Upper bounds for numerical radius inequalities involving off-diagonal operator matrices

Abstract

In this article, we establish some upper bounds for numerical radius inequalities, including those of $2\times 2$ operator matrices and their off-diagonal parts. Among other inequalities, it is shown that if , then $${\omega }^r\left(T\right)\le 2^{r-2}\Vert f^{2r}\left(\right|X\left|\right)+g^{2r}\left(\right|Y^{\ast }\left|\right){\Vert }^{\frac{1}{2}}\Vert f^{2r}\left(\right|Y\left|\right)+g^{2r}\left(\right|X^{\ast }\left|\right){\Vert }^{\frac{1}{2}}$$ and $${\omega }^r\left(T\right)\le 2^{r-2}\Vert f^{2r}\left(\right|X\left|\right)+f^{2r}\left(\right|Y^{\ast }\left|\right){\Vert }^{\frac{1}{2}}\Vert g^{2r}\left(\right|Y\left|\right)+g^{2r}\left(\right|X^{\ast }\left|\right){\Vert }^{\frac{1}{2}},$$ where $X,Y$ are bounded linear operators on a Hilbert space $\mathcal{H}$, $r\ge 1$, and $f$, $g$ are nonnegative continuous functions on $[0,\infty )$ satisfying the relation $f\left(t\right)g\left(t\right)=t$ ($t\in [0,\infty )$). Moreover, we present some inequalities involving the generalized Euclidean operator radius of operators $T_1,\dots ,T_n$.