Finite-rank perturbations of positive operators and isometries
Volume 173 / 2006
Abstract
We completely characterize the ranks of $A-B$ and $A^{1/2}-B^{1/2}$ for operators $A$ and $B$ on a Hilbert space satisfying $A\geq B\geq 0$. Namely, let $l$ and $m$ be nonnegative integers or infinity. Then $l=\mathop{\rm rank} (A-B)$ and $m=\mathop{\rm rank} (A^{1/2}-B^{1/2})$ for some operators $A$ and $B$ with $A\geq B\geq 0$ on a Hilbert space of dimension $n$ ($1\leq n\leq \infty$) if and only if $l=m=0$ or $0< l\leq m\leq n$. In particular, this answers in the negative the question posed by C. Benhida whether for positive operators $A$ and $B$ the finiteness of $\mathop{\rm rank} (A-B)$ implies that of $\mathop{\rm rank} (A^{1/2}-B^{1/2})$.
For two isometries, we give necessary and sufficient conditions in order that they be finite-rank perturbations of each other. One such condition says that, for isometries $A$ and $B$, $A-B$ has finite rank if and only if $A=(I+F)B$ for some unitary operator $I+F$ with finite-rank $F$. Another condition is in terms of the parts in the Wold–Lebesgue decompositions of the nonunitary isometries $A$ and $B$.
