Extensions of operators and the Dirichlet problem in potential theory.

*(English)*Zbl 0626.31003Let \(U\subset\mathbb{R}^m\) be a bounded open set and \(\mathcal H(U)\) be the space of harmonic functions on \(U\). Define now,

\[ H(U)=\{h\in C(\bar U): h|_ U\in {\mathcal H}(U)\},\quad H(\partial U)=H(U)|_{\partial U}. \]

Thus \(f\in H(\partial U)\) if and only if there is a solution of the classical Dirichlet problem for \(f\). On \(H(\partial U)\), the operator \(T\) of the classical Dirichlet problem is defined, of course, by \(T(h|_{\partial U})=h|_U\), \(h\in H(U)\), and one is in fact interested in a study of positive linear extensions of \(T\) from \(H(\partial U)\) to \(C(\partial U)\) and, possibly, to a larger space than \(C(\partial U).\)

M. V. Keldysh [Dokl. Akad. Nauk SSSR 32, 308–309 (1941; Zbl 0061.23104)] proved the following theorem: All positive linear extensions of the operator \(T\) from \(H(\partial U)\) to \(C(\partial U)\) coincide. The main purpose of the present paper is to find a suitable abstract setting appropriate for a better understanding of the nature of the Keldysh theorem.

To this end, in Section 2, a question of uniqueness of extensions of operators on Riesz spaces is analyzed. The “domain of uniqueness” is characterized in terms that admit applications to potential theory. In Section 3, a more special situation, namely that of function spaces, is investigated. Validity of an abstract Keldysh theorem is shown to be equivalent to various other conditions. Also relations to Korovkin type theorems are studied. In Section 4 it is shown how conclusions obtained in an abstract setting can be used to prove results already known as well as new results concerning a Keldysh type theorem in abstract potential theory.

Most of the results of the present paper are included in the author’s text [“The first boundary value problem in potential theory” (Czech), Fac. Math. Phys., Charles Univ., Prague, 1–120 (1983)].

\[ H(U)=\{h\in C(\bar U): h|_ U\in {\mathcal H}(U)\},\quad H(\partial U)=H(U)|_{\partial U}. \]

Thus \(f\in H(\partial U)\) if and only if there is a solution of the classical Dirichlet problem for \(f\). On \(H(\partial U)\), the operator \(T\) of the classical Dirichlet problem is defined, of course, by \(T(h|_{\partial U})=h|_U\), \(h\in H(U)\), and one is in fact interested in a study of positive linear extensions of \(T\) from \(H(\partial U)\) to \(C(\partial U)\) and, possibly, to a larger space than \(C(\partial U).\)

M. V. Keldysh [Dokl. Akad. Nauk SSSR 32, 308–309 (1941; Zbl 0061.23104)] proved the following theorem: All positive linear extensions of the operator \(T\) from \(H(\partial U)\) to \(C(\partial U)\) coincide. The main purpose of the present paper is to find a suitable abstract setting appropriate for a better understanding of the nature of the Keldysh theorem.

To this end, in Section 2, a question of uniqueness of extensions of operators on Riesz spaces is analyzed. The “domain of uniqueness” is characterized in terms that admit applications to potential theory. In Section 3, a more special situation, namely that of function spaces, is investigated. Validity of an abstract Keldysh theorem is shown to be equivalent to various other conditions. Also relations to Korovkin type theorems are studied. In Section 4 it is shown how conclusions obtained in an abstract setting can be used to prove results already known as well as new results concerning a Keldysh type theorem in abstract potential theory.

Most of the results of the present paper are included in the author’s text [“The first boundary value problem in potential theory” (Czech), Fac. Math. Phys., Charles Univ., Prague, 1–120 (1983)].

Reviewer: Themistocles M. Rassias (Athens)

##### MSC:

31D05 | Axiomatic potential theory |

31B20 | Boundary value and inverse problems for harmonic functions in higher dimensions |

46E15 | Banach spaces of continuous, differentiable or analytic functions |