Jamshidi, M., Fatehi, F. (2017). Projection Inequalities and Their Linear Preservers. Wavelet and Linear Algebra, 4(2), 61-67. doi: 10.22072/wala.2017.63024.1115

Mina Jamshidi; Farzad Fatehi. "Projection Inequalities and Their Linear Preservers". Wavelet and Linear Algebra, 4, 2, 2017, 61-67. doi: 10.22072/wala.2017.63024.1115

Jamshidi, M., Fatehi, F. (2017). 'Projection Inequalities and Their Linear Preservers', Wavelet and Linear Algebra, 4(2), pp. 61-67. doi: 10.22072/wala.2017.63024.1115

Jamshidi, M., Fatehi, F. Projection Inequalities and Their Linear Preservers. Wavelet and Linear Algebra, 2017; 4(2): 61-67. doi: 10.22072/wala.2017.63024.1115

Projection Inequalities and Their Linear Preservers

^{1}Graduate University of Advanced Technology, Kerman, Iran.

^{2}University of Sussex, Brighton, United Kingdom.

Abstract

This paper introduces an inequality on vectors in $\mathbb{R}^n$ which compares vectors in $\mathbb{R}^n$ based on the $p$-norm of their projections on $\mathbb{R}^k$ ($k\leq n$). For $p>0$, we say $x$ is $d$-projectionally less than or equal to $y$ with respect to $p$-norm if $\sum_{i=1}^k\vert x_i\vert^p$ is less than or equal to $ \sum_{i=1}^k\vert y_i\vert^p$, for every $d\leq k\leq n$. For a relation $\sim$ on a set $X$, we say a map $f:X \rightarrow X$ is a preserver of that relation, if $x\sim y$ implies $f(x)\sim f(y)$, for every $x,y\in X$. All the linear maps that preserve $d$-projectional equality and inequality are characterized in this paper.