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.
Categories: math.FA cs.IT math-ph math.IT math.MP math.NT math.OC