On convergence of absolutely continuous spectral measures for truncations of limit Jacobi matrices (Q1175362)

Let \(A_ \mu\) be the Jacobi matrix with spectral measure \(\mu\) (\(\mu\geq 0,\text{ supp}\mu\subset\mathbb{R}\)) and \(T\) be the matrix of the right shift in the one-sided space of sequences \(\ell_ 2\). The mapping of the Jacobi matrix \(A_ \mu\mapsto T^* A_ \mu T=A_{\mu_ 1}\) defines a transform of the spectral measure \(\mu\mapsto\mu_ 1\). The author studies the iteration properties of this transform of measures: \(\mu\mapsto\mu_ n\), \(A_{\mu_ n}=(T^*)^ n A_ \mu T^ n\), \(n\geq 1\), and in particular obtains some convergence conditions of the sequence \((\mu_ n)\).