### Abstract

A J-frame is a frame F for a Krein space (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 H. With every J-frame the so-called J-frame operator is associated, which is a self-adjoint operator in the Krein space 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 2×2 block operator representation. The J-frame bounds of 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 2×2 block representation. Moreover, this 2×2 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.

Original language | English |
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Pages (from-to) | 623-649 |

Number of pages | 27 |

Journal | Opuscula Mathematica |

Volume | 38 |

Issue number | 5 |

DOIs | |

Publication status | Published - 13 Jun 2018 |

### Keywords

- frame
- Krein space
- block operator matrix
- spectrum

## Cite this

Giribet, J., Langer, M., Leben, L., Maestripieri, A., Martinez Peria, F., & Trunk, C. (2018). Spectrum of

*J*-frame operators.*Opuscula Mathematica*,*38*(5), 623-649. https://doi.org/10.7494/OpMath.2018.38.5.623