Spectrum of J -frame operators

LEBEN, LESLIE; TRUNK, CARSTEN; GIRIBET, JUAN; MAESTRIPIERI, ALEJANDRA; LANGER, MATTHIAS; MARTÍNEZ PERÍA, FRANCISCO

Opuscula Mathematica

AGH University of Science and Technology Faculty of Applied Mathematics

Lugar: Cracovia; Año: 2018 vol. 5 p. 597 - 623

1232-9274

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.