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The classical as well as noncommutative Korovkin-type theorems deal with the convergence of positive linear maps with respect to different modes of convergence, like norm or weak operator convergence etc. In this article, new versions of Korovkin-type theorems are proved using the notions of convergence induced by strong, weak and uniform eigenvalue clustering of matrix sequences with growing order. Such modes of convergence were originally considered for the special case of Toeplitz matrices and...

Consider a one dimensional Schrödinger operator $\tilde{A}=-\ddot{u}+V\cdot u$ with a periodic potential $V(\cdot )$, defined on a suitable subspace of $L^2(ℝ)$. Its spectrum is the union of closed intervals, and in general these intervals are separated by open intervals (spectral gaps). The Borg theorem states that we have no gaps if and only if the periodic potential $V(\cdot )$ is constant almost everywhere. In this paper we consider families of Finite Difference approximations of the operator $\tilde{A}$, which depend upon two parameters $n$, i.e., the number...