Spectrum of J-frame operators

Giribet, Juan and Langer, Matthias and Leben, Leslie and Maestripieri, Alejandra and Martinez Peria, Francisco and Trunk, Carsten (2018) Spectrum of J-frame operators. Opuscula Mathematica, 38 (5). pp. 623-649. ISSN 2300-6919

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    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.