Let \(V\) be a finite-dimensional inner product space. If \(T\) and \(U\) are positive linear operators on \(V\), prove that \((T + U)\) is positive. Give an example which shows that \(TU\) need not be positive.
Solution
1. The sum of two positive operators is positive.
If \(T\) and \(U\) are positive, then both are self-adjoint, so \[ (T+U)^*=T^*+U^*=T+U. \] For every nonzero \(x\), \[ ((T+U)x\mid x) =(Tx\mid x)+(Ux\mid x)>0. \] Hence \(T+U\) is positive.
2. A product of positive operators need not be positive.
On \(\mathbb R^2\) with the standard inner product, take \[ T=\begin{pmatrix}1&0\\0&2\end{pmatrix}, \qquad U=\begin{pmatrix}2&1\\1&2\end{pmatrix}. \] The matrix \(T\) is positive because its eigenvalues are \(1,2\). The matrix \(U\) is positive because its eigenvalues are \(1,3\).
But \[ TU =\begin{pmatrix}2&1\\2&4\end{pmatrix}, \] which is not symmetric. Therefore \(TU\) is not self-adjoint, and hence cannot be positive.