Spectrum of J-frame operators

Giribet, Juan; Langer, Matthias GND; Leben, Leslie GND; Maestripieri, Alejandra; Martínez Pería, Francisco; Trunk, Carsten GND

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.


Citation style:
Giribet, J., Langer, M., Leben, L., Maestripieri, A., Martínez Pería, F., Trunk, C., 2018. Spectrum of J-frame operators. Opuscula mathematica, Opuscula mathematica 38, 2018, 623–649. https://doi.org/10.7494/OpMath.2018.38.5.623
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