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A Priori Estimates

  • S. G. Krein

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.

Keywords

Banach Space Domain Versus Adjoint Equation Unique Solvability Finite Dimensional Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Birkhäuser Boston, Inc. 1982

Authors and Affiliations

  • S. G. Krein
    • 1
  1. 1.Department of MathematicsVoronezh UniversityVoronezhUSSR

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