An analog of Hölder's inequality for the spectral radius of Hadamard products

Abstract

We prove new inequalities related to the spectral radius $\rho$ of Hadamard products (denoted by $\circ$) of complex matrices. Let $p,q\in [1,\infty ]$ satisfy $\frac{1}{p}+\frac{1}{q}=1$, we show an analog of Hölder's inequality on the space of $n\times n$ complex matrices $$\rho (A\circ B)\le \rho (|A{|}^{\circ p}{)}^{\frac{1}{p}}\rho (|B{|}^{\circ q}{)}^{\frac{1}{q}}\text{for all A,B∈Cn×n,}$$ where $|\cdot |$ denotes entry-wise absolute values, and $(\cdot {)}^{\circ p}$ represents the entry-wise Hadamard power. We derive a sharper inequality for the special case $p=q=2$. Given $A,B\in {\mathbb{C}}^{n\times n}$, for some $\beta \in (0,1]$ depending on $A$ and $B$, $$\rho (A\circ B)\le \beta \rho (|A\circ A|{)}^{\frac{1}{2}}\rho (|B\circ B|{)}^{\frac{1}{2}}.$$ Analysis for another special case $p=1$ and $q=\infty$ is also included.

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