# Spectrum of $J$-frame operators

## TL;DR

This paper characterizes $J$-frame operators in Krein spaces using a $2 imes 2$ block operator form, enabling spectrum analysis, bounds, and the construction of a square root for these operators.

## Contribution

It introduces a $2\times 2$ block operator representation for $J$-frame operators, facilitating spectral analysis and the construction of associated $J$-frames.

## Key findings

- Characterization of $J$-frame operators via $2\times 2$ block form
- Recovery of $J$-frame bounds from numerical ranges
- Construction of a square root for $J$-frame operators

## Abstract

A $J$-frame is a frame $\mathcal{F}$ for a Krein space $(\mathcal{H}, [\, , \,])$ which is compatible with the indefinite inner product $[\, , \, ]$ in the sense that it induces an indefinite reconstruction formula that resembles those produced by orthonormal bases in $\mathcal{H}$. With every $J$-frame the so-called $J$-frame operator is associated, which is a self-adjoint operator in the Krein space $\mathcal{H}$. The $J$-frame operator plays an essential role in the indefinite reconstruction formula. In this paper we characterize the class of $J$-frame operators in a Krein space by a $2\times 2$ block operator representation. The $J$-frame bounds of $\mathcal{F}$ are then recovered as the suprema and infima of the numerical ranges of some uniformly positive operators which are build from the entries of the $2\times 2$ block representation. Moreover, this $2\times 2$ block representation is utilized to obtain enclosures for the spectrum of $J$-frame operators, which finally leads to the construction of a square root. This square root allows a complete description of all $J$-frames associated with a given $J$-frame operator.