Finite-rank perturbations of positive operators and isometries

Man-Duen Choi¹ ; Pei Yuan Wu²

¹Department of Mathematics University of Toronto Toronto, Ontario M5S 2E4, Canada
²Department of Applied Mathematics National Chiao Tung University Hsinchu 300, Taiwan

Studia Mathematica, Tome 173 (2006) no. 1, pp. 73-79

Résumé

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 $B$.

Détail

DOI : 10.4064/sm173-1-5

Keywords: completely characterize ranks a b b operators hilbert space satisfying geq geq namely nonnegative integers infinity mathop rank a b mathop rank b operators geq geq hilbert space dimension leq leq infty only leq leq particular answers negative question posed benhida whether positive operators finiteness mathop rank a b implies mathop rank b isometries necessary sufficient conditions order finite rank perturbations each other condition says isometries a b has finite rank only unitary operator finite rank nbsp another condition terms parts wold lebesgue decompositions nonunitary isometries nbsp