Problem 2

open

Let \(f:[0,1]^2\to\mathbb{R}\) be a continuous strictly convex function.

Suppose \(X\) and \(Y\) are commuting positive contractions on a Hilbert space \(K\), and \(P\in B(K)\) is an orthogonal projection such that \(PXP\) and \(PYP\) commute.

Assume that

\[ P\,f(X,Y)\,P = P\,f(PXP,PYP)\,P. \]

Does it follow that \(P\) reduces both \(X\) and \(Y\)? Equivalently,

\[ PX=XP, \qquad PY=YP. \]

Source: Boris Bilich.

operator theory convexity Hilbert spaces positive contractions

LaTeX source

Let \(f:[0,1]^2\to\mathbb{R}\) be a continuous strictly convex function.

Suppose \(X\) and \(Y\) are commuting positive contractions on a Hilbert space \(K\), and \(P\in B(K)\) is an orthogonal projection such that \(PXP\) and \(PYP\) commute.

Assume that

\[
P\,f(X,Y)\,P
=
P\,f(PXP,PYP)\,P.
\]

Does it follow that \(P\) reduces both \(X\) and \(Y\)? Equivalently,

\[
PX=XP,
\qquad
PY=YP.
\]

Added June 1, 2026.