p-adic Ghobber-Jaming Uncertainty Principle

Abstract: Let ${\tau_j}{j=1}^n$ and ${\omega_k}{k=1}^n$ be two orthonormal bases for a finite dimensional p-adic Hilbert space $\mathcal{X}$. Let $M,N\subseteq {1, \dots, n}$ be such that \begin{align*} \displaystyle \max_{j \in M, k \in N}|\langle \tau_j, \omega_k \rangle|<1, \end{align*} where $o(M)$ is the cardinality of $M$. Then for all $x \in \mathcal{X}$, we show that \begin{align} (1) \quad \quad \quad \quad |x|\leq \left(\frac{1}{1-\displaystyle \max_{j \in M, k \in N}|\langle \tau_j, \omega_k \rangle|}\right)\max\left{\displaystyle \max_{j \in M^c}|\langle x, \tau_j\rangle |, \displaystyle \max_{k \in N^c}|\langle x, \omega_k\rangle |\right}. \end{align} We call Inequality (1) as \textbf{p-adic Ghobber-Jaming Uncertainty Principle}. Inequality (1) is the p-adic version of uncertainty principle obtained by Ghobber and Jaming \textit{[Linear Algebra Appl., 2011]}. We also derive analogues of Inequality (1) for non-Archimedean Banach spaces.