By Ingold L.

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PX ≤ λPY for some λ > 0. X ⊆ Y. PX PY = PY PX = PX . PY − PX is a projection (it is PY∩X ⊥ ). 3 The set Proj(H) of projections in L(H) is a complete lattice (Boolean algebra). PX ∧ PY = PX ∩Y , PX ∨ PY = P(X +Y)− , PX⊥ = PX ⊥ = I − PX . i PXi = P∩Xi and i PXi = P( Xi )− . If P and Q commute, then P ∧Q = P Q and P ∨ Q = P + Q − P Q. If (Pi ) is an increasing net of projections, then Pi → i Pi strongly; if (Pi ) is decreasing, then Pi → i Pi strongly. Note that L(H)+ is not a lattice unless H is one-dimensional.

3 The set Proj(H) of projections in L(H) is a complete lattice (Boolean algebra). PX ∧ PY = PX ∩Y , PX ∨ PY = P(X +Y)− , PX⊥ = PX ⊥ = I − PX . i PXi = P∩Xi and i PXi = P( Xi )− . If P and Q commute, then P ∧Q = P Q and P ∨ Q = P + Q − P Q. If (Pi ) is an increasing net of projections, then Pi → i Pi strongly; if (Pi ) is decreasing, then Pi → i Pi strongly. Note that L(H)+ is not a lattice unless H is one-dimensional. 1 0 1 1 For example, P = and Q = 12 have no least upper bound 0 0 1 1 1 0 3 1 in (M2 )+ .

G. if S = −R). We do always have (R + S)T = RT + ST . One must be careful in deﬁning commutation relations among partially deﬁned operators: ST = T S implies D(ST ) = D(T S). This is a very strong requirement: T does not even commute with the 0 operator in this sense unless T is everywhere deﬁned. 3 Definition. Let T be a partially deﬁned operator on H, and S ∈ L(H). Then S and T are permutable if ST ⊆ T S. More generally, if S ⊆ L(H), T is permutable with S if T is permutable with every S ∈ S.