Let A (x) be a norm continuous family of bounded self-adjoint operators on a separable Hilbert space H and let A(x)(n) be the orthogonal compressions of A (x) to the span of first n elements of an orthonormal basis of H. The problem considered here is to approximate the spectrum of A(x) using the sequence of eigenvalues of A(x)(n). We show that the bounds of the essential spectrum and the discrete spectral values outside the bounds of essential spectrum of A(x) can be approximated uniformly on all compact subsets by the sequence of eigenvalue functions of A(x)(n). The known results, for a bounded self-adjoint operator, are translated into the case of a norm continuous family of operators. Also an attempt is made to predict the existence of spectral gaps that may occur between the bounds of essential spectrum of A(0) = A and study the effect of norm continuous perturbation of operators in the prediction of spectral gaps. As an example, gap issues of some block Toeplitz-Laurent operators are discussed. The pure linear algebraic approach is the main advantage of the results here
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 indeed the Korovkin-type approach, in the setting of preconditioning large linear systems with Toeplitz structure, is well known. Here we extend this finite-dimensional ap- proach to the infinite-dimensional context of operators acting on separable Hilbert spaces. The asymptotics of these preconditioners are evaluated and analyzed using the concept of completely positive maps. It is observed that any two limit points, under Kadison' BW-topology, of the same sequence of preconditioners are equal modulo compact opera- tors. Moreover, this indicates the role of preconditioners in the spectral approximation of bounded self-adjoint operators
Let H be a complex separable Hilbert space, and let A be a bounded self-adjoint operator on H. Consider the orthonormal basis B = {e 1 , e 2 , . . .} and the projection P n of H onto the finite-dimensional subspace spanned by the first n elements of B. The finite-dimensional truncations A n = P n AP n shall be regarded as a sequence of finite matrices by restricting their domains to P n (H) for each n. Many researchers used the sequence of eigenvalues of A n to obtain information about the spectrum of A. But in many situations, these A n 's need not be simple enough to make the computations easier. The natural question Can we use some simpler sequence of matrices B n instead of A n ? is addressed in this article. The notion of preconditioners and their convergence in the sense of eigenvalue clustering are used to study the problem. The connection between preconditioners and compact perturbations of operators is identified here. The usage of preconditioners in the spectral gap prediction problems is also discussed. The examples of Toeplitz and block Toeplitz operators are considered as an application of these results. Finally, some future possibilities are discussed.
In this talk, I wish to discuss the linear algebraic techniques for approximating the spectrum of bounded random self-adjoint operators on separable Hilbert spaces. The random version of the truncation
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