Regularity of domains of parameterized families of closed linear operators
Volume 80 / 2003
Abstract
The purpose of this paper is to provide a method of reduction of some problems concerning families $A_t=(A(t))_{t\in {\cal T}}$ of linear operators with domains $({\cal D}_t)_{t\in {\cal T}}$ to a problem in which all the operators have the same domain ${\cal D}$. To do it we propose to construct a family $({\mit \Psi }_t)_{t\in {\cal T}}$ of automorphisms of a given Banach space $X$ having two properties: (i) the mapping $t\mapsto {\mit \Psi }_t$ is sufficiently regular and (ii) ${\mit \Psi }_t({\cal D})={\cal D}_t$ for $t\in {\cal T}$. Three effective constructions are presented: for elliptic operators of second order with the Robin boundary condition with a parameter; for operators in a Hilbert space for which eigenspaces form a complete orthogonal system of closed linear subspaces; and for a class of closed operators having bounded inverses.
