Noncommutative Mathematics for Quantum Systems
62 Noncommutative Mathematics for Quantum Systems Definition 1.7.11 Let XandY be two normal operators on a Hilbert spaceH, not n ...
Independence and L ́evy Processes in Quantum Probability 63 and 〈 1 ,h(Y) 1 〉 = ∫ Ch(y)dν(y) for all h ∈ Cb(C), that is, L(Y, 1 ...
64 Noncommutative Mathematics for Quantum Systems generated by X and Y, then W is unitary. Proof Define W on simple tensors of b ...
Independence and L ́evy Processes in Quantum Probability 65 Remark 1.7.14 It follows that the joint law of two monotonically ind ...
66 Noncommutative Mathematics for Quantum Systems This example shows that convolution from the left by a Dirac mass is in genera ...
Independence and L ́evy Processes in Quantum Probability 67 then we can check that this equals ψ(x,y) + x ∫ψ(x,y′) z−y′ dν(y ′) ...
68 Noncommutative Mathematics for Quantum Systems GX+Y(z) =〈Ω,(z−X−Y)−^1 Ω〉= 〈 1 ,(z−MxP 2 −My)−^11 〉 = 〈 1 , 1 z−y + xGν(z) (z− ...
Independence and L ́evy Processes in Quantum Probability 69 Proposition 1.7.21 Letμandνbe two probability measures onR+, ν 6 =δ ...
70 Noncommutative Mathematics for Quantum Systems Similarly, we get ImzGν(z) > 0 forz ∈C−. It follows that the functions in f ...
Independence and L ́evy Processes in Quantum Probability 71 after some tedious, but straightforward, computation. Remark 1.7.22 ...
72 Noncommutative Mathematics for Quantum Systems Therefore, G√XY√X(z) = ∫ R+×R+ ( 1 +g 1 (x) z−y +h 1 (x) ) dμ⊗ν(x,y) = ∫ R+ Gν ...
Independence and L ́evy Processes in Quantum Probability 73 either, and convolution from the left by a Dirac mass is in general ...
74 Noncommutative Mathematics for Quantum Systems Definition 1.7.30 Let XandY be two normal operators on a Hilbert spaceH, not n ...
Independence and L ́evy Processes in Quantum Probability 75 Proof Under the identification C⊕L^2 (C,μ) 0 ⊕L^2 (C,ν) 0 ∼= L^2 (C, ...
76 Noncommutative Mathematics for Quantum Systems g 1 (Ny)f 1 (Nx), fn(Nx)gn(Ny)···f 1 (Nx)g 1 (Ny), and gn(Ny)fn− 1 (Nx)···f 1 ...
Independence and L ́evy Processes in Quantum Probability 77 This is an isometry, since 〈 W α 1 f 1 g 1 ,W α 2 f 2 g 2 ...
78 Noncommutative Mathematics for Quantum Systems This proves the first formula. The second formula follows by symmetry. Letf,g∈ ...
Independence and L ́evy Processes in Quantum Probability 79 Proposition 1.7.35 Letμandνbe two probabilities onRand define operat ...
80 Noncommutative Mathematics for Quantum Systems Gν(z), it can not vanish forzwith Imz<0 either. The functionsz−^1 x andz−xx ...
Independence and L ́evy Processes in Quantum Probability 81 Substituting Equation (1.7.11) into this expression, we get the desi ...
«
1
2
3
4
5
6
7
8
9
10
»
Free download pdf