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Spectrum of $J$-frame operators
- Source :
- Opuscula Math. 38 (2018), 623-649
- Publication Year :
- 2017
-
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.
- Subjects :
- Mathematics - Functional Analysis
47B50, 47A10, 46C20, 42C15
Subjects
Details
- Database :
- arXiv
- Journal :
- Opuscula Math. 38 (2018), 623-649
- Publication Type :
- Report
- Accession number :
- edsarx.1703.03665
- Document Type :
- Working Paper
- Full Text :
- https://doi.org/10.7494/OpMath.2018.38.5.623