Exponential numbers of linear operators in normed spaces Original Research Article

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, short parallelAl xshort parallel ≠ short parallelxshort parallel}, or 0 if short parallelAk xshort parallel = short parallelxshort parallel 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) ≤ (Cp/2n+p/2−1)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 [n/2]2 − [n/2] + 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.