The Extreme Points of Order Intervals of Positive Operators

The Extreme Points of Order Intervals of Positive Operators

Consider the order interval of operators [0, A] = { X|0 ≤ X ≤ A}. In finite dimensions (or if A is invertible) then the extreme points of [0, A] are the shorted operators (generalized Schur complements) of A. This is false in the general infinite dimensional case. We give an example arising from the...

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Veröffentlicht in:Advances in applied mathematics 1994-09, Vol.15 (3), p.360-370
Hauptverfasser: Green, W.L., Morley, T.D.
Format: Artikel
Sprache:eng
Schlagworte:
Exact sciences and technology
Functional analysis
Mathematical analysis
Mathematics
Operator theory
Potential theory
Sciences and techniques of general use
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Zusammenfassung:Consider the order interval of operators [0, A] = { X|0 ≤ X ≤ A}. In finite dimensions (or if A is invertible) then the extreme points of [0, A] are the shorted operators (generalized Schur complements) of A. This is false in the general infinite dimensional case. We give an example arising from the discretization of the biharmonic equation. We also give necessary and sufficient conditions for an extreme point X 0 to be a short of A.
ISSN:0196-8858
1090-2074
DOI:10.1006/aama.1994.1013