On spectra of some completely positive maps

Let ∑ i = 1 ∞ A i A i ∗ and ∑ i = 1 ∞ A i ∗ A i converge in the strong operator topology. We study the map Φ A defined on the Banach space of all bounded linear operators B ( H ) by Φ A ( X ) = ∑ i = 1 ∞ A i X A i ∗ and its restriction Φ A | K ( H ) to the Banach space of all compact operators K ( H...

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Bibliographic Details
Published in:Positivity : an international journal devoted to the theory and applications of positivity in analysis 2024-04, Vol.28 (2), p.22, Article 22
Main Authors: Li, Yuan, Gao, Shuhui, Zhao, Cong, Ma, Nan
Format: Article
Language:English
Subjects:
Banach spaces
Calculus of Variations and Optimal Control
Optimization
Econometrics
Eigenvalues
Fixed points (mathematics)
Fourier Analysis
Hilbert space
Linear operators
Mathematics
Mathematics and Statistics
Operator Theory
Potential Theory
Spectra
Topology
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Summary:Let ∑ i = 1 ∞ A i A i ∗ and ∑ i = 1 ∞ A i ∗ A i converge in the strong operator topology. We study the map Φ A defined on the Banach space of all bounded linear operators B ( H ) by Φ A ( X ) = ∑ i = 1 ∞ A i X A i ∗ and its restriction Φ A | K ( H ) to the Banach space of all compact operators K ( H ) . We first consider the relationship between the boundary eigenvalues of Φ A | K ( H ) and its fixed points. Also, we show that the spectra of Φ A and Φ A | K ( H ) are the same sets. In particular, the spectra of two completely positive maps involving the unilateral shift are described.
ISSN:1385-1292
1572-9281
DOI:10.1007/s11117-024-01037-4