A note on containment of operators

Abstract

Two new types of containment of operators on Hilbert space, namely partial containment and semi-containment are introduced. We show in Proposition 11 that A is semi-contained in B if and only if the map P(B) → P(A) for polynomials P extends to be an ultra-weakly continuous completely positive map from all bounded operators on the underlying Hilbert space of B. We show in Theorem 15 that if an isometry is semi-contained in a contraction T, then T has a non-zero invariant subspace on which T is isometric. The semi-equivalence class of the simple unilateral shift is characterized in Theorem 18, and we show that a unilateral positive-weighted shift semi-contains the unilateral shift if and only the weights are eventually 1.