$K-$frames for Super Hilbert Spaces

TL;DR

This paper explores the relationships between K-frames in separate Hilbert spaces and their direct sums, including minimal frames and orthonormal bases, advancing the understanding of frame theory in super Hilbert spaces.

Contribution

It introduces new insights into how K-frames and L-frames relate within direct sum Hilbert spaces, including minimal frames and orthonormal bases.

Findings

Characterization of K-frames for direct sums of Hilbert spaces

Analysis of K⊕L-minimal frames in super Hilbert spaces

Study of K⊕L-orthonormal bases in the context of super Hilbert spaces

Abstract

Let $H_1$ and $H_2$ be two Hilbert spaces, $K$ and $L$ be bounded operatrors on $H_1$ and $H_2$ respectively. In this paper we study the relationship between $K$-frames for $H_1$ and $L$-frames for $H_2$ and $K\oplus L$-frames for $H_1\oplus H_2$. The $K\oplus L$-minimal frames and $K\oplus L$-orthonormal bases for $H_1\oplus H_2$ are also studied.