Spectrum of J-frame operators

Opuscula Mathematica

Juan Giribet¹ 

关键词: frame;    Krein space;    block operator matrix;    spectrum;   

DOI  :  10.7494/OpMath.2018.38.5.623

学科分类：数学（综合）

来源: AGH University of Science and Technology Press

【 摘 要 】

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 operatorswhich 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.

【 授权许可】

Free