@article{ENFLO1995225,
    Abstract = {Let X be a real or complex normed space, A be a linear operator in the space X, and x ϵ X. We put E(X, A, x) = min{l : l>0, ∥Al x∥ ≠ ∥x∥}, or 0 if ∥Ak x∥ = ∥x∥ for all integer k>0. Then let E(X, A) = supx E(X, A, x) and E(X) = supA E(X, A). If dim X ≥ 2 then E(X) ≥ dim X + 1. A space X is called E-finite if E(X) < ∞. In this case dim X < ∞, and we set dim X = n. The main results are following. If X is polynomially normed of a degree p, then it is E-finite; moreover, E(X) ≤ Cpn+p−1 (over R), and E(X) ≤ (Cp2n+p2−1)2 (over C). If X is Euclidean complex, then n2 − n + 2 ≤ E(X) ≤ n2 − 1 for n ≥ 3; in particular, E(X) = 8 if n = 3. Also, E(X) = 4 if n = 2. If X is Euclidean real, then [n2]2 − [n2] + 2 ≤ E(X) ≤ n(n + 1)2, and E(X) = 3 if n = 2. Much more detailed information on E-numbers of individual operators in the complex Euclidean space is obtained. If A is not nilpotent, then E(X, A) ≤ 2ns − s2, where s is the number of nonzero eigenvalues. For any operator A we prove that E(X, A) ≤ n2 − n + t, where t is the number of distinct moduli of nonzero nonunitary eigenvalues. In some cases E-numbers are ``small'' and can be found exactly. For instance, E(X, A) ≤ 2 if A is normal, and this bound is achieved. The topic is closely connected with some problems related to the number-theoretic trigonometric sums.},
    Author = {Enflo, P. and Gurarii, V.I. and Lomonsov, V. and Lyubich, Yu.I.},
    File = {Exponential numbers of linear operators in normed spaces - 1-s2.0-002437959300217N-main.pdf},
    ISSN = {0024-3795},
    Journal = {Linear Algebra and its Applications},
    Pages = {225-260},
    Title = {Exponential numbers of linear operators in normed spaces},
    URL = {https://www.sciencedirect.com/science/article/pii/002437959300217N},
    Volume = {219},
    Year = {1995},
    bdsk-url-1 = {https://www.sciencedirect.com/science/article/pii/002437959300217N},
    bdsk-url-2 = {https://doi.org/10.1016/0024-3795(93)00217-N},
    date-added = {2023-01-26 11:24:53 +0100},
    date-modified = {2023-01-26 11:24:53 +0100},
    doi = {10.1016/0024-3795(93)00217-N}
}

@article{ENFLO1995225, Abstract = {Let X be a real or complex normed space, A be a linear operator in the space X, and x ϵ X. We put E(X, A, x) = min{l : l>0, ∥Al x∥ ≠ ∥x∥}, or 0 if ∥Ak x∥ = ∥x∥ for all integer k>0. Then let E(X, A) = supx E(X, A, x) and E(X) = supA E(X, A). If dim X ≥ 2 then E(X) ≥ dim X + 1. A space X is called E-finite if E(X) < ∞. In this case dim X < ∞, and we set dim X = n. The main results are following. If X is polynomially normed of a degree p, then it is E-finite; moreover, E(X) ≤ Cpn+p−1 (over R), and E(X) ≤ (Cp2n+p2−1)2 (over C). If X is Euclidean complex, then n2 − n + 2 ≤ E(X) ≤ n2 − 1 for n ≥ 3; in particular, E(X) = 8 if n = 3. Also, E(X) = 4 if n = 2. If X is Euclidean real, then [n2]2 − [n2] + 2 ≤ E(X) ≤ n(n + 1)2, and E(X) = 3 if n = 2. Much more detailed information on E-numbers of individual operators in the complex Euclidean space is obtained. If A is not nilpotent, then E(X, A) ≤ 2ns − s2, where s is the number of nonzero eigenvalues. For any operator A we prove that E(X, A) ≤ n2 − n + t, where t is the number of distinct moduli of nonzero nonunitary eigenvalues. In some cases E-numbers are ``small'' and can be found exactly. For instance, E(X, A) ≤ 2 if A is normal, and this bound is achieved. The topic is closely connected with some problems related to the number-theoretic trigonometric sums.}, Author = {Enflo, P. and Gurarii, V.I. and Lomonsov, V. and Lyubich, Yu.I.}, File = {Exponential numbers of linear operators in normed spaces - 1-s2.0-002437959300217N-main.pdf}, ISSN = {0024-3795}, Journal = {Linear Algebra and its Applications}, Pages = {225-260}, Title = {Exponential numbers of linear operators in normed spaces}, URL = {https://www.sciencedirect.com/science/article/pii/002437959300217N}, Volume = {219}, Year = {1995}, bdsk-url-1 = {https://www.sciencedirect.com/science/article/pii/002437959300217N}, bdsk-url-2 = {https://doi.org/10.1016/0024-3795(93)00217-N}, date-added = {2023-01-26 11:24:53 +0100}, date-modified = {2023-01-26 11:24:53 +0100}, doi = {10.1016/0024-3795(93)00217-N} }