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arXiv · 1703.03665

Spectrum of $J$-frame operators

Juan Ignacio Giribet·Matthias Langer·Leslie Leben·Alejandra Maestripieri·Francisco Martínez Pería·Carsten Trunk
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Abstract

A $J$-frame is a frame $\mathcal{F}$ for a Krein space $(\mathcal{H}, [\, , \,])$ 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 $\mathcal{H}$. 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.

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Juan Ignacio Giribet, Matthias Langer, Leslie Leben, Alejandra Maestripieri, Francisco Martínez Pería, Carsten Trunk. 2017-03-10. Spectrum of $J$-frame operators. https://doi.org/10.7494/opmath.2018.38.5.623

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