The purpose of this paper is to provide a method of reduction of some problems concerning families $A_t={\left(A\left(t\right)\right)}_{t\in }$ of linear operators with domains ${{(}_t)}_{t\in }$ to a problem in which all the operators have the same domain . To do it we propose to construct a family ${\left({\Psi }_t\right)}_{t\in }$ of automorphisms of a given Banach space X having two properties: (i) the mapping $t\mapsto {\Psi }_t$ is sufficiently regular and (ii) ${\Psi }_t\left(\right){=}_t$ for t ∈ . Three effective constructions are presented: for elliptic operators of second order with the Robin boundary condition with a parameter;...