Invariant Subspaces for T-Invariant Linear Operators

Abstract

1 — Some examples. We consider the eigenvalue problems:

$$\matrix{ {{\Delta _2}{\rm{u + }}\lambda {\rm{u}} = 0} & {{\rm{in}}\,\Omega ,} & {{\rm{u}} = 0} & {{\rm{on}}\,\partial \Omega } \cr } $$

(1)

and

$$\matrix{ {{\Delta _4}{\rm{u}} - \lambda {\rm{u}} = 0} & {{\rm{in}}\,\Omega ,} & {{\rm{u}} = {{\rm{u}}_{{{\rm{x}}_1}}} = {{\rm{u}}_{{{\rm{x}}_2}}} = 0} & {{\rm{on}}\,\partial \Omega } \cr } ,$$

(2)

where Ω is the rectangular domain of the plane IR² defined by

$$\Omega = \left\{ {\left( {{{\rm{x}}_1},\,{{\rm{x}}_2}} \right):\left| {{{\rm{x}}_1}} \right| < {\rm{a,}}\,\left| {{{\rm{x}}_2}} \right| < {\rm{b}}} \right\}.$$