A characterization of linear operators that preserve isolation numbers

Abstract

We obtain  characterizations of Boolean linear operators that preserve some of the isolation numbers  of Boolean matrices.  In particular, we show that the following are equivalent:  (1) $T$ preserves the isolation number of all matrices; (2) $T$  preserves the set of matrices with isolation number one and the set of those with isolation number $k$ for some $2\leq k\leq \min\{m,n\}$; (3) for $1\leq k\leq \min\{m,n\}-1$, $T$ preserves matrices
with isolation number $k$, and those with isolation number $k+1$, (4) $T$ maps $J$ to $J$ and preserves the set of matrices of isolation number 2;  (5)  $T$ is a $(P,Q)$-operator, that is, for fixed permutation matrices $P$ and $Q$,  $m\times n$ matrix $X,$~ $T(X)=PXQ$ or, $m=n$ and  $T(X)=PX^tQ$ where $X^t$ is the transpose of $X$.