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By Bobylev N. A., Bulatov V.

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5 The lattice operations are not distributive in general: if P, Q, R are projections in L(H), then it is not true in general that (P ∨ Q) ∧ R = (P ∧ R) ∨ (Q ∧ R) (this equality does hold if P, Q, R commute). 6 Proposition. Let P, Q, R be projections in L(H) with P ⊥ Q and P ≤ R. Then (P + Q) ∧ R = P ∨ (Q ∧ R). 3, since if Pi → P weakly, for any ξ ∈ H we have Pi ξ 2 = Pi ξ, ξ → P ξ, ξ = P ξ 2 . The set of projections is strongly closed. The weak closure is the positive portion of the closed unit ball.

A partial isometry is an operator U ∈ L(X , Y) such that U ∗ U is a projection P . A partial isometry U is an isometry from N (U )⊥ onto R(U ), and P = U ∗ U = PN (U )⊥ ; U U ∗ is also a projection Q = PR(U ) . The projections P and Q are called the initial and final projections, or source and range projections, of U . 4 Proposition. Let S, T ∈ L(H) with S ∗ S ≤ T ∗ T . Then there is a unique W ∈ L(H) with W ∗ W ≤ QT (hence W ≤ 1), and S = W T . If R ∈ L(H) commutes with S, T , and T ∗ , then RW = W R.

Then Si T → ST in norm. 5 If Y is a Hilbert space and {ηi : i ∈ Ω} an orthonormal basis, for a finite subset F ⊆ Ω let PF be the projection onto the span of {ηi : i ∈ F }. Then PF is a finite-rank projection, and PF → I strongly. e. T is a norm-limit of finite-rank operators. ] Thus K(X , Y) is precisely the norm-closure of the set C(X , Y) of bounded finite-rank operators. As a corollary, the adjoint of a compact operator is compact. If T is compact, T ∗ T is too and so is f (T ∗ T ) for any continuous function f vanishing at 0.

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