Maps Preserving Schatten p-Norms of Convex Combinations. (14th January 2014)

• Record Type: Journal Article
• Authors: Kuo, David Li-Wei; Tsai, Ming-Cheng; Wong, Ngai-Ching; Zhang, Jun
• Other Names: Peralta Antonio M. Academic Editor.
• Is Part Of: Abstract and applied analysis. Volume 2014(2014)
• Publication Date: 2014-01-14
• Subjects: Mathematical analysis -- Periodicals; Mathematical analysis; Applied Mathematics; Mathematical Analysis; Periodicals; 515.05
• DOI: 10.1155/2014/520795
• Languages: English
• ISSNs: 1085-3375

Abstract

Abstract : We study maps ϕ of positive operators of the Schatten p -classes (1 < p < + ∞ ), which preserve the p -norms of convex combinations, that is, ∥ t ρ + ( 1 - t ) σ ∥ p = ∥ t ϕ ( ρ ) + ( 1 - t ) ϕ ( σ ) ∥ p, ∀ ρ, σ ∈ 𝒮 p + ( H ) 1, t ∈ [ 0, 1 ] . They are exactly those carrying the form ϕ ( ρ ) = U ρ U * for a unitary or antiunitary U . In the case p = 2, we have the same conclusion whenever it just holds ∥ ρ + σ ∥ 2 = ∥ ϕ ( ρ ) + ϕ ( σ ) ∥ 2 for all the positive Hilbert-Schmidt class operators ρ, σ of norm 1 . Some examples are demonstrated.