Continuous K-g-fusion frames in Hilbert spaces

This paper aims at introducing the concept of c-K-g-fusion frames, which are generalizations of K-g-fusion frames, proving some new results on c-K-g-fusion frames in Hilbert spaces, defining duality of c-K-g-fusion frames and characterizing the kinds of the duals, and discussing the perturbation of c-K-g-fusion frames.Publisher's Versio

Dilations for Systems of Imprimitivity acting on Banach Spaces

Motivated by a general dilation theory for operator-valued measures, framings and bounded linear maps on operator algebras, we consider the dilation theory of the above objects with special structures. We show that every operator-valued system of imprimitivity has a dilation to a probability spectral system of imprimitivity acting on a Banach space. This completely generalizes a well-kown result which states that every frame representation of a countable group on a Hilbert space is unitarily equivalent to a subrepresentation of the left regular representation of the group. The dilated space in general can not be taken as a Hilbert space. However, it can be taken as a Hilbert space for positive operator valued systems of imprimitivity. We also prove that isometric group representation induced framings on a Banach space can be dilated to unconditional bases with the same structure for a larger Banach space This extends several known results on the dilations of frames induced by unitary group representations on Hilbert spaces.Comment: 21 page

Continuous Rankin Bound for Hilbert and Banach Spaces

Let $(\Omega, \mu)$ be a measure space and $\{\tau_\alpha\}_{\alpha\in \Omega}$ be a normalized continuous Bessel family for a real Hilbert space $\mathcal{H}$. If the diagonal $\Delta := \{(\alpha, \alpha):\alpha \in \Omega\}$ is measurable in the measure space $\Omega\times \Omega$, then we show that \begin{align} (1) \quad\quad\quad\quad \sup _{\alpha, \beta \in \Omega, \alpha\neq \beta}\langle \tau_\alpha, \tau_\beta\rangle \geq \frac{-(\mu\times\mu)(\Delta)}{(\mu\times\mu)((\Omega\times\Omega)\setminus\Delta)}. \end{align} We call Inequality (1) as continuous Rankin bound. It improves 76 years old result of Rankin [\textit{Ann. of Math., 1947}]. It also answers one of the questions asked by K. M. Krishna in the paper [Continuous Welch bounds with applications, \textit{Commun. Korean Math. Soc., 2023}]. We also derive Banach space version of Inequality (1).Comment: 6 Pages, 0 Figure