The Equation Adjoint to the Factored Equation

  • S. G. Krein


Assume that N(A) is closed and consider the operator Ã*, the adjoint to the operator à from Ẽ = E/N(A) to F. Then Ã* is defined on υ(Ã*) ⊂ F* and takes its value in Ẽ*. Using the general theorems, Ẽ* = N(A), where the orthogonal complement is taken in E* Let us show that υ(Ã*) and υ(A*) are the same. Indeed, let g ∈ υ(A*) ⊂ F*, i.e., the functional g(Ax) is bounded on υ(A):
$$ \left| {g(Ax)} \right| \leqslant c{\left\| x \right\|_E}\quad (x \in D(A)) $$


Banach Space Linear Equation Dimensional Case Homogeneous Equation Orthogonal Complement 
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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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