[Paper Review] Adjoint of Pair Frames

Abolhassan Fereydooni, Ahmad Safapour+1 more|arXiv (Cornell University)|Aug 9, 2012

Mathematical Analysis and Transform Methods12 references3 citations

TL;DR

This paper generalizes the concept of $(p,q)$-pair frames to $(\ell,\ell^*)$-pair frames in Banach spaces and introduces the adjoint (conjugate) of a pair frame for the dual space. It establishes conditions under which the adjoint exists, proving that if $(G,F)$ is an $(\ell,\ell^*)$-pair Bessel or frame for a Banach space $\mathrm{X}$, then $(F,G)$ forms a corresponding pair Bessel or frame for the dual space $\mathrm{X^*}$, with invertibility preserved under appropriate operator conditions.

ABSTRACT

The concept of (p,q)-pair frames is generalized to (l,l^*)-pair frames. Adjoint (conjugate) of a pair frames for dual space of a Banach space is introduced and some conditions for the existence of adjoint (conjugate) of pair frames are presented.

Motivation & Objective

• To generalize the notion of $(p,q)$-pair frames to $(\ell,\ell^*)$-pair frames in reflexive Banach spaces.
• To define and investigate the adjoint (conjugate) of a pair frame for the dual space of a Banach space.
• To establish conditions under which the adjoint of a pair frame exists and preserves frame properties such as Bessel sequences and invertibility.
• To explore the duality between pair frames in a Banach space and their counterparts in the dual space via adjoint construction.
• To extend known results on Schauder frames, atomic decompositions, and Banach frames to the context of pair frames and their adjoints.

Proposed method

• Introduces the concept of $(\ell,\ell^*)$-pair Bessel and $(\ell,\ell^*)$-pair frames using bounded linear operators $T: \ell \to \mathrm{X}$ and analysis operators $U_G: \mathrm{X} \to \mathbb{C}^\mathbb{I}$, where $\ell$ is a Banach sequence space.
• Defines the adjoint of a pair frame via the adjoint of the synthesis operator $T^*$ and the adjoint of the analysis operator $U_G^*$, leading to the construction of a new pair $(\{W^*g_i\}, \{Vf_i\})$ in the dual space.
• Uses the duality between $\ell$ and $\ell^*$, especially when $\ell$ is a BK-space or Schauder sequence space, to ensure continuity and isometric isomorphism of duals.
• Applies the theory of unconditional convergence and permutation invariance to handle both unconditional and general cases of pair frames.
• Employs operator-theoretic tools such as boundedness, invertibility, and composition of operators to preserve frame properties under adjoint transformation.
• Relies on established results from functional analysis, including the identification of $\ell^*$ with the space of coefficient sequences $\ell^{\circledast}$ via the Riesz representation theorem for functionals on $\ell$.

Experimental results

Research questions

• RQ1 Under what conditions does the adjoint of a $(\ell,\ell^*)$-pair frame exist in the dual space of a Banach space?
• RQ2 Can a pair frame for a Banach space $\mathrm{X}$ be used to construct a corresponding pair frame for its dual space $\mathrm{X^*}$ via adjoint operations?
• RQ3 How do the properties of Bessel sequences and frame operators transform under the adjoint construction?
• RQ4 What is the relationship between the invertibility of the frame operator $S_{FG}$ and the invertibility of its adjoint $S^*_{TG}$?
• RQ5 In what cases does the adjoint of a Schauder frame or atomic decomposition remain a Schauder frame or atomic decomposition in the dual space?

Key findings

• If $(G,F)$ is an $(\ell,\ell^*)$-pair Bessel for a Banach space $\mathrm{X}$, then $(F,G)$ is an $(\ell^*,\ell)$-pair Bessel for the dual space $\mathrm{X^*}$, with the frame operator $S_{FG}^*$ being well-defined and bounded.
• The adjoint of a pair frame $(G,F)$, constructed via $\{W^*g_i\}$ and $\{Vf_i\}$ for bounded operators $V, W$, remains a pair frame if $V$ and $W$ are invertible.
• When $\ell$ is a Schauder sequence space (CB-space) or RCB-space, the dual $\ell^*$ is isometrically isomorphic to $\ell^{\circledast}$, enabling a canonical identification crucial for duality.
• If $T: \ell \to \mathrm{X}$ is a bounded operator and $(G,T)$ is a Banach frame for $\mathrm{X}$, then there exists a family $H = \{h_i\} \subset \mathrm{X^*}$ such that $(H, T_G)$ is a Banach frame for $\mathrm{X^*}$ with respect to $\ell^*$.
• An $(\ell,\ell^*)$-atomic decomposition for $\mathrm{X}$ exists if and only if the dual pair $(F,G)$ forms an $(\ell^*,\ell)$-atomic decomposition for $\mathrm{X^*}$, under the same conditions.
• The adjoint of a pair frame preserves unconditionality: if $(G,F)$ is an unconditional pair frame, then $(F,G)$ is an unconditional pair frame for $\mathrm{X^*}$, provided $\ell$ is an unconditional Banach sequence space.