# UNITARILY INVARIANT NORM INEQUALITIES INVOLVING G1 OPERATORS

• Bakherad, Mojtaba (Department of Mathematics Faculty of Mathematics University of Sistan and Baluchestan)
• Accepted : 2017.12.21
• Published : 2018.07.31
• 157 5

#### 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 $${\parallel}f(A)Xg(B){\pm}g(B)Xf(A){\parallel}_2{\leq}{\Large{\parallel}}{\frac{(I+{\mid}A{\mid})X(I+{\mid}B{\mid})+(I+{\mid}B{\mid})X(I+{\mid}A{\mid})}{^dA^dB}}{\Large{\parallel}}_2$$, 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, Re(f) > 0 and Re(g) > 0.

#### Keywords

$G_1$ operator;unitarily invariant norm;commutator operator;the Hilbert-Schmidt;analytic function

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