  −   P , ▁where ▁ 1 ▁is ▁the ▁identity ▁operator . ▁It ▁follows ▁from ▁the ▁above ▁that ▁T P ▁= ▁P T ▁if ▁and ▁only ▁if ▁both ▁Ran ▁P ▁and ▁Ran ( 1   −   P ) ▁are ▁invariant ▁under ▁T . ▁In ▁that ▁case , ▁T ▁has ▁matrix ▁representation ▁ ▁Col lo qu ial ly , ▁a ▁projection ▁that ▁comm utes ▁with ▁T ▁" di ag onal izes " ▁T . ▁▁ ▁In variant ▁sub space ▁problem ▁▁ ▁The ▁invariant ▁sub space ▁problem ▁concerns ▁the ▁case ▁where ▁V ▁is ▁a ▁separ able ▁Hil bert ▁space ▁over ▁the ▁complex ▁numbers , ▁of ▁dimension ▁> ▁ 1 , ▁and ▁T ▁is ▁a ▁bounded ▁operator . ▁The ▁problem ▁is ▁to ▁decide ▁whether ▁every ▁such ▁T ▁has ▁a ▁non - t rivial , ▁closed , ▁invariant ▁sub space . ▁This ▁problem ▁is ▁un sol ved ▁. ▁ ▁In ▁the ▁more ▁general ▁case ▁where ▁V ▁is ▁hypoth es ized ▁to ▁be ▁a ▁Ban ach ▁space , ▁there ▁is ▁an ▁example ▁of ▁an ▁operator ▁without ▁an ▁invariant ▁sub space ▁due ▁to ▁Per ▁En f lo ▁( 1 9 7 6 ). ▁ ▁A ▁concrete ▁example ▁of ▁an ▁operator ▁without ▁an ▁invariant ▁sub space ▁was ▁produced ▁in ▁ 1 9 8 5 ▁by ▁Charles ▁Read . ▁▁ ▁In variant - sub space ▁lattice ▁▁ ▁Given ▁a ▁non empty ▁ Σ ▁ ⊂ ▁L ( V ), ▁the ▁invariant ▁sub spaces ▁invariant ▁under ▁each ▁element ▁of ▁ Σ ▁form ▁a ▁lattice , ▁sometimes ▁called ▁the ▁invariant -