Noncommutative Donoho-Stark-Elad-Bruckstein-Ricaud-Torrésani Uncertainty Principle

Let $\{{\tau }_n{\}}_{n=1}^{\mathrm{\infty }}$ and $\{{\omega }_m{\}}_{m=1}^{\mathrm{\infty }}$ be two modular Parseval frames for a Hilbert C*-module $\mathcal{E}$. Then for every $x\in \mathcal{E}\setminus \{0\}$, we show that \begin{align} (1) \quad \quad \quad \quad |θ_τx |_0 |θ_ωx |0 \geq \frac{1}{\sup{n, m \in $\mathbb{N}$} |\langle τ_n, ω_m\rangle |^2}. \end{align} We call Inequality (1) as $\mathbf{\text{Noncommutative}}\mathbf{\text{Donoho-Stark-Elad-Bruckstein-Ricaud-Torr}}\acute{\mathbf{\text{e}}}\mathbf{\text{sani}}\mathbf{\text{Uncertainty}}\mathbf{\text{Principle}}$. Inequality (1) is the noncommutative analogue of breakthrough Ricaud-Torrésani uncertainty principle $\mathit{\text{[IEEE}}\mathit{\text{Trans.}}\mathit{\text{Inform.}}\mathit{\text{Theory,}}\mathit{\text{2013]}}$. In particular, Inequality (1) extends Elad-Bruckstein uncertainty principle $\mathit{\text{[IEEE}}\mathit{\text{Trans.}}\mathit{\text{Inform.}}\mathit{\text{Theory,}}\mathit{\text{2002]}}$ and Donoho-Stark uncertainty principle $\mathit{\text{[SIAM}}\mathit{\text{J.}}\mathit{\text{Appl.}}\mathit{\text{Math.,}}\mathit{\text{1989]}}$.