Norm Inequalities for Hilbert space operators with Applications

Abstract: 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 $1971$.