Opuscula Math. 38, no. 5 (2018), 623-649
https://doi.org/10.7494/OpMath.2018.38.5.623

Opuscula Mathematica

# Spectrum of J-frame operators

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

Abstract. 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.

Keywords: frame, Krein space, block operator matrix, spectrum.

Mathematics Subject Classification: 47B50, 47A10, 46C20, 42C15.

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