If $Τ < U$ and $0 < S$, it need not be that $ST < SU$.

Question

Let $V$ be a finite-dimensional inner product space. If $Τ$ and $U$ are linear operators on $V$, we write $Τ < U$ it $U - T$ isa positive operator. Prove the following :

(a) $T < U$ and $U < Τ$ is impossible.

(b) If $Τ < U$ and $U < S$, then $Τ < S$.

(c) If $Τ < U$ and $0 < S$, it need not be that $ST < SU$.

A positive linear operator is one such that $Τ = Τ^*$ and $(Τα|α) > 0$ for all $\alpha \neq 0$.

I have done part a and b.

Need hints for part c.

Answer 1

Hint: look for examples where $S$ and $U - T$ do not commute.