By Hamilton W.R.

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**Example text**

8 Corollary. If T, N ∈ L(H), N is normal, T commutes with N , and f is a bounded Borel function on σ(N ), then T and T ∗ commute with f (N ). In particular, under the hypotheses, if N = U |N | is the polar decomposition, then T (and also T ∗ ) commutes with Re(N ), Im(N ), U , and |N |. 7 Unbounded Operators It is beyond the scope of this volume to give a complete treatment of unbounded operators. However, at some points in the theory of von Neumann algebras, it is very useful to work with unbounded operators, so we brieﬂy discuss the parts of the theory relevant to our applications.

Conversely, if T is closed and symmetric and R(T − λI) is dense for some λ ∈ C \ R, then T is self-adjoint. 4. 4 Proposition. Let T be a self-adjoint densely deﬁned operator on H which is one-one on D(T ) with dense range. Then the “inverse” T −1 , with domain R(T ), is self-adjoint. 5 Theorem. Let T be a closed operator on H. Then T ∗ T is densely deﬁned and self-adjoint; I + T ∗ T maps D(T ∗ T ) one-to-one onto H, and (I + T ∗ T )−1 and T (I + T ∗ T )−1 are everywhere deﬁned, bounded, and of norm ≤ 1, and (I + T ∗ T )−1 ≥ 0.

The function ξ → η is a bounded self-adjoint idempotent operator PX , called the (orthogonal) projection onto X . PX ≥ 0, PX2 = PX , PX = 1, and σ(PX ) = {0, 1}. Conversely, if P ∈ L(H) satisﬁes P = P ∗ = P 2 , then P is a projection: P = PX , where X = R(P ). There is thus a one-one correspondence between projections in L(H) and closed subspaces of H. 2 Proposition. If X , Y are closed subspaces of H, then the following are equivalent: (i) (ii) (iii) (iv) (v) PX ≤ PY (as elements of L(H)+ ). PX ≤ λPY for some λ > 0.