Subnormal operator

Summary

A subnormal operator is a bounded operator on a Hilbert space that is the restriction of a normal operator to an invariant subspace. Every normal operator is subnormal, and every subnormal operator is hyponormal. The unilateral shift is the basic example, and multiplication by $z$ on spaces of analytic functions gives many more. Scott Brown showed that subnormal operators have nontrivial invariant subspaces.