Unitarily invariant norm inequalities involving $G_1$ operators

Abstract

In this paper, we present some upper bounds for unitarily invariant norms inequalities. Among other inequalities, we show some upper bounds for the Hilbert-Schmidt norm. In particular, we prove \begin{align*} \|f(A)Xg(B)\pm g(B)Xf(A)\|_2\leq \left\|\frac{(I+|A|)X(I+|B|)+(I+|B|)X(I+|A|)}{d_Ad_B}\right\|_2, \end{align*} where $A,B,X\in {\mathbb{M}}_n$ such that $A$, $B$ are Hermitian with $\sigma (A)\cup \sigma (B)\subset \mathbb{D}$ and $f,g$ are analytic on the complex unit disk $\mathbb{D}$, $g(0)=f(0)=1$, $\text{Re}(f)>0$ and $\text{Re}(g)>0$.

To appear in Commun. Korean Math. Society

References in corpus (1)

• Cartesian decomposition and Numerical radius inequalities