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Search: All articles in the CMB digital archive with keyword spectral theor

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1. CMB 2016 (vol 59 pp. 354)

Li, Chi-Kwong; Tsai, Ming-Cheng
Factoring a Quadratic Operator as a Product of Two Positive Contractions
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 \begin{equation*} aI \oplus bI \oplus \begin{pmatrix} aI & P \cr 0 & bI \cr \end{pmatrix} \quad \text{on} \quad H_1\oplus H_2\oplus (H_3\oplus H_3) \end{equation*} for some $a, b\in [0,1]$ and strictly positive operator $P$ with $\|P\| \le |\sqrt{a} - \sqrt{b}|\sqrt{(1-a)(1-b)}.$ Also, we give a necessary condition for a bounded linear operator $T$ with operator matrix $ \big( \begin{smallmatrix} T_1 & T_3 \\ 0 & T_2\cr \end{smallmatrix} \big) $ on $H\oplus K$ that can be written as a product of two positive contractions.

Keywords:quadratic operator, positive contraction, spectral theorem
Categories:47A60, 47A68, 47A63

2. CMB 2013 (vol 57 pp. 463)

Bownik, Marcin; Jasper, John
Constructive Proof of Carpenter's Theorem
We give a constructive proof of Carpenter's Theorem due to Kadison. Unlike the original proof our approach also yields the real case of this theorem.

Keywords:diagonals of projections, the Schur-Horn theorem, the Pythagorean theorem, the Carpenter theorem, spectral theory
Categories:42C15, 47B15, 46C05

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