# Approximation by interpolation spectral subspaces of operators with discrete spectrum

### Abstract

The paper describes approximation properties of interpolation

spectral subspaces of an unbounded operator $A$ with discrete

spectrum $\sigma(A)$ in a Banach space $\mathfrak X$, as well as

ones corresponding subspaces ${\mathcal E}_{q,p}^{\nu}(A)$ of

analytic vectors relative to $A$. Some properties of subspaces

${\mathcal E}_{q,p}^{\nu}(A)$ are established, including the

possibility of their identification with the interpolation subspaces

obtained by the real method of interpolation. A relation between spectral subspaces and subspaces ${\mathcal

E}_{q,p}^{\nu}(A)$ of analytic vectors of $A$ is also

established.

We prove the inequalities that provide a sharp estimate of

errors for the best approximations by interpolation spectral

subspaces, as well as the subspaces ${\mathcal E}_{q,p}^{\nu}(A)$.

Such inequalities fully characterize the subspace of elements from

$\mathfrak X$ in relation to rapidity of approximations. The

obtained estimates of spectral approximation errors are expressed

in terms of the quasi-norms of the approximation spaces $\mathcal

{B}_{q,p,\tau}^{s}(A)$ associated with the given operator $A$. In

this regard, the $E$-functional is used that plays a similar role

as the module of smoothness in the function theory.

We use the so-called normalization factor to write the constants

in the estimates of spectral approximation errors. This normalization

factor is determined by the parameters $\tau$ and $s$ of the

approximation spaces $\mathcal {B}_{q,p,\tau}^{s}(A)$ and has a

special form in the case $\tau(1+s)=2$.

Applications to spectral approximations of the regular elliptic

operators with variable smooth coefficients in the space

$L_q(\Omega)$ over an open bounded set $\Omega\subset\mathbb{R}^n$

and some self-adjoint ordinary elliptic differential operators in

a bounded interval $\Omega=(a,b)$ are shown.

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