Schatten $p$-norm and numerical radius inequalities with applications

We develop a new refinement of the Kato’s inequality and using this refinement we obtain several upper bounds for the numerical radius of a bounded linear operator as well as the product of operators, which improve the well known existing bounds. Further, we obtain a necessary and sufficient condition for the positivity of $2\times 2$ certain block matrices and using this condition we deduce an upper bound for the numerical radius involving a contraction operator. Furthermore, we study the Schatten $p$-norm inequalities for the sum of two $n\times n$ complex matrices via singular values and from the inequalities we obtain the $p$-numerical radius and the classical numerical radius bounds. We show that for every $p>0$, the $p$-numerical radius $w_p(\cdot ):{\mathcal{M}}_n(\mathbb{C})\to \mathbb{R}$ satisfies $w_p(T)\le \frac{1}{2}\sqrt{{\parallel \mathrm{\mid }T{\mathrm{\mid }}^{2(1-t)}+\mathrm{\mid }{T}^{\ast }{\mathrm{\mid }}^{2(1-t)}\parallel }^{}\parallel \mathrm{\mid }T{\mathrm{\mid }}^{2t}+\mathrm{\mid }{T}^{\ast }{\mathrm{\mid }}^{2t}{\parallel }_{p\mathrm{/}2}^{}}$ for all $t\in [0,1]$. Considering $p\to \mathrm{\infty }$, we get a nice refinement of the well known classical numerical radius bound $w(T)\le \sqrt{\frac{1}{2}\parallel T^{\ast }T+T{T}^{\ast }\parallel }\mathrm{.}$ As an application of the Schatten $p$-norm inequalities we develop a bound for the energy of graph. We show that $\mathcal{E}(G)\ge \frac{2m}{\sqrt{{\max }_{1\le i\le n}\left\{{\sum }_{j,v_i\sim v_j}{d}_j\right\}}},$ where $\mathcal{E}(G)$ is the energy of a simple graph $G$ with $m$ edges and $n$ vertices $v_1,v_2,\dots ,v_n$ such that degree of $v_i$ is $d_i$ for each $i=1,2,\dots ,n\mathrm{.}$