正交投影的子矩阵

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, $Q=\\left(\\begin{array}{cc}AB\\\\ B^*D\\end{array}\\right),$ 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) ${\\mathcal {R}}(P)\\perp{\\mathcal {R}}(Q) \\Leftrightarrow \\dim {\\mathcal {R}}(A)=0.$}

SUBMATRIX OF ORTHOGONAL PROJECTION

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(\begin{array}{cc}AB\\B^*D\end{array}\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.