The condition numbers CN(T) = ∥T∥ · ∥T−1∥ of Toeplitz and analytic n × n matrices T are studied. It is shown that the supremum of CN(T) over all such matrices with ∥T∥ ≤ 1 and the given minimum of eigenvalues r = min |λi| > 0 behaves as the corresponding supremum over all n × n matrices (i.e., as \(\frac{1}{{r^n }}\) (Kronecker)), and this equivalence is uniform in n and r. The proof is based on a use of the Sarason-Sz.Nagy-Foias commutant lifting theorem. Bibliography: 2 titles.
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N. Nikolski, “Condition numbers of large matrices and analytic capacities,” St.Petersburg Math. J., 17, 641–682 (2006).
N. Nikolski, Operators, Functions, and Systems: an Easy Reading, Vol. 1, Amer. Math. Soc. Monographs and Surveys (2002).
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Published in Zapiski Nauchnykh Seminarov POMI, Vol. 355, 2008, pp. 173–179.
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Zarouf, R. Toeplitz condition numbers as an H ∞ interpolation problem. J Math Sci 156, 819–823 (2009). https://doi.org/10.1007/s10958-009-9294-5
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DOI: https://doi.org/10.1007/s10958-009-9294-5