Advances in Operator Theory and Inequalities

FURTHER GENERALIZED NUMERICAL RADIUS
INEQUALITIES FOR HILBERT SPACE OPERATORS

Vuk Stojiljković

$^1$ Mathematical Institute of Serbian Academy of Sciences and Arts,
Kneza Mihaila 36, Belgrade 11000, Serbia
[email protected]

We introduce a new generalized numerical radius, $w_{h,g}^{Re}(A)$, which is defined based on the generalized real and imaginary parts of an operator, as defined by Kittaneh and Stojiljković in a recent paper. We will demonstrate that this new quantity, $w_{h,g}^{Re}(A)$, is a norm on the $C^*$-algebra of bounded linear operators, $B(H)$, and that it is equivalent to the standard operator norm, $||\cdot||$. A key strength of this new concept is its generality; it encompasses and refines existing definitions and inequalities, including those previously established by Sheikhhosseini et al. and Kittaneh. Furthermore, we will explore various new inequalities, including those involving powers of operators and operator matrices, providing extensions and refinements to previous results in the field. The adaptability of $w_{h,g}^{Re}(A)$ through the functions $h$ and $g$ suggests promising applications and significant contributions to the ongoing refinement and extension of operator inequalities in functional analysis. Results in this presentation are based on the recent paper submitted by Kittaneh and Stojiljković.