Undetermined Equations

  • S. G. Krein


Consider the equation
$$ Ax + Bu = y\left( {x \in D\left( A \right),u \in D\left( B \right),y \in F} \right)$$
where A and B are linear operators acting from Banach spaces E and G, respectively, into the Banach space F. If R(A) ∩ R(B) ≠ {θ}, then equation (A,B) is manifestly not uniquely solvable. Indeed, let Ax0 = Bu0 (x0 ∈ P(A), U0D(B)). Then the pair (X0,-u0) is a solution of the homogeneous equation (A,B). Due to such behaviour, it is natural to say that equation (A,B) is undetermined.


Banach Space Linear Operator Homogeneous Equation Unique Solvability Determinative Equation 
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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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