Advice on Statistical Analysis for Circulation Research by Hideo Kusuoka and Julien I.E. Hoffman

By Hideo Kusuoka and Julien I.E. Hoffman

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2) Let v be an isometry on a Hilbert space H ˜ let H0 be any subspace of H. For all n ∈ N define H[0,n] := span{v m H0 : m = 0, . . , n} =: H0 ⊕ D1 ⊕ . . Dn . ˜ = H ˆ := span{v m H0 : m ∈ N0 }. Then there is a choice Assume that H ∞ sequence (Γn )n=1 initiated on H0 with Γ1 : H0 → H0 , Γn+1 : Dn → Dn∗ for all n, such that we have the following block matrix in Hessenberg form for the isometry v:   Γ1 D1∗ Γ2 D1∗ D2∗ Γ3 . . D1∗ . . Dm−1∗ Γm . .  D1 −Γ1∗ Γ2 −Γ1∗ D2∗ Γ3 . . −Γ1∗ D2∗ . . Dm−1∗ Γm .

Ad ∈ B(H) there is a stochastic map from Od to B(H) mapping skn . . sk1 s∗j1 . . s∗jm to akn . . ak1 a∗j1 . . a∗jm for all families of indices ki , ji ∈ {1, . . , d} and all n, m ∈ N0 . Proof: Realize H as a Markovian subspace such that s∗k |H = a∗k , for example ak · a∗k . Then by constructing a coupling representation π of Od from Z = the map Od z → pH π(z)|H does the job. 2) for a proof which does not use dilation theory. 10 Cyclicity Lemma: For Θ = ˜ sk · s∗k and a Markovian subspace H ⊂ H: ˆ [0,n] = span {skn .

E. we get a Gram matrix. 4. It is instructive to think of a Gram matrix as a kind of covariance matrix, which is actually true when the χi are realized as centered random variables. e. 1): (A, φ) = (A1 , φ1 ) ⊗ (A2 , φ2 ). If (H, π, Ω) arises from the GNS-construction of (A, φ), then H = H1 ⊗H2 and Ω = Ω1 ⊗Ω2 , where the indexed quantities arise from the GNS-construction of (A1 , φ1 ) and (A2 , φ2 ). If a ∈ A with Hilbert space norm a φ = π(a)Ω = 1, then we can speak of entanglement of π(a)Ω. We define aH1 := (π(a)Ω)H1 and call aH1 ∈ T (H1 ) the covariance operator of a ∈ A.

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