A Priori Estimates

Abstract

As we have shown in § 3, if A is an arbitrary operator with a dense domain and if equation (A) is correctly solvable on R(A), i.e., if one has the estimate

$$ \parallel x{\parallel _E} \leqslant k\parallel Ax{\parallel _F}\;\;(x \in D(A)) $$

((7.1))

then the adjoint equation (A*) is everywhere solvable. To obtaint the estimate (7.1) one need not know for which right-hand sides equation (A) is solvable. Looking from the point of view of the theory of equations, the content of (7.1) is: if x is any solution of equation (A), the it satisfies $\parallel x{\parallel _E} \leqslant k\parallel y{\parallel _F}$. This is the reason why such estimates are known under the name of a priori estimates.