Unexpected Uncertainty Principle for Disc Banach Spaces

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 \|θ_f x\|_0\|θ_g x\|_0 \geq \frac{1}{\left(\displaystyle\sup_{n,m \in \mathbb{N} }|f_n(ω_m)|\right)^p\left(\displaystyle\sup_{n, m \in \mathbb{N}}|g_m(τ_n)|\right)^p}, \end{align} where \begin{align*} & θ_f: \mathcal{D}(θ_f) \ni x \mapsto θ_fx := \{f_n(x)\}_{n=1}^\infty\in \ell^p(\mathbb{N}), \quad θ_g: \mathcal{D}(θ_g) \ni x \mapsto θ_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.

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