Let X be a real or complex normed space, A be a linear operator in the space X, and x is an element of X. We put E(X, A, x) = min{l : l > 0, parallel to A(l)x parallel to not equal parallel to x parallel to}, or 0 if parallel to A(k)x parallel to = parallel to x parallel to for all integer k > 0. Then let E(X, A) = sup(x), E(X, A, x) and E(X) = sup(A) E(X, A). If dim X greater than or equal to 2 then E(X) greater than or equal to dim X + 1. A space X is called E-finite if E(X) < infinity. In this case dim X < infinity, 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) less than or equal to C-n+p-1(p) (over R), and E(X) less than or equal to (C-n+p/2-1(p/2))(2) (over C). If X is Euclidean complex, then n(2) - n + 2 less than or equal to E(X) less than or equal to n(2) - 1 for n greater than or equal to 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 less than or equal to E(X) less than or equal to 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) less than or equal to 2ns - s(2), where s is the number of nonzero eigenvalues. For any operator A we prove that E(X, A) less than or equal to n(2) - 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) less than or equal to 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.
