J. Giribet, M. Langer, L. Leben, A. Maestripieri, F. Martínez Pería and C. Trunk,
Spectrum of J-frame operators,
Opuscula Math. 38 (2018), 623–649
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.