[Paper Review] Detailed characterization of unconditional convergence and invertibility of multipliers

Diana T. Stoeva, Péter Balázs|arXiv (Cornell University)|Jul 5, 2010

Mathematical Analysis and Transform Methods16 references7 citations

TL;DR

This paper provides a comprehensive characterization of unconditional convergence and invertibility for multipliers $ M_{m, ilde{\Phi},\Psi} $ in Hilbert spaces, analyzing all combinations of sequence types (non-Bessel, Bessel, frames, Riesz bases) and symbol properties (semi-normalized, bounded, unbounded). It establishes a complete classification—'always', 'possible', or 'not possible'—for invertibility and unconditional convergence, supported by explicit examples and tabulated results.

ABSTRACT

In this paper we investigate the possibility of unconditional convergence and invertibility of multipliers $M_{m,Φ,Ψ}$ depending on the properties of the sequences $Ψ$,$Φ$ and $m$. We characterize a complete set of conditions for the invertibility and the unconditional convergence of multipliers, and collect those results in tables. We either prove that unconditional convergence and invertibility is not possible, that one or both of these conditions are always the case for the given parameters, or we give examples for the feasible combinations. We give a full list of examples for all conditions.

Motivation & Objective

• To systematically determine under which conditions multipliers $ M_{m,\Phi,\Psi} $ are unconditionally convergent and invertible.
• To classify all combinations of sequence types (non-Bessel, Bessel, frames, Riesz bases) and symbol properties (bounded, semi-normalized, unbounded).
• To provide a full set of examples for each feasible combination, including cases where invertibility or convergence is guaranteed, impossible, or conditional.
• To unify and extend prior results on frame multipliers by incorporating Riesz bases, non-Bessel sequences, and general symbol behavior.
• To support practical applications in audio and acoustics by clarifying theoretical limits of time-frequency filtering via multipliers.

Proposed method

• Define multipliers $ M_{m,\Phi,\Psi} $ as $ f \mapsto \sum_n m_n \langle f, \phi_n \rangle \psi_n $, operating on Hilbert space $ \mathcal{H} $.
• Use frame theory concepts: Bessel sequences, frames, Riesz bases, and their bounds $ A, B $, and dual sequences.
• Apply operator-theoretic tools: analysis and synthesis operators $ U_\Phi, T_\Phi $, and the frame operator $ S_\Phi $.
• Leverage known results: unconditional convergence via Bessel property (Proposition 2.1), invertibility via boundedness and bounded belowness (Propositions 2.6, 2.7).
• Construct explicit examples using orthonormal bases $ (e_n) $, scaling sequences $ m_n $, and modified sequences $ \Psi $ to realize all combinations.
• Tabulate results across 10 tables (Tables 1–10), categorizing outcomes as 'ALWAYS', 'POSSIBLE', or 'NOT POSSIBLE' for invertibility and unconditional convergence.

Experimental results

Research questions

• RQ1 Under what conditions on $ \Phi $, $ \Psi $, and $ m $ is the multiplier $ M_{m,\Phi,\Psi} $ unconditionally convergent?
• RQ2 When is $ M_{m,\Phi,\Psi} $ invertible, and what properties of $ \Phi $, $ \Psi $, and $ m $ guarantee or preclude invertibility?
• RQ3 Can multipliers be both unconditionally convergent and invertible for all combinations of sequence types (e.g., Riesz basis with non-Bessel sequence) and symbol types?
• RQ4 Are there cases where invertibility or unconditional convergence is impossible regardless of symbol choice?
• RQ5 What are explicit examples demonstrating each feasible combination of convergence and invertibility outcomes?

Key findings

• Unconditional convergence of $ M_{m,\Phi,\Psi} $ is guaranteed if $ \Phi $ and $ m\Psi $ are both Bessel sequences, as per Proposition 2.1.
• Invertibility of $ M_{m,\Phi,\Psi} $ is not possible if $ \Phi $ is a non-Bessel sequence, regardless of $ \Psi $ and $ m $, due to lack of boundedness.
• When $ \Phi $ and $ \Psi $ are both Riesz bases, $ M_{m,\Phi,\Psi} $ is unconditionally convergent and invertible if $ m $ is semi-normalized and bounded away from zero.
• For $ \Phi = (e_n) $, $ \Psi = (e_1, \frac{1}{2}e_2, e_1, \frac{1}{4}e_3, \dots) $, and $ m = (\frac{1}{2}, 1, \frac{1}{4}, \frac{1}{3}, \dots) $, the multiplier is unconditionally convergent and non-invertible.
• The identity operator $ I $ and the operator $ G_1 $ (with symbol $ m_n = \frac{1}{n} $) serve as canonical examples for unconditionally convergent, non-invertible multipliers.
• When $ \Phi $ is a Riesz basis and $ \Psi $ is a Bessel non-frame sequence, $ M_{m,\Phi,\Psi} $ can be unconditionally convergent and non-invertible, as shown in Example 4.8.1 with $ m = (1) $.