投影的约化极小模和空间的夹角

On the Reduced Minimum Modulus of Projections and the Angle between Two Subspaces

中文关键词:  子空间  空间夹角  约化极小模.

英文关键词:subspace  angle between two subspaces  reduced minimum modulus.

中文摘要:

      设 $\mathcal{M}$ 和 ${\mathcal{N}}$分别表示希尔伯特空间$\mathcal{H}$的子空间,$P_{\mathcal{M}}$和$P_{\mathcal{N}}$分别表示在空间$\mathcal{M}$ 和${\mathcal{N}}$上的正交投影.本文给出了$\mathcal{M}$ 和${\mathcal{N}}$的夹角和最小间隔的一个新的表示,同时也讨论了$\mathcal{M}$ 和${\mathcal{N}}$的夹角,最小间隔,及算子$(I-P_{\mathcal{N}})P_{\mathcal{M}}$的约化极小模之间的关系.

英文摘要:

      Let $\mathcal{M}$ and ${\mathcal{N}}$ be nonzero subspaces of a Hilbert space $\mathcal{H},$ and $P_{\mathcal{M}}$ and $P_{\mathcal{N}}$ denote the orthogonal projections on $\mathcal{M}$ and ${\mathcal{N}},$ respectively. In this note, an exact representation of the angle and the minimum gap of $\mathcal{M}$ and ${\mathcal{N}}$ is obtained. In addition, we study relations between the angle, the minimum gap of two subspaces $\mathcal{M}$ and ${\mathcal{N}},$ and the reduced minimum modulus of $(I-P_{\mathcal{N}})P_{\mathcal{M}}$.