Putnam’s inequality for 𝑝-hyponormal operators

Abstract

The purpose of this paper is to show the following: Let 0 > p > 1 2 0 > p > \frac {1}{2} . If T is a p-hyponormal operator on a Hilbert space, then \[ ‖ ( T ∗ T ) p − ( T T ∗ ) p ‖ ≤ p π ∬ σ ( T ) ρ 2 p − 1 d ρ d θ . \left \| {{{({T^ \ast }T)}^p} - {{(T{T^ \ast })}^p}} \right \| \leq \frac {p}{\pi }\iint _{\sigma (T)} {{\rho ^{2p - 1}}d\rho d\theta }. \] That is, Putnam’s inequality holds for p-hyponormal operators.