A Priori Estimates

  • S. G. Krein


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


Banach Space Domain Versus Adjoint Equation Unique Solvability Finite Dimensional Space 
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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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