In the latter formulation, measurements are described using the spectral measure of | \ Psi \ rangle, if the system is prepared in | \ Psi \ rangle prior to the measurement.
12.
So in this case the solution to the Hamburger moment problem is unique and " ? ", being the spectral measure of " T ", has finite support.
13.
This is essentially the result of Feldheim, that any stable random vector can be characterized by a spectral measure \ Lambda ( a finite measure on \ mathbb S ) and a shift vector \ delta \ in \ mathbb R ^ d.
14.
Is a polynomial of degree " n " & minus; 1 and these polynomials are orthonormal with respect to the spectral measure corresponding to the first basis vector \ delta _ 1 ( n ) = \ delta _ { 1, n }.
15.
In the classical case " T " was a compact self-adjoint operator; in this case " T " is just a self-adjoint bounded operator with 0 d " " T " d " I . The abstract theory of spectral measure can therefore be applied to " T " to give the eigenfunction expansion for " D ".
16.
Random matrices interpretation : if A and B are some independent n by n Hermitian ( resp . real symmetric ) random matrices such that at least one of them is invariant, in law, under conjugation by any unitary ( resp . orthogonal ) matrix and such that the empirical spectral measures of A and B tend respectively to \ mu and \ nu as n tends to infinity, then the empirical spectral measure of A + B tends to \ mu \ boxplus \ nu.
17.
Random matrices interpretation : if A and B are some independent n by n Hermitian ( resp . real symmetric ) random matrices such that at least one of them is invariant, in law, under conjugation by any unitary ( resp . orthogonal ) matrix and such that the empirical spectral measures of A and B tend respectively to \ mu and \ nu as n tends to infinity, then the empirical spectral measure of A + B tends to \ mu \ boxplus \ nu.
18.
Random matrices interpretation : if A and B are some independent n by n non negative Hermitian ( resp . real symmetric ) random matrices such that at least one of them is invariant, in law, under conjugation by any unitary ( resp . orthogonal ) matrix and such that the empirical spectral measures of A and B tend respectively to \ mu and \ nu as n tends to infinity, then the empirical spectral measure of AB tends to \ mu \ boxtimes \ nu.
19.
Random matrices interpretation : if A and B are some independent n by n non negative Hermitian ( resp . real symmetric ) random matrices such that at least one of them is invariant, in law, under conjugation by any unitary ( resp . orthogonal ) matrix and such that the empirical spectral measures of A and B tend respectively to \ mu and \ nu as n tends to infinity, then the empirical spectral measure of AB tends to \ mu \ boxtimes \ nu.
