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Form Absolute Value

  • James LottesEmail author
Chapter
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Part of the Springer Theses book series (Springer Theses)

Abstract

This chapter presents and investigates a new absolute value of the operator associated with a sesquilinear form. Properties of the inner product and norm associated with the new absolute value generalize useful properties associated with the “energy” inner product and norm (to which they reduce in the symmetric case), without degrading for highly nonsymmetric operators. The “energy” inner product plays a key role in the analysis of symmetric problems, and its absence for nonsymmetric problems has been a significant barrier in their analysis. The new absolute value presented here fills this gap, and is a key tool used in the convergence theory of the next chapter. Existence and uniqueness are proved for operators with positive real part, while necessary and sufficient conditions are given for arbitrary matrices. Practical computation methods are also discussed.

Keywords

Hilbert Space Energy Norm Polar Decomposition Positive Real Part Matrix Case 
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

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Google Inc.Mountain ViewUSA

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