Factoring a Quadratic Operator as a Product of Two Positive Contractions

Li, Chi-Kwong ; Tsai, Ming-Cheng

Canadian mathematical bulletin, Tome 59 (2016) no. 2, pp. 354-362

Résumé

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 $$aI\,\oplus \,bI\,\oplus \left( \begin{matrix}aI & P\\0 & bI\\ \end{matrix} \right)\,\text{on}\,{{H}_{1}}\,\oplus \,{{H}_{2}}\,\oplus \,\left( {{H}_{3\,}}\,\oplus \,{{H}_{3}} \right)$$ for some $a,\,b\,\in \,\left[ 0,\,1 \right]$ and strictly positive operator $P$ with $\left\| P \right\|\,\le \,\left| \sqrt{a}-\sqrt{b} \right|\sqrt{\left( 1-a \right)\left( 1-b \right)}$ . Also, we give a necessary condition for a bounded linear operator $T$ with operator matrix $\left( \begin{matrix} {{T}_{1}} & {{T}_{3}}\\ 0 & {{T}_{2}}\\\end{matrix} \right)$ on $H\,\oplus \,K$ that can be written as a product of two positive contractions.

Détail

DOI : 10.4153/CMB-2015-049-4

Mots-clés : 47A60, 47A68, 47A63, quadratic operator, positive contraction, spectral theorem

Bibliographie

[1] [1] Amrein, W. O. and Sinha, K. B., On pairs of projections in a Hilbert space. Linear Algebra Appl. 208/209(1994), 425–435. Google Scholar | DOI

[2] [2] Ballantine, C. S., Products of positive definite matrices. I. Pacific J. Math. 23(1967), 427–433. Google Scholar | DOI

[3] [3] Ballantine, C. S., Products of positive definite matrices. II. Pacific J. Math. 24(1968), 7–17. Google Scholar | DOI

[4] [4] Ballantine, C. S., Products of positive definite matrices. III. J. Algebra 10(1968), 174–182. Google Scholar | DOI

[5] [5] Ballantine, C. S., Products of positive definite matrices. IV. Linear Algebra Appl. 3(1970), 79–114. Google Scholar | DOI

[6] [6] Berberian, S. K., Approximate proper vectors.Proc. Amer. Math. Soc. 13(1962), 111–114. Google Scholar | DOI

[7] [7] Bôttcherand, A. Spitkovsky, I. M., A gentle guide to the basics of two projections theory.Linear Algebra Appl. 432(2010), 1412–1459. Google Scholar | DOI

[8] [8] Corach, G. and Maestripieri, A., Products of orthogonal projections and polar decompositions. Linear Algebra Appl. 434(2011), 1594–1609. Google Scholar | DOI

[9] [9] Foias, C. and Frazho, A. E., The commutant lifting approach to interpolation problems.Operator Theory: Advances and Applications, 44, Birkhâuser-Verlag, Basel, 1990. Google Scholar

[10] [10] Fujii, J. I., Fujii, M., Izumino, S., Kubo, F., and Nakamoto, R., Strang's inequality. Math.Japon. 37(1992), no. 3, 479–486. Google Scholar

[11] [11] Halmos, P. R., Two subspaces. Trans. Amer. Math. Soc. 144(1969), 381–389. Google Scholar | DOI

[12] [12] Radjavi, H. and Williams, J. P., Products of self-adjoint operators.Michigan Math. J. 16(1969), 177–185. Google Scholar | DOI

[13] [13] Tso, S.-H. and Wu, P. Y., Matricial ranges of quadratic operators. Rocky Mountain J. Math. 29(1999), 1139–1152. Google Scholar | DOI

[14] [14] Wu, P. Y., Products of positive semidefinite matrices.Linear Algebra Appl. 111(1988), 53–61. Google Scholar | DOI

[15] [15] Wu, P. Y., The operator factorization problems. Linear Algebra Appl. 117(1989), 35–63. Google Scholar | DOI