Operator limit of Wigner matrices I

We consider the Wigner matrix $W_n$ of dimension $n\times n$ as $n\to \mathrm{\infty }$. The objective of this paper is two folds: first we construct an operator $\mathcal{W}$ on a suitable Hilbert space $\mathcal{H}$ and then define a suitable notion of convergence such that the matrices $W_n$ converge in that notion of convergence to $\mathcal{W}$. We further investigate some properties of $\mathcal{W}$ and $\mathcal{H}$. We show that $\mathcal{H}$ is a nontrivial extension of $L^2[0,1]$ with respect to the Lebesgue measure and the spectral measure of $\mathcal{W}$ at any function $f\in L^2[0,1]$ is almost surely the semicircular law.