Norm Inequalities for Hilbert space operators with Applications

Several unitarily invariant norm inequalities and numerical radius inequalities for Hilbert space operators are studied. We investigate some necessary and sufficient conditions for the parallelism of two bounded operators. For a finite rank operator $A,$ it is shown that \begin{eqnarray*} \|A\|_{p} &\leq &\left(\textit{rank} \, A\right)^{1/{2p}} \|A\|_{2p} \,\, \leq \,\, \left(\textit{rank} \, A\right)^{{(2p-1)}/{2p^2}} \|A\|_{2p^2}, \quad \textit{for all $p\geq 1$} \end{eqnarray*} where $\|\cdot\|_p$ is the Schatten $p$-norm. If $\{ \lambda_n(A) \}$ is a listing of all non-zero eigenvalues (with multiplicity) of a compact operator $A$, then we show that \begin{eqnarray*} \sum_{n} \left|\lambda_n(A)\right|^{p} &\leq& \frac12 \| A\|_{ p}^{ p} + \frac12 \| A^2\|_{p/2}^{p/2}, \quad \textit{for all $p\geq 2$} \end{eqnarray*} which improves the classical Weyl's inequality $\sum_{n} \left|\lambda_n(A)\right|^{p} \leq \| A\|_{ p}^{ p}$ [Proc. Nat. Acad. Sci. USA 1949]. For an $n\times n$ matrix $A$, we show that the function $p\to n^{-{1}/{p}}\|A\|_p$ is monotone increasing on $p\geq 1,$ complementing the well known decreasing nature of $p\to \|A\|_p.$ \indent As an application of these inequalities, we provide an upper bound for the sum of the absolute values of the zeros of a complex polynomial. As another application we provide a refined upper bound for the energy of a graph $G$, namely, $\mathcal{E}(G) \leq \sqrt{2m\left(\textit{rank Adj(G)} \right)},$ where $m$ is the number of edges, improving on a bound by McClelland in $1971$.