Fr\'echet frames, general definition and expansions

TL;DR

This paper introduces a comprehensive framework for Fréchet frames in Banach and Fréchet spaces, providing definitions, expansions, and conditions for frame properties and operator range complementedness.

Contribution

It generalizes the concept of frames to Fréchet spaces using decreasing Banach space sequences and sequence spaces, offering new theoretical insights.

Findings

Defined $(X_1, heta, X_2)$-frames in Banach spaces.

Established frame expansions with convergence in specific norms.

Provided necessary and sufficient conditions for frame properties and operator range complementedness.

Abstract

We define an {\it $(X_1,\Theta, X_2)$-frame} with Banach spaces $X_2\subseteq X_1$, $|\cdot|_1 \leq |\cdot|_2$, and a $BK$-space $(\Theta, \snorm[\cdot])$. Then by the use of decreasing sequences of Banach spaces ${X_s}_{s=0}^\infty$ and of sequence spaces ${\Theta_s}_{s=0}^\infty$, we define a general Fr\' echet frame on the Fr\' echet space $X_F=\bigcap_{s=0}^\infty X_s$. We give frame expansions of elements of $X_F$ and its dual $X_F^*$, as well of some of the generating spaces of $X_F$ with convergence in appropriate norms. Moreover, we give necessary and sufficient conditions for a general pre-Fr\' echet frame to be a general Fr\' echet frame, as well as for the complementedness of the range of the analysis operator $U:X_F\to\Theta_F$.