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Arbitrary operator A on a Banach space X which is the generator of C_0-groupwith certain growth condition at infinity is considered. The relationshipbetween its exponential type entire vectors and its spectral subspaces isfound. Inverse theorems on connection between the degree of smoothness ofvector $x\in X$ with respect to operator A, the rate of convergence to zero ofthe best approximation of x by exponential type entire vectors for operator A,and the k-module of continuity are established. Also, a generalization of theBernstein-type inequality is obtained. The results allow to obtainBernstein-type inequalities in weighted L_p spaces.

Inverse theorems in the theory of approximation of vectors in a Banach space with exponential type entire vectors