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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.

On the eigenvalues of a 2 x 2 block operator matrix

Muminov M. I., Rasulov T. H.

Opuscula Mathematica

2015

Vol. 35, no. 3

371--395

A 2 x 2 block operator matrix H acting in the direct sum of one- and two-particle subspaces of a Fock space is considered. The existence of infinitely many negative eigenvalues of H22 (the second diagonal entry of H) is proved for the case where both of the associated Friedrichs models have a zero energy resonance. For the number N(z) of eigenvalues of H22 lying below z < 0, the following asymptotics is found [formula]. Under some natural conditions the infiniteness of the number of eigenvalues located respectively inside, in the gap, and below the bottom of the essential spectrum of H is proved.