Page 107 - Special Topic Session (STS) - Volume 3

STS518 Steen T.
chosen vector state). If is unbounded (i.e. non-continuous) the spectral
distribution µ is similarly defined by the equation

[()] = ∫ () ()

ℝ
which is then required to hold for any bounded Borel function ∶ ℝ → ℝ,
and where () is defined in terms of spectral calculus (see e.g. [17]). In this
case µ generally has unbounded support.

Each of the five notions of independence mentioned above gives rise to a
corresponding notion of (additive) convolution of two probability measures µ
and on ℝ, roughly defined as the (spectral) distribution of + , where
and are two Hermitian operators such that µ = µ, µ = , and and are

independent in the considered sense (see Theorem 2.1 below).
Among the four non-classical notions of independence, free
independence, introduced by Voiculescu in the 1980’s, is by far the most
developed and well-studied. This is mainly due to its applications in the theory
of operator algebras (which was Voiculescu’s original motivation for
introducing the concept; see [22]) and its striking connection to the theory of
random matrices, which we shall indicate below. In the Hilbert space
framework outlined above, two Hermitian bounded operators and are
freely independent, if, for any sequence , , , . .. of polynomials in one
1
2
3
variable and for any in , it holds that

and that

and similar conditions for products starting with polynomial expressions in y.
In words the requirement is that any product of centered polynomial
expressions, alternating in x and y, must have expectation equal to 0.

2. Free additive convolution
2.1 Theorem & Definition ([21],[7]). Let µ and ν be (Borel-) probability
measures on ℝ. Then there exists a Hilbert space ℋ, a unit vector ξ in ℋ and
(possibly unbounded) Hermitian operators x and y acting on ℋ, such that the

following conditions hold (with the notation introduced above)
(i) µ = µ, µ = ν.
x
y
(ii) x and y are freely independent with respect to E .
ξ
The conditions (i) and (ii) uniquely determine (in particular) the spectral
distribution of the Hermitian operator x + y, which may thus be denoted by
µ ν and referred to as the free (additive) convolution of µ and ν.

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