Approximation of positive operators by analytic vectors - Dmytryshyn, M.I. (Carpathian Mathematical Publications, 2020)
We give the estimates of approximation errors while approximating of a positive operator A in a Banach space by analytic vectors. Our main results are formulated in the form of Bernstein and Jackson type inequalities with explicitly calculated constants. We consider the classes of invariant subspaces E_q,p^ u,α(A) of analytic vectors of A and the special scale of approximation spaces B_q,p,τ^s,α(A) associated with the complex degrees of positive operator. The approximation spaces are determined by E-functional, that plays a similar role as the module of smoothness. We show that the approximation spaces can be considered as interpolation spaces generated by K-method of real interpolation. The constants in the Bernstein and Jackson type inequalities are expressed using the normalization factor.