1. CMB 2016 (vol 59 pp. 354)
 Li, ChiKwong; Tsai, MingCheng

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
$\P\ \le \sqrt{a}  \sqrt{b}\sqrt{(1a)(1b)}.$ 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 SchurHorn theorem, the Pythagorean theorem, the Carpenter theorem, spectral theory Categories:42C15, 47B15, 46C05 
