Approximation of positive operators by analytic vectors

Abstract

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 ${\mathcal E}_{q,p}^{\nu,\alpha}(A)$ of analytic vectors of $A$ and the special scale of approximation spaces $\mathcal {B}_{q,p,\tau}^{s,\alpha}(A)$ 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.

Bernstein-Jackson-type inequalities with exact constants in Orlicz spaces

“…It is important to determine exact estimates of the constants in the Bernstein and Jackson type inequalities, which allow us to estimate the best approximation errors by analytic vectors of an operator in a Banach space [8,9].…”

Approximation by interpolation spectral subspaces of operators with discrete spectrum

“…Note that exact estimates for approximation errors of spectral approximations for unbounded operators in Banach spaces, using the Besov-type quasi-norms and normalization factor N ′ θ,r = [rθ(1−θ)] 1/r for 1 ≤ r < ∞ and N ′ θ,∞ = 1, are given in [9]. N θ,r is also used in [5] to study the approximation problem by invariant subspaces of analytic vectors of positive operators in Banach spaces. The calculated constants in estimates are asymptotically exact in the sense that for a fixed θ (0 < θ < 1) the following limit lim r→∞…”