Operator frame for $Hom_{\mathcal{A}}^{\ast}(\mathcal{X})$

TL;DR

This paper introduces the concept of operator frames for adjointable operators on Hilbert pro-$C^{\

Contribution

It extends the theory of operator frames to Hilbert pro-$C^{\ast}$-modules, including analysis, synthesis, stability, tensor products, and duals.

Findings

Defined operator frames for $Hom_{\mathcal{A}}^{\ast}(\mathcal{X})$

Studied stability under perturbations

Explored tensor products and duals of operator frames

Abstract

The concept of operator frame can be considered as a generalization of frame. Firstly, we introduce the notion of operator frame for the set of all adjointable operators $Hom_{\mathcal{A}}^{\ast}(\mathcal{X})$ on a Hilbert pro-$C^{\ast}$-module $\mathcal{X}.$ The analysis operator, the synthesis operator and the frame operator are presented. Secondly, we study the stability of operator frame under small perturbations. We also study the tensor product of operator frame for Hilbert pro-$C^{\ast}$-modules. Finally, we establish its dual and some properties.