Approximately orthogonality preserving maps in Krein spaces (Q2297348)

Mappings which preserve (exactly and approximately) orthogonality relations have been widely investigated in Hilbert spaces as well as in some more general structures. The present paper moves the problem to Krein spaces, by which we mean a linear space \(\mathcal{K}\) with indefinite inner product \([\cdot,\cdot]\), admitting a suitable canonical decomposition \(\mathcal{K}^{+}[\oplus]\mathcal{K}^{-}\) such that \((\mathcal{K}^{+},[\cdot,\cdot])\) and \((\mathcal{K}^{-},-[\cdot,\cdot])\) are Hilbert spaces. Two vectors \(x,y\in\mathcal{K}\) are orthogonal (denoted by \(x[\bot]y\)) iff \([x,y]=0\) and approximately orthogonal or \(\varepsilon\)-orthogonal (denoted by \(x[\bot]^{\varepsilon}y\)), with \(\varepsilon\in[0,1)\), iff \(|[x,y]|\leq\varepsilon\|x\|\,\|y\|\) (where \(\|\cdot\|\) is a suitable norm corresponding to \([\cdot,\cdot]\)). To present the main result of the paper, let \(T\) be a nonzero linear operator between two Krein spaces \(\mathcal{K}_1\) and \(\mathcal{K}_2\), preserving in a sense the structure of both spaces. Suppose that \(T\) is approximately orthogonality preserving, i.e., \(x[\bot]y\) implies \(Tx[\bot]^{\varepsilon}Ty\) for all \(x,y\in\mathcal{K}\). Then, for some positive number \(\gamma\), we get \(T[x,y]\approx \gamma [x,y]\) (in some sense), which means that \(T\) approximately preserves the indefinite inner product \([\cdot,\cdot]\). This is a generalization of the result from the reviewer's paper [J. Math. Anal. Appl. 304, No. 1, 158--169 (2005; Zbl 1090.46017)].