Factoring a quadratic operator as a product of two positive contractions

2014/05/16 by Chi-Kwong Li, Li, Chi-Kwong, Ming-Cheng Tsai +1
Mathematics · #47A60 #47A63 #47A68 #FOS: Mathematics #Functional Analysis (math.FA) #math.FA #msc:47A60 #msc:47A63 #msc:47A68

paper · pdf · doi:10.48550/arxiv.1405.4042

9 pages

arxiv created 2014/05/16 · arxiv updated 2014/05/19

Abstract

Let T be a quadratic operator on a complex Hilbert space H. We show that T can be written as a product of two positive contractions if and only if T is of the form aI ⊕ bI ⊕\beginpmatrix aI P \cr 0 bI \cr \endpmatrix on H₁⊕ H₂⊕ (H₃⊕ H₃) for some a, b∈ [0,1] and strictly positive operator P with ‖P‖ ≤ |√(a) - √(b)|√((1-a)(1-b)). Also, we give a necessary condition for a bounded linear operator T with operator matrix \beginpmatrix T₁ & T₃ 0 & T₂\cr\endpmatrix on H⊕ K that can be written as a product of two positive contractions.

Related

• Product of two positive contractions
• On the parallel sum of positive operators, forms, and functionals
• An operator is a product of two quasi-nilpotent operators if and only if\n it is not semi-Fredholm
• Factorizations of Contractions
• Perturbations of operators similar to contractions and the commutator equation