TY  - JOUR
ID  - Giribet2018
LB  - Giribet2018
AU  - Giribet, Juan
AU  - Langer, Matthias
AU  - Leben, Leslie
AU  - Maestripieri, Alejandra
AU  - Martínez Pería, Francisco
AU  - Trunk, Carsten
TI  - Spectrum of J-frame operators
ST  - Spectrum of J-frame operators
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2018
VL  - 38
IS  - 5
SP  - 623
EP  - 649
DO  - https://doi.org/10.7494/OpMath.2018.38.5.623
UR  - https://doi.org/10.7494/OpMath.2018.38.5.623
AB  - A \(J\)-frame is a frame \(\mathcal{F}\) for a Krein space \((\mathcal{H},[\cdot,\cdot ])\) which is compatible with the indefinite inner product \([\cdot,\cdot ]\) in the sense that it induces an indefinite reconstruction formula that resembles those produced by orthonormal bases in \(\mathcal{H}\). With every \(J\)-frame the so-called \(J\)-frame operator is associated, which is a self-adjoint operator in the Krein space \(\mathcal{H}\). 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 \(2\times 2\) block operator representation. The \(J\)-frame bounds of \(\mathcal{F}\) 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 \(2\times 2\) block representation. Moreover, this \(2\times 2\) 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.
KW  - frame
KW  - Krein space
KW  - block operator matrix
KW  - spectrum
SN  - 1232-9274
ER  -