MathDB

Let

H

be a complex Hilbert space. The bounded linear operator

A

is called positive if

\langle Ax, x\rangle \geq 0

for all

x\in H

. Let

\sqrt A

be the positive square root of

A

, i.e. the uniquely determined positive operator satisfying

(\sqrt{A})^2=A

. On the set of positive operators we introduce the

A\circ B=\sqrt A B\sqrt B

operation. Prove that for a given pair

A, B

of positive operators the identity

(A\circ B)\circ C=A\circ (B\circ C)

holds for all positive operator

C

if and only if

AB=BA

Problem Statement