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Spectrum of J-frame operators

Giribet J., Langer M., Leben L., Maestripieri A., Peria F. M., Trunk C.

Opuscula Mathematica

2018

Vol. 38, no. 5

623--649

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 x 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 2x2 block representation. Moreover, this 2x2 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.

Towards theory of C-symmetries

Kuzhel S., Sudilovskaya V.

Opuscula Mathematica

2017

Vol. 37, no. 1

65--80

The concept of C-symmetry originally appeared in PT-symmetric quantum mechanics is studied within the Krein spaces framework.

Eigenvalue estimates for operators with finitely many negative squares

Behrndt J., Mows R., Trunk C.

Opuscula Mathematica

2016

Vol. 36, no. 6

717--734

Let A and B be selfadjoint operators in a Krein space. Assume that the resolvent difference of A and B is of rank one and that the spectrum of A consists in some interval I ⊂ R of isolated eigenvalues only. In the case that A is an operator with finitely many negative squares we prove sharp estimates on the number of eigenvalues of B in the interval I. The general results are applied to singular indefinite Sturm-Liouville problems.