Norm Inequalities for Hilbert space operators with Applications

TL;DR

This paper establishes new inequalities for Hilbert space operators involving unitarily invariant norms, eigenvalues, and Schatten norms, with applications to polynomial zeros and graph energy bounds.

Contribution

It introduces novel inequalities for operator norms and eigenvalues, improving classical bounds and providing new applications in polynomial and graph theory.

Findings

Derived inequalities for Schatten p-norms of finite rank operators.

Improved bounds for eigenvalue sums of compact operators.

Provided applications to polynomial zeros and graph energy estimates.

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 $1971$.