Can spectral value sets of Toeplitz band matrices jump ?

The spectral value set sp B,C ε A of a bounded linear operator A on l2 is the union of the spectra of all operators of the form A+ BKC, where‖K‖ < ε andB,C are fixed bounded linear operators. It turns out that small changes of ε may cause drastic changes of the set sp ε A. We conjecture that this can never happen if B or C is compact andA is given by an infinite Toeplitz band matrix. In the present paper, this conjecture is proved for certain interesting operators B and C and for several classes of Toeplitz band matrices, including Hessenberg matrices and matrices of small bandwidth. Our approach is based on working with the monodromy group of the Riemann surface associated with the generating function of the matrix. © 2002 Elsevier Science Inc. All rights reserved. AMS classification: Primary: 47B35; Secondary: 15A18; 30F10; 93B03; 93D09

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