Banach Journal of Mathematical Analysis

On the measure of the spectrum of direct integrals

Anton A. Kutsenko

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We obtain the estimate of the Lebesgue measure of the spectrum for the direct integral of matrix-valued functions. These estimates are applicable for a wide class of discrete periodic operators. For example these results give new and sharp spectral bounds for 1D periodic Jacobi matrices and 2D discrete periodic Schrodinger operators.

Article information

Banach J. Math. Anal., Volume 9, Number 2 (2015), 1-8.

First available in Project Euclid: 19 December 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47A10: Spectrum, resolvent
Secondary: 47B36: Jacobi (tridiagonal) operators (matrices) and generalizations 34L15: Eigenvalues, estimation of eigenvalues, upper and lower bounds

Direct integral Jacobi matrix spectral estimates measure of spectrum


Kutsenko, Anton A. On the measure of the spectrum of direct integrals. Banach J. Math. Anal. 9 (2015), no. 2, 1--8. doi:10.15352/bjma/09-2-1.

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