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ZEROS OF ANALYTIC FUNCTIONS AND NORMS OF INVERSE MATRICES

Abstract : Let k(n) be the smallest constant such that for any n-dimensional normed space X and any invertible linear operator T is an element of L(X) we have vertical bar det(T)vertical bar.vertical bar parallel to T(-1)parallel to <= k(n) parallel to T parallel to(n-1) Let A(+) be the Banach space of all analytic functions f(z) = Sigma(k >= 0) a(k) z(k) on the unit disk D with absolutely convergent Taylor series, and let parallel to f parallel to A(+) = Sigma(k >= 0) vertical bar a(k)vertical bar; define phi(n) on (D) over bar (n) by phi(n)(lambda(1) ,..., lambda(n)) = inf{parallel to f parallel to A(+) -vertical bar f(0)vertical bar; f(z) = g(z) pi(n)(i=1) (lambda(i) - z), g is an element of A(+), g(0) = 1}. We show that k(n) = sup {phi(n)(lambda(1) ,..., lambda(n)) ; (lambda(1) ,..., lambda(n)) is an element of (D) over bar (n)}. Moreover, if S is the left shift operator on the space l(infinity) : S (x(0), x(1) ,..., x(p) ,...) = (x(1) ,..., x(p), ...) and if J(n) (S) denotes the set of all S-invariant n-dimensional subspaces of l(infinity) on which S is invertible, we have k(n) = sup{vertical bar det(S vertical bar(E))vertical bar parallel to(S vertical bar(E))(-1)parallel to ; E is an element of J(n)(S)}. J. J. Schiller (1970) proved that k(n) <= root en and conjectured that k(n) = 2, for n >= 2. In fact k(3) > 2 and using the preceding results, we show t h a t, up to a logarithmic factor, k(n) is of the order of root n when n -> + infinity.
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E. Gluskin, M. Meyer, Alain Pajor. ZEROS OF ANALYTIC FUNCTIONS AND NORMS OF INVERSE MATRICES. Israël Journal of Mathematics, Hebrew University Magnes Press, 1994, 87 (1-3), pp.225--242. ⟨10.1007/BF02772996⟩. ⟨hal-00693561⟩

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