By Fulton W.

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Clearly, (λ − G r )−1 is positive since (λ − Z )−1 and B(λ) are so; and hence [σ ]h is positive. Thus, [ρ]h = tr([ρ]h ) = λtr([σ ]h ) + (1 − r )tr( J ([σ ]h )) (as tr(( J + Z )([σ ]h )) = 0) ≥ λ tr([σ ]h ) = λ ||[σ ]h ||. 14 we can decompose any [ρ]h as [ρ]h = [ρ1 ]h − [ρ2 ]h with ||[ρ]h || = ||[ρ1 ]h || + ||[ρ2 ]h ||, where [ρ1 ]h and [ρ2 ]h are positive. We complete the proof of the theorem by appealing to the Hille–Yosida theorem, and also noting that positivity of (λ − G r )−1 implies (r ) ✷ positivity of T∗,t .

Of B(H) such that T has the form Rn∗ x Rn , T (x) = n where the sum converges strongly. 9 is of the form K = H ⊗ k for some separable k, and π in that theorem can be taken to be π(x) = U ∗ (x ⊗ 1k )U, where U is a unitary. 9, and also choose orthonormal bases { f i }, {en } of H and k respectively. By defining Rn to be the unique bounded operator satisfying u, Rn v = (u ⊗ en ), Rv , (u, v ∈ H), we have for x ∈ B(H), u, T (x)v = Ru, (x ⊗ 1)Rv Ru, f i ⊗ en = f i ⊗ en , (x ⊗ 1)Rv i,n = Rn u, f i f i , x Rn v i,n Rn u, x Rn v .

Let X be a locally convex, sequentially complete, linear topological space and (Tt )t≥0 be a one-parameter family of continuous linear operators from X to itself satisfying Tt Ts = Tt+s , T0 = I, and limt→t0 Tt x = Tt0 x for all x ∈ X and t0 ≥ 0. Such a family is called a one-parameter semigroup of class C0 (or strongly continuous) of operators on X . The family Tt is called equi-continuous if given any continuous seminorm p on X , there exists a continuous seminorm q on X such that p(Tt x) ≤ q(x) for all t ≥ 0 and x ∈ X .

### Algebraic Curves. An Introduction to Algebraic Geometry by Fulton W.

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