Some operator inequalities for unitary invariant norms

CRISTINA CANO, IRENE MOSCONI AND DEMETRIO STOJANOFF

REVISTA DE LA UNIóN MATEMáTICA ARGENTINA

UMA

Año: 2005 vol. 46 p. 53 - 66

0041-6932

Let $L(\mathcal {H})$ be the algebra of bounded operators on a complex separable Hilbert space $\mathcal {H}$.  Let $N$ be a unitary invariant norm  defined on a norm ideal $\mathcal {I} \subseteq  L(\mathcal {H})$. Given two positive invertible operators $P, Q \in L(\mathcal {H})$ and $k \in (-2,2]$, we show that $N\big(PTQ^{-1} +P^{-1}TQ + kT\big) \ge (2+k) N(T)$, $T\in\mathcal {I}$. This extends Zhang's inequality for matrices. We prove that this inequality is equivalent to two particular cases of  itself, namely $P=Q$ and $Q=P\inv$. We also characterize  those numbers $k$ such that the map  $\Upsilon : L(\mathcal {H}) \to L(\mathcal {H})$ given by $\Upsilon (T) = PTQ^{-1} +P^{-1}TQ + kT $  is invertible, and we estimate the induced norm of $\Upsilon^{-1}$  acting on the norm ideal $\mathcal {I}$. We compute sharp constants for the involved inequalities in several particular cases.