﻿ 矩阵算子的广义正交问题的研究

# 矩阵算子的广义正交问题的研究Study on Generalized Orthogonal Problems of Matrix Operators

Abstract: This paper considers that in the operator space, taking T1, T2 and as n × n matrices, the equivalent conditions of T1 and T2 satisfying the orthogonal Birkhoff, isospheric orthogonal and Roberts are given.

1. 引言

2. 算子空间中n×n矩阵广义正交的等价条件

$‖x+\alpha y‖\ge ‖x‖$

$‖x+y‖=‖x-y‖$

$‖x+\alpha y‖=‖x-\alpha y‖$

${‖x+y‖}^{2}={‖x‖}^{2}+{‖y‖}^{2}$

1) 正定性： $‖A‖\ge 0$，当 $A=0$ 时等号成立。

2) 齐次性：任意给出 $k\in C,A\in {C}^{n×n}$，都有 $‖kA‖=|k|‖A‖$

3) 三角不等式：任意给出 $A,B\in {C}^{n×n}$，都有 $‖A+B‖\le ‖A‖+‖B‖$

4) 任意给出 $A,B\in {C}^{n×n}$，都有 $‖AB‖\le ‖A‖‖B‖$

$\begin{array}{c}{‖{T}_{1}+\lambda {T}_{2}‖}^{2}\ge {‖\left({T}_{1}+\lambda {T}_{2}\right)x‖}^{2}={‖{T}_{1}x‖}^{2}+{\lambda }^{2}{‖{T}_{2}x‖}^{2}+2\lambda \mathrm{Re}〈{T}_{1}x,{T}_{2}x〉\\ ={‖{T}_{1}x‖}^{2}+{\lambda }^{2}{‖{T}_{2}x‖}^{2}\ge {‖{T}_{1}x‖}^{2}={‖{T}_{1}‖}^{2}\end{array}$

${T}_{1}x=‖{T}_{1}‖x$$\mathrm{Re}〈{T}_{1}x,{T}_{2}x〉=0$

$\begin{array}{c}{‖{T}_{1}+{T}_{2}‖}^{2}={‖\left({T}_{1}+{T}_{2}\right)x‖}^{2}={‖{T}_{1}x‖}^{2}+{‖{T}_{2}x‖}^{2}+2\mathrm{Re}〈{T}_{1}x,{T}_{2}x〉\\ ={‖{T}_{1}x‖}^{2}+{‖{T}_{2}x‖}^{2}={‖{T}_{1}‖}^{2}+{‖{T}_{2}‖}^{2}\end{array}$

$\begin{array}{c}{‖{T}_{1}-{T}_{2}‖}^{2}={‖\left({T}_{1}-{T}_{2}\right)x‖}^{2}={‖{T}_{1}x‖}^{2}+{‖{T}_{2}x‖}^{2}-2\mathrm{Re}〈{T}_{1}x,{T}_{2}x〉\\ ={‖{T}_{1}x‖}^{2}+{‖{T}_{2}x‖}^{2}={‖{T}_{1}‖}^{2}+{‖{T}_{2}‖}^{2}\end{array}$

$\left({‖\left({T}_{1}+{T}_{2}\right)x‖}^{2}-{‖\left({T}_{1}-{T}_{2}\right)x‖}^{2}\right)=4\mathrm{Re}〈{T}_{1}x,{T}_{2}x〉=0$

$\begin{array}{c}{‖{T}_{1}+\lambda {T}_{2}‖}^{2}={‖\left({T}_{1}+\lambda {T}_{2}\right)x‖}^{2}={‖{T}_{1}x‖}^{2}+{|\lambda |}^{2}{‖{T}_{2}x‖}^{2}+2\mathrm{Re}\stackrel{˜}{\lambda }〈{T}_{1}x,{T}_{2}x〉\\ ={‖{T}_{1}x‖}^{2}+{|\lambda |}^{2}{‖{T}_{2}x‖}^{2}={‖{T}_{1}‖}^{2}+{|\lambda |}^{2}{‖{T}_{2}‖}^{2}\end{array}$

$\begin{array}{c}{‖{T}_{1}-\lambda {T}_{2}‖}^{2}={‖\left({T}_{1}-\lambda {T}_{2}\right)x‖}^{2}={‖{T}_{1}x‖}^{2}+{|\lambda |}^{2}{‖{T}_{2}x‖}^{2}-2\mathrm{Re}\stackrel{˜}{\lambda }〈{T}_{1}x,{T}_{2}x〉\\ ={‖{T}_{1}x‖}^{2}+{|\lambda |}^{2}{‖{T}_{2}x‖}^{2}={‖{T}_{1}‖}^{2}+{|\lambda |}^{2}{‖{T}_{2}‖}^{2}\end{array}$

$\left({‖\left({T}_{1}+\lambda {T}_{2}\right)x‖}^{2}-{‖\left({T}_{1}-\lambda {T}_{2}\right)x‖}^{2}\right)=4\mathrm{Re}\stackrel{˜}{\lambda }〈{T}_{1}x,{T}_{2}x〉=0$

$\begin{array}{c}{‖{T}_{1}+{T}_{2}‖}^{2}={‖\left({T}_{1}+{T}_{2}\right)x‖}^{2}={‖{T}_{1}x‖}^{2}+{‖{T}_{2}x‖}^{2}+2\mathrm{Re}〈{T}_{1}x,{T}_{2}x〉\\ ={‖{T}_{1}x‖}^{2}+{‖{T}_{2}x‖}^{2}={‖{T}_{1}‖}^{2}+{‖{T}_{2}‖}^{2}\end{array}$

${‖\left({T}_{1}+{T}_{2}\right)x‖}^{2}={‖{T}_{1}x‖}^{2}+{‖{T}_{2}x‖}^{2}$ (1)

${‖\left({T}_{1}+{T}_{2}\right)x‖}^{2}={‖{T}_{1}x‖}^{2}+{‖{T}_{2}x‖}^{2}+2\mathrm{Re}〈{T}_{1}x,{T}_{2}x〉$ (2)

3. 结论

[1] 吴森林, 计东海. 正交性相关问题的研究[D]: [硕士学位论文]. 哈尔滨: 哈尔滨理工大学, 2006: 34-46.

[2] Birkhoff, G. (1935) Orthogonality in Linear Metric Spaces. Duke Mathematical Journal, 1, 169-172.
https://doi.org/10.1215/S0012-7094-35-00115-6

[3] James, R.C. (1945) Orthogonality in Normed Linear Spaces. Duke Mathematical Journal, 12, 291-301.
https://doi.org/10.1215/S0012-7094-45-01223-3

[4] Roberts, B.D. (1934) On the Geomrtry of Abstract Vector Spaces. Tohoku Mathematic Journal, 39, 42-59.

[5] 任芳国, 高莹. 随机矩阵的范数[J]. 东北师大学报(自然科学版), 2012, 44(1): 28-31.

[6] Gustafson, K. (1970) The Toeplitz-Hausdorff Theorem for Linear Operators. Proceedings of the American Mathematical Society, 25, 203-204.
https://doi.org/10.1090/S0002-9939-1970-0262849-9

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