**Soso Tsotniashvili, David Zarnadze**

## Selfadjoint Operators and
Generalized Central Algorithms in Frechet Spaces

**Abstract:**

The paper gives an extension of the fundamental principles of selfadjoint
operators in Fr\'{e}chet--Hilbert spaces, countable-Hilbert and nuclear Fr\'{e}chet
spaces. Generalizations of the well known theorems of von Neumann,
Hellinger-Toeplitz, Friedrichs and Ritz are obtained. Definitions of generalized
central and generalized spline algorithms are given. The restriction $A^{\infty}$
of a selfadjoint operator $A$ defined on a dense set $D(A)$ of the Hilbert space
$H$ to the Frechet space $D(A^{\infty})$ is substantiated. The extended Ritz
method is used for obtaining an approximate solution of the equation $A^{\infty}
u=f$ in the Frechet space $D(A^{\infty})$. It is proved that approximate
solutions of this equation constructed by the extended Ritz method do not depend
on the number of norms that generate the topology of the space $D(A^{\infty})$.
Hence this approximate method is both a generalized central and generalized
spline algorithm.

Examples of selfadjoint and positive definite elliptic differential operators
satisfying the above conditions are given. The validity of theoretical results
in the case of a harmonic oscillator operator is confirmed by numerical
calculations.

**Keywords:**

Selfadjoint operator, best approximation, generalized central algorithm,
extended Ritz method, energetic Fr\'{e}chet space.

**MSC 2000:** 7B25, 65D15, 41A65, 65J10, 65L60, 68Q25