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.