On the mappings connected with parallel addition of nonnegative operators

On the mappings connected with parallel addition of nonnegative operators We study a mapping $$\tau _G$$ τ G of the cone $${\mathbf B}^+({\mathcal H})$$ B + ( H ) of bounded nonnegative self-adjoint operators in a complex Hilbert space $${\mathcal H}$$ H into itself. This mapping is defined as a strong limit of iterates of the mapping $${\mathbf B}^+({\mathcal H})\ni X\mapsto \mu _G(X)=X-X:G\in {\mathbf B}^+({\mathcal H})$$ B + ( H ) ∋ X ↦ μ G ( X ) = X - X : G ∈ B + ( H ) , where $$G\in {\mathbf B}^+({\mathcal H})$$ G ∈ B + ( H ) and X : G is the parallel sum. We find explicit expressions for $$\tau _G$$ τ G and describe its properties. In particular, it is shown that $$\tau _G$$ τ G is sub-additive, homogeneous of degree one, and its image coincides with the set of its fixed points which is a subset of $${\mathbf B}^+({\mathcal H})$$ B + ( H ) , consisting of all Y such that $$\mathrm{ran\,} Y^{\frac{1}{2}}\cap \mathrm{ran\,} G^{\frac{1}{2}}=\{0\}$$ ran Y 1 2 ∩ ran G 1 2 = { 0 } . Relationships between $$\tau _G$$ τ G and Lebesgue type decomposition of nonnegative self-adjoint operator are established and applications to the properties of unbounded self-adjoint operators with trivial intersections of their domains are given. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

On the mappings connected with parallel addition of nonnegative operators

Positivity , Volume 21 (1) – May 12, 2016
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Publisher
Springer International Publishing
Copyright
Copyright © 2016 by Springer International Publishing
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-016-0421-5
Publisher site
See Article on Publisher Site

Abstract

We study a mapping $$\tau _G$$ τ G of the cone $${\mathbf B}^+({\mathcal H})$$ B + ( H ) of bounded nonnegative self-adjoint operators in a complex Hilbert space $${\mathcal H}$$ H into itself. This mapping is defined as a strong limit of iterates of the mapping $${\mathbf B}^+({\mathcal H})\ni X\mapsto \mu _G(X)=X-X:G\in {\mathbf B}^+({\mathcal H})$$ B + ( H ) ∋ X ↦ μ G ( X ) = X - X : G ∈ B + ( H ) , where $$G\in {\mathbf B}^+({\mathcal H})$$ G ∈ B + ( H ) and X : G is the parallel sum. We find explicit expressions for $$\tau _G$$ τ G and describe its properties. In particular, it is shown that $$\tau _G$$ τ G is sub-additive, homogeneous of degree one, and its image coincides with the set of its fixed points which is a subset of $${\mathbf B}^+({\mathcal H})$$ B + ( H ) , consisting of all Y such that $$\mathrm{ran\,} Y^{\frac{1}{2}}\cap \mathrm{ran\,} G^{\frac{1}{2}}=\{0\}$$ ran Y 1 2 ∩ ran G 1 2 = { 0 } . Relationships between $$\tau _G$$ τ G and Lebesgue type decomposition of nonnegative self-adjoint operator are established and applications to the properties of unbounded self-adjoint operators with trivial intersections of their domains are given.

Journal

PositivitySpringer Journals

Published: May 12, 2016

References

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