Continuous frames in Krein spaces

TL;DR

This paper extends the concept of continuous frames from Hilbert spaces to Krein spaces, defining their properties, operators, and duals, and illustrating their construction via Gram operators.

Contribution

It introduces the definition of continuous frames in Krein spaces, analyzes their properties, and connects them to Hilbert space frames through Gram operators.

Findings

Frame operator in Krein spaces involves fundamental symmetry.

Continuous frames can be transferred from Hilbert to Krein spaces.

The paper establishes a frame decomposition theorem in Krein spaces.

Abstract

The purpose of this paper is to propose a definition of continuous frames of rank n for Krein spaces and to study their basic properties. Similarly to the Hilbert space case, continuous frames are characterized by the analysis, the pre-frame and the frame operator, where the latter gives rise to a frame decomposition theorem. The paper includes a discussion of similar, dual and Parseval frames and of reproducing kernels. In addition, the importance of the fundamental symmetry in the formula for the frame operator in a Krein space is clarified. As prime examples, it is shown how to transfer continuous frames for Hilbert spaces to Krein spaces arising from a possibly non-regular Gram operator.