Finite-rank perturbations of positive operators and isometries

Man-Duen Choi, and Pei Yuan Wu. "Finite-rank perturbations of positive operators and isometries." Studia Mathematica 173.1 (2006): 73-79. <http://eudml.org/doc/286535>.

@article{Man2006,
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 ≥ B ≥ 0. Namely, let l and m be nonnegative integers or infinity. Then l = rank(A - B) and $m = rank(A^\{1/2\} - B^\{1/2\})$ for some operators A and B with A ≥ B ≥ 0 on a Hilbert space of dimension n (1 ≤ n ≤ ∞) if and only if l = m = 0 or 0 < l ≤ m ≤ n. In particular, this answers in the negative the question posed by C. Benhida whether for positive operators A and B the finiteness of rank(A - B) implies that of $rank(A^\{1/2\} - B^\{1/2\})$. 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.},
author = {Man-Duen Choi, Pei Yuan Wu},
journal = {Studia Mathematica},
keywords = {finite-rank perturbation; positive operator; isometry; Wold–Lebesgue decomposition},
language = {eng},
number = {1},
pages = {73-79},
title = {Finite-rank perturbations of positive operators and isometries},
url = {http://eudml.org/doc/286535},
volume = {173},
year = {2006},
}

TY - JOUR
AU - Man-Duen Choi
AU - Pei Yuan Wu
TI - Finite-rank perturbations of positive operators and isometries
JO - Studia Mathematica
PY - 2006
VL - 173
IS - 1
SP - 73
EP - 79
AB - 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 ≥ B ≥ 0. Namely, let l and m be nonnegative integers or infinity. Then l = rank(A - B) and $m = rank(A^{1/2} - B^{1/2})$ for some operators A and B with A ≥ B ≥ 0 on a Hilbert space of dimension n (1 ≤ n ≤ ∞) if and only if l = m = 0 or 0 < l ≤ m ≤ n. In particular, this answers in the negative the question posed by C. Benhida whether for positive operators A and B the finiteness of rank(A - B) implies that of $rank(A^{1/2} - B^{1/2})$. 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.
LA - eng
KW - finite-rank perturbation; positive operator; isometry; Wold–Lebesgue decomposition
UR - http://eudml.org/doc/286535
ER -