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The Equation Adjoint to the Factored Equation

  • S. G. Krein

Abstract

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):
$$ \left| {g(Ax)} \right| \leqslant c{\left\| x \right\|_E}\quad (x \in D(A)) $$
.

Keywords

Banach Space Linear Equation Dimensional Case Homogeneous Equation Orthogonal Complement 
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.

Copyright information

© Birkhäuser Boston, Inc. 1982

Authors and Affiliations

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

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