Projection Inequalities and Their Linear Preservers

Document Type : Research Paper


1 Graduate University of Advanced Technology, Kerman, Iran.

2 University of Sussex, Brighton, United Kingdom.



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.


[1] T. Ando, Majorization, doubly stochastic matrices and comparison of eigenvalues, Linear Algebra Appl., 118 (1989), 163-248.
[2] A. Armandnejad, S. Mohtashami and M. Jamshidi, On linear preservers of g-tridiagonal majorization on R^n, Linear Algebra Appl., 459 (2014), 145-153.
[3] A. Armandnejad and M. Jamshidi, Multiplicative isomorphisma at invertible matrices, Miskolc Math. Notes, 15(2) (2014), 287-292.
[4] G. Bennet, Majorization versus power majorization, Anal. Math., 12(4) (1986), 283-286.
[5] A. Armandnejad and A. Salemi, The structure of linear preservers of gs-majorization, Bull. Iran. Math. Soc., 32(2) (2006), 31-42.
[6] G. Dahl, Matrix majorization, Linear Algebra Appl., 288 (1999), 53-73.
[7] A.W.  Marshall and  Ingram Olkin, Inequalities: Theory of Majorization and Its Applications, Academic Press, 1979.