p-adic Ghobber-Jaming Uncertainty Principle

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 τ_j, ω_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 τ_j, ω_k \rangle|}\right)\max\left{\displaystyle \max_{j \in M^c}|\langle x, τ_j\rangle |, \displaystyle \max_{k \in N^c}|\langle x, ω_k\rangle |\right}. \end{align} We call Inequality (1) as $\mathbf{\text{p-adic}}\mathbf{\text{Ghobber-Jaming}}\mathbf{\text{Uncertainty}}\mathbf{\text{Principle}}$. Inequality (1) is the p-adic version of uncertainty principle obtained by Ghobber and Jaming $\mathit{\text{[Linear}}\mathit{\text{Algebra}}\mathit{\text{Appl.,}}\mathit{\text{2011]}}$. We also derive analogues of Inequality (1) for non-Archimedean Banach spaces.