Invertibility of multipliers

Abstract

In the present paper the unconditional convergence and the invertibility of multipliers is investigated. Multipliers are operators created by (framelike) analysis, multiplication by a fixed symbol, and resynthesis. Sufficient and/or necessary conditions for unconditional convergence and invertibility are determined depending on the properties of the analysis and synthesis sequences, as well as the symbol. Examples which show that the given assertions cover different classes of multipliers are given. If a multiplier is invertible, a formula for the inverse operator is determined. The case when one of the sequences is a Riesz basis is completely characterized.

Canonical forms of unconditionally convergent multipliers

“…This is because, by [4], if the multiplier $M_{m,\Phi ,\Psi }$ is invertible, $m\in {\ell }^{\infty }$, and $\Phi$ and $\Psi$ are Bessel for $\mathrm{scriptH}$, then $\Phi$ and $\Psi$ must be frames for $\mathrm{scriptH}$. Using this connection we can determine certain classes, as in the unconditional case, where we give an affirmative answer of [${normalQ}_{normal\mathrm{Inv1}}$]:…”

“…By Stoeva and Balazs [4] we know that the invertibility of multipliers is connected to a frame condition (under some assumptions). Furthermore, we consider the following example:…”

“…By Stoeva and Balazs [4] we know that the invertibility of multipliers is connected to the frame condition of the involved sequences. So in this case we can ask:

[${normalQ}_{normal\mathrm{Inv1}}$] If $M_{m,\Phi ,\Psi }$ is unconditionally convergent and invertible, do sequences $\left(c_n\right)$ and $\left(d_n\right)$ exist so that $M_{m,\Phi ,\Psi }\stackrel{\nabla }{=}{M}_{\left(1\right),\left(c_n{\varphi }_n\right),\left(d_n{\psi }_n\right)}$, where $\left(c_n{\varphi }_n\right)$ and $\left(d_n{\psi }_n\right)$ are frames for $\mathrm{scriptH}$?

We determine classes of multipliers for which [${normalQ}_{normal\mathrm{Inv1}}$] has affirmative answer, see Section 4.…”

Continuous Atomic Systems for Subspaces

“…For basic results on frames, we refer the reader to the references [10,12,14,21,22,29,30]. Here, it should be noted that H which satisfied in the equality (2) is important in the theory of frames and in many situations, the study of such family of operators is of considerable importance; see for example, [3,[5][6][7]9,11,19,20,23,25,27,28,31,32,[34][35][36][37][38][39][40]42,43]. Therefore, it is expected that the study of the family of bounded operators which satisfied in the equality (3) plays a crucial role in the theory of frames in Hilbert space.…”

A Class of Algorithms for Time-Frequency Multiplier Estimation

“…Remark 3: The multiplier estimated by our approach are far from being invertible [38], as they intrinsically depend on the sounds used as input and output signals. Then, inverting the role of the input and output signals will lead to different sounds.…”