Is there a notion of "limiting infinite-dimensional operator" corresponding to a limiting spectral distribution?

Given a $n\times n$ matrix $A_n$, the average eigenvalue can easily be computed by $$ave_i(\lambda_i(A_n)) = \frac 1n\operatorname{tr}(A_n).$$

Suppose that we have a sequence of positive-definite matrices $A_1,\dots,A_n,\dots$ such that the spectral distribution of a matrix $A_n$, $$s.d.(A_n) = \frac 1n\sum_{i=1}^n \delta_{\lambda_i(A_n)},$$ where $\delta$ is the dirac-delta, converges (as $n\to\infty$) to a limiting spectral distribution compactly supported in $(0,\infty)$. In this case the notion of "average eigenvalue" is well-defined as the expected value of the underlying random variable.

It would be useful to me to be able to think about this limiting distribution also as an operator itself, in order to prove inequalities regarding the average eigenvalue of various functions of $A_n$ and show that they hold in the limit.

General question: does it make sense to represent this limiting distribution with an infinite-dimensional operator endowed with a probability distribution on its eigenvalues?

I think it will help to clarify my question with a very simple example:

Let's say that in addition to the positive-definite $A_n$ above, we have another sequence of positive-definite matrices $B_n$ converging to a limiting spectral distribution which is also compactly support in $(0,\infty)$.

For each pair of matrices $(A_n,B_n)$ it is not hard to show that if $\alpha>1$, $$ave_i( \lambda_i\left( (A_n+\alpha B_n)^{-1} \right) )<ave_i\left( \lambda_i\left( (A_n+B_n)^{-1} \right) \right).$$

It should certainly also be the case that this strict inequality between the average-eigenvalues should hold for the limiting spectral distributions. If this limiting distribution could be represented as an operator with sufficiently nice properties, the same finite-dimensional proof of the above inequality should extend to the infinite-dimensional positive-definite operator case.

• You say you want to avoid free probablity theory: does this mean you want to avoid the notion of a NC-probability space? That setup would be my first thought for a setting where you try to realize the limit of these operators in a suitable sense Oct 9 '16 at 17:04
• A hasty thought, which might not be correct: can you try to realize the sequence $(A_n)$ as an element of a "tracial ultraproduct" in the sense of von Neumann algebras? Oct 9 '16 at 17:05
• I'm interested in any straightforward interpretation of things. What I'm really saying is that I'm not very familiar with the machinery of free probability. If it's necessary for the answer, I would be grateful to hear about how to proceed if it were accompanied with a reference to the relevant theorems required. Oct 9 '16 at 17:06
• I'm afraid I don't understand your suggestion, @Yemon. Oct 9 '16 at 17:10
• I'm going to be too busy for a while to write a proper answer, sorry. Hopefully other people may be able to contribute Oct 9 '16 at 17:18