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$K-$frames for Super Hilbert Spaces

Najib Khachiaa

TL;DR

This work studies how $K$-frames for $H_1$ and $L$-frames for $H_2$ relate to $K\oplus L$-frames for the super Hilbert space $H_1\oplus H_2$, and investigates $K\oplus L$-minimal frames and $K\oplus L$-orthonormal bases. By analyzing synthesis/analysis/frame operators and projecting onto components, the authors establish that a $K\oplus L$-frame on the direct sum implies component $K$- and $L$-frames, and they derive exact operator decompositions and duality relations across the direct sum. They further provide necessary and sufficient conditions for $K\oplus L$-minimal frames, and elucidate when $K\oplus L$-orthonormal bases can arise, including the roles of (co-)isometries and subspace orthogonality. The results yield a structured framework for multicomponent frame theory in super Hilbert spaces, with implications for joint reconstruction under componentwise operator constraints.

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.

$K-$frames for Super Hilbert Spaces

TL;DR

This work studies how -frames for and -frames for relate to -frames for the super Hilbert space , and investigates -minimal frames and -orthonormal bases. By analyzing synthesis/analysis/frame operators and projecting onto components, the authors establish that a -frame on the direct sum implies component - and -frames, and they derive exact operator decompositions and duality relations across the direct sum. They further provide necessary and sufficient conditions for -minimal frames, and elucidate when -orthonormal bases can arise, including the roles of (co-)isometries and subspace orthogonality. The results yield a structured framework for multicomponent frame theory in super Hilbert spaces, with implications for joint reconstruction under componentwise operator constraints.

Abstract

Let and be two Hilbert spaces, and be bounded operatrors on and respectively. In this paper we study the relationship between -frames for and -frames for and -frames for . The -minimal frames and -orthonormal bases for are also studied.
Paper Structure (6 sections, 32 theorems, 21 equations)

This paper contains 6 sections, 32 theorems, 21 equations.

Table of Contents

  1. Introduction and preliminaries
  2. Main results
  3. $K$-Frames, $L$-Frames and $K\oplus L$-frames
  4. $K$-duaity, $L$-duality and $K\oplus L$-duality
  5. $K\oplus L$-minimal frames
  6. $K\oplus L$-orthonormal bases

Key Result

Theorem 1.1

4 Let $K\in B(H_1,H)$, $L\in B(H_2,H)$, the following statements are equivalent:

Theorems & Definitions (69)

  • Theorem 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Example 1.6
  • Remark 1.7
  • Proposition 1.8
  • Proposition 1.9
  • Proposition 1.10
  • ...and 59 more