Unexpected Uncertainty Principle for Disc Banach Spaces (2404.00910v1)

Abstract: Let $({f_n}{n=1}^\infty, {\tau_n}{n=1}^\infty)$ and $({g_n}{n=1}^\infty, {\omega_n}{n=1}^\infty)$ be unbounded continuous p-Schauder frames ($0<p<1$) for a disc Banach space $\mathcal{X}$. Then for every $x \in ( \mathcal{D}(\theta_f) \cap\mathcal{D}(\theta_g))\setminus{0}$, we show that \begin{align}\label{UB} (1) \quad \quad \quad \quad |\theta_f x|0|\theta_g x|_0 \geq \frac{1}{\left(\displaystyle\sup{n,m \in \mathbb{N} }|f_n(\omega_m)|\right)^p\left(\displaystyle\sup_{n, m \in \mathbb{N}}|g_m(\tau_n)|\right)^p}, \end{align} where \begin{align*} & \theta_f: \mathcal{D}(\theta_f) \ni x \mapsto \theta_fx := {f_n(x)}{n=1}^\infty\in \ell^p(\mathbb{N}), \quad \theta_g: \mathcal{D}(\theta_g) \ni x \mapsto \theta_gx := {g_n(x)}{n=1}^\infty\in \ell^p(\mathbb{N}). \end{align*} Inequality (1) is unexpectedly different from both bounded uncertainty principle arXiv:2308.00312v1 and unbounded uncertainty principle arXiv:2312.00366v1 for Banach spaces.