▁if ▁M ▁is ▁a ▁maximal ▁left ▁ideal , ▁since ▁a ▁sub space ▁V ▁ ⊂ ▁A / M ▁is ▁an ▁invariant ▁under ▁{ Φ '( a ) ▁| ▁a ▁ ∈ ▁A } ▁if ▁and ▁only ▁if ▁its ▁pre image ▁under ▁the ▁quot ient ▁map , ▁V ▁+ ▁M , ▁is ▁a ▁left ▁ideal ▁in ▁A . ▁ ▁Al most - in variant ▁half spaces ▁ ▁Rel ated ▁to ▁invariant ▁sub spaces ▁are ▁so - called ▁almost - in variant - half spaces ▁( AI HS ' s ). ▁ ▁A ▁closed ▁sub space ▁ ▁of ▁a ▁Ban ach ▁space ▁ ▁is ▁said ▁to ▁be ▁almost - in variant ▁under ▁an ▁operator ▁ ▁if ▁ ▁for ▁some ▁finite - dimensional ▁sub space ▁; ▁equival ently , ▁ ▁is ▁almost - in variant ▁under ▁ ▁if ▁there ▁is ▁a ▁finite - rank ▁operator ▁ ▁such ▁that ▁, ▁i . e . ▁if ▁ ▁is ▁invariant ▁( in ▁the ▁usual ▁sense ) ▁under ▁. ▁In ▁this ▁case , ▁the ▁minimum ▁possible ▁dimension ▁of ▁ ▁( or ▁rank ▁of ▁) ▁is ▁called ▁the ▁defect '. ▁ ▁Clear ly , ▁every ▁finite - dimensional ▁and ▁finite - cod im ensional ▁sub space ▁is ▁almost - in variant ▁under ▁every ▁operator . ▁ ▁Thus , ▁to ▁make ▁things ▁n ont rivial , ▁we ▁say ▁that ▁ ▁is ▁a ▁half space ▁whenever ▁it ▁is ▁a ▁closed ▁sub space ▁with ▁infinite ▁dimension ▁and ▁infinite ▁cod im ension . ▁ ▁The ▁A I HS ▁problem ▁asks ▁whether ▁every ▁operator ▁adm its