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    • Keywords


      Operator; perturbation; essential spectrum; spectral gap; Toeplitz operators and matrices.

    • Abstract


      Let $A(x)$ be a norm continuous family of bounded self-adjoint operators on a separable Hilbert space $\mathbb{H}$ and let $A(x)_n$ be the orthogonal compressions of $A(x)$ to the span of first 𝑛 elements of an orthonormal basis of $\mathbb{H}$. 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.

    • Author Affiliations


      K Kumar1 M N N Namboodiri1 S Serra-Capizzano2

      1. Department of Mathematics, Cochin University of Science and Technology (CUSAT), Cochin 682 022, India
      2. Department of Science and High Technology, Universita Insubria – Como Campus, via Valleggio, 11, 22100 Como, Italy
    • Dates

  • Proceedings – Mathematical Sciences | News

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      Posted on July 25, 2019

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