摘要:
设H是n维复Hilbert空间,Q是定义在H上的正交投影. 任给H的子空间M, 设dimM=r, 在空间分解 H=M⊕M⊥下, Q=(ABB∗D), 其中A∈B(M),B∈B(M⊥,M),D∈B(M⊥). 利用算子分块的技巧, 对空间进一步分解, 讨论了Q的子矩阵A,B,D的性质及其之间的关系, 并进一步讨论了M上的正交投影P与Q之间的关系. 得到了(i) R(P)∩R(Q)=\{0\}⇔dimR(A)=dimR(B), (ii) R(P)+R(Q)=H⇔dimR(D)=n−r, (iii) R(P)⊥R(Q)⇔dimR(A)=0.}
Abstract:
Let H be a n-dimensional complex Hilbert space, and Q be an orthogonal projection on H. If M is a subspace of H and dimM=r, then under the space decomposition H=M⊕M⊥, Q=(ABB∗D),where A∈B(M),B∈B(M⊥,M),D∈B(M⊥). In this paper, using of the technique of block operator matrix, the properties and relations between A,B and D are given. Furtherly, the relations between P and Q are discussed, where P is an orthogonal projection on M, (i) R(P)∩R(Q)=\{0\}⇔dimR(A)=dimR(B), (ii) R(P)+R(Q)=H⇔dimR(D)=n−r, (iii) R(P)⊥R(Q)⇔dimR(A)=0 are obtained.
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