Some stability properties for analytic operator functions

Let $\mathfrak G$ be a connected, finite-dimensional, complex analytic manifold; let T(lambda) be an analytic function defined on $\mathfrak G$, whose values are $J$-biexpanding operators on a $J$-space $H$. Let $\mathfrak R(A)$ denote the range of $A$. The following assertions are proved: 1. The lineals $\mathfrak R(\sqrt{T(\lambda)^*JT(\lambda)-J})\equiv\mathfrak R$ and $\mathfrak R(\sqrt{T(\lambda)JT(\lambda)^*-J})\equiv\mathfrak R_*$ do not depend on $\lambda$. 2. For arbitrary $\lambda,\mu\in\mathfrak G$ we have $\mathfrak R(T(\lambda)-T(\mu))\subset\mathfrak R_*$, $\mathfrak R(T(\lambda)^*-T(\mu)^*)\subset\mathfrak R$.