Approximately orthogonality preserving maps in Krein spaces

Abstract

In this paper, we investigate approximately orthogonality preserving maps in the setting of Krein spaces. More precisely, suppose that $${\mathcal {K}}_1$$ and $${\mathcal {K}}_2$$ are two Krein spaces and that $$T:{\mathcal {K}}_1\rightarrow {\mathcal {K}}_2$$ is a nonzero linear $$\varepsilon $$-orthogonality preserving map for some $$\varepsilon \in [0, 1)$$ such that $$T({\mathcal {K}}_1^{\pm })\subseteq {\mathcal {K}}_2^{\pm }$$. We show that T is injective and continuous and there exists $$\gamma >0$$ such that $$|[T(x),T(y)]-\gamma ^2 [x,y]|\le \delta \min \{\gamma ^2 \Vert x\Vert \Vert y\Vert , \Vert T(x)\Vert \Vert T(y)\Vert \},$$ for $$x,y\in {\mathcal {K}}_1$$ with $$\delta =12\varepsilon \big (\frac{1}{1-\varepsilon } +\sqrt{\frac{1+\varepsilon }{1-\varepsilon }}\big ).$$ We also give some conditions under which the Pythagorean equality holds true in a Krein space.