 2 上三角型算子矩阵的闭值域性问题研究

Research on Closedness Range of 2 × 2Upper Triangular Operator Matrices

摘要: 设 H₁ 和 H₂ 是无穷维可分的 Hilbert 空间, 记M = ( _0  B^A  C )为H₁⊕H₂上的上三角型算子矩阵. 本文基于空间分解法, 利用矩阵元 A, B, C 的值域和零空间性质研究了算子矩阵 M 的值域 闭性, 并给出了ρ_cr(M) = ρ_cr(A)∩ρ_cr(B)成立的条件, 其中 ρ_cr(M) = {λ∈ℂ : R(M - λI) = R(M - λI)}

Abstract: Let H₁ and H₂ be infinite dimensional separable Hilbert spaces and M = ( _0  B^A  C ) be a 2 × 2 upper triangular operator matrix acting on H₁⊕H₂. In this paper, the closedness of the range R(M ) is described by using the range and the null spaces of A, B, C and the spatial decomposition method. In addition, the conditions under which ρ_cr(M) = ρ_cr(A)∩ρ_cr(B) holds are given, where ρ_cr(M) = {λ∈ℂ : R(M - λI) = R(M - λI)}.