Triangularizability of operators with increasing spectrum

We establish finite- and infinite-dimensional versions of the following assertion. If M is a matrix with the property that whenever P and Q are diagonal projections with P⩽Q, the spectrum of PMP (considered as an operator on the range of P) is contained in that of QMQ (considered as an operator on the range of Q), then there is a permutation matrix U such that U−1MU is triangular.