Operators with equal norm and trace

Question

Suppose $T:\mathcal H\to\mathcal H$ is a self adjoint postive operator with equal norm and trace. So all the Schatten p-norms are equal. How does $T$ look like?

Answer 1

The fact that $T$ is positive together with finite trace tells you that $T$ is trace-class, so compact. Thus $$ T=\sum_j \lambda_j P_j, $$ where $\lambda_1\geq\lambda_2\geq\cdots$ are the eigenvalues and $\{P_j\}$ are pairwise orthogonal rank-one projections. We have $$ \|T\|=\lambda_1,\ \ \ \operatorname{Tr}(T)=\sum_j\lambda_j. $$ If both are equal then $\lambda_2=0$, and so $T=\lambda P$ for $\lambda\in\mathbb C$ and $P$ a rank-one projection.