Factoring a Quadratic Operator as a Product of Two Positive Contractions

Chi-Kwong Li, College of William & Mary, Dept Math, Williamsburg, VA 23187 USA
Ming-Cheng Tsai, Natl Sun Yat Sen Univ, Dept Appl Math, Kaohsiung 80424, Taiwan

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 circle plus bI circle plus [GRAPHICS] on H-1 circle plus H-2 circle plus (H-3 circle plus H-3) for some a, b is an element of [0, 1] and strictly positive operator P with parallel to P parallel to <= vertical bar root a - root b vertical bar root(1 - a) (1 - b). Also, we give a necessary condition for a bounded linear operator T with operator matrix [GRAPHICS] on H circle plus K that can be written as a product of two positive contractions.