Introduction to Continuous biframes in Hilbert spaces and their tensor products

TL;DR

This paper introduces the concept of continuous biframes in Hilbert spaces, generalizing discrete biframes, and explores their properties, representations, duals, and tensor product extensions.

Contribution

It presents the first formal definition of continuous biframes, provides representation theorems, and extends the concept to tensor products of Hilbert spaces.

Findings

Established the representation theorem for continuous biframes.

Characterized continuous biframes using invertible operators.

Extended the concept to tensor products of Hilbert spaces.

Abstract

We introduce the notion of a continuous biframe in a Hilbert space which is a generalization of discrete biframe in Hilbert space. Representation theorem for this type of generalized frame is verified and some characterizations of this biframe with the help of a invertible operator is given. Here we also introduce the concept of continuous biframe for the tensor products of Hilbert spaces and give an example. Further, we study dual continuous biframe and continuous biframe Bessel multiplier in Hilbert spaces and their tensor products.