Contractions Satisfying the Absolute Value Property $$ |A|^2 \leq |A^2| $$

Abstract

Let B(H) denote the algebra of operators on a complex Hilbert space H, and let U denote the class of operators \( A \in B(H) \) which satisfy the absolute value condition \( |A|^2 \leq |A^2| \). It is proved that if \( A \in \mathcal{U} \) is a contraction, then either A has a nontrivial invariant subspace or A is a proper contraction and the nonnegative operator \( D = |A^2| - |A|^2 \) is strongly stable. A Putnam-Fuglede type commutativity theorem is proved for contractions A in \( \mathcal{U} \), and it is shown that if normal subspaces of \( |A|^2 \leq |A^2| \). It is proved that if \( A \in \mathcal{U} \) are reducing, then every compact operator in the intersection of the weak closure of the range of the derivation \( \delta_{A}(X) = AX - XA \) with the commutant of A* is quasinilpotent.