正交投影的子矩阵
Submatrix of Orthogonal Projection
-
摘要: 设\mathcal H是n维复Hilbert空间,Q是定义在\mathcal H上的正交投影. 任给\mathcal H的子空间\mathcal M, 设\dim\mathcal M=r, 在空间分解 \mathcal H=\mathcal M\oplus\mathcal M^\perp下, Q=\left(\beginarrayccAB\\ B^*D\endarray\right), 其中A\in\mathcal B(\mathcal M), B\in\mathcal B(\mathcal M^\perp,\mathcal M), D\in\mathcal B(\mathcal M^\perp). 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\mathcal M上的正交投影P与Q之间的关系. 得到了(i) \mathcal R(P)\cap\mathcal R(Q)=\0\ \Leftrightarrow \dim \mathcal R(A)=\dim \mathcal R(B), (ii) \mathcal R(P)+\mathcal R(Q)=\mathcal H \Leftrightarrow \dim \mathcal R(D)=n-r, (iii) \mathcal R(P)\perp\mathcal R(Q) \Leftrightarrow \dim \mathcal R(A)=0.Abstract: Let \mathcal H be a n-dimensional complex Hilbert space, and Q be an orthogonal projection on \mathcal H. If \mathcal M is a subspace of \mathcal H and \dim\mathcal M=r, then under the space decomposition \mathcal H=\mathcal M\oplus\mathcal M^\perp, Q=\left(\beginarrayccAB\\B^*D\endarray\right), where A\in\mathcal B(\mathcal M), B\in\mathcal B(\mathcal M^\perp,\mathcal M), D\in\mathcal B(\mathcal M^\perp). 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 \mathcal M, (i) \mathcal R(P)\cap\mathcal R(Q)=\0\ \Leftrightarrow \dim \mathcal R(A)=\dim \mathcal R(B), (ii) \mathcal R(P)+\mathcal R(Q)=\mathcal H \Leftrightarrow \dim \mathcal R(D)=n-r, (iii) \mathcal R(P)\perp\mathcal R(Q) \Leftrightarrow \dim \mathcal R(A)=0 are obtained.
下载: