On sequence spaces for Fr\'echet frames

TL;DR

This paper investigates the construction of specialized sequence spaces to establish frame structures, such as Banach and Fréchet frames, for Banach and Fréchet spaces, enhancing understanding of their functional analysis properties.

Contribution

It introduces a method to construct sequence spaces that serve as frames for Banach and Fréchet spaces, extending the theory of frames in functional analysis.

Findings

Constructed sequence spaces serve as frames for Banach and Fréchet spaces.

Established conditions for $ ilde{ heta}$-frames and Banach frames.

Extended frame theory to Fréchet spaces.

Abstract

We analyze the construction of a sequence space $\widetilde{\Theta}$, resp. a sequence of sequence spaces, in order to have $\{g_i\}$ as a $\widetilde{\Theta}$-frame or Banach frame for a Banach space $X$, resp. pre-$F$-frame or $F$-frame for a Fr\'echet space $X_F=\cap_{s\in {\mathbb N}_0} X_s$, where $\{X_s\}_{s\in {\mathbb N}_0}$ is a sequence of Banach spaces.