Inverse theorems in the theory of approximation of vectors in a Banach space with exponential type entire vectors - Torba, S. M. (Methods of Functional Analysis and Topology, 2010)
An arbitrary operator A on a Banach space X which is a generator of a C_0-group with a certain growth condition at infinity is considered. A relationship between its exponential type entire vectors and its spectral subspaces is found. Inverse theorems on the connection between the degree of smoothness of a vector xX with respect to the operator A, the rate of convergence to zero of the best approximation of x by exponential type entire vectors for operator A, and the k-module of continuity with respect to A are established. Also, a generalization of the Bernstein-type inequality is obtained. The results allow to obtain Bernstein-type inequalities in weighted L_p spaces.