 Since "P" = "P"("K";"T") is bounded and commutes with "T" it enables "T" to be expressed in the form "U" ⊕ "V" where "U" = "T"|"PX" and "V" = "T"|(1−"P")"X".Both "PX" and (1 − "P")"X" are invariant subspaces of "T" moreover "σ"("U") = "K" and "σ"("V") = "σ"("T") \ "K".A key property is mutual orthogonality.