Geodesic
A geometric approach to inequalities for the Hilbert–Schmidt norm
Filomat, Tome 37 (2023) no. 30, p. 10435

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DOI

We show that if X and Y are two non-zero Hilbert–Schmidt operators, then for any λ ≥ 0, cos2 ΘX,Y ≤ 1 1 + λ √ cosΘ|X∗ |,|Y∗ | √ cosΘ|X|,|Y| |〈X,Y〉| ∥X∥ 2 ∥Y∥ 2 + λ 1 + λ cosΘ|X∗ |,|Y∗ | cosΘ|X|,|Y| ≤ cosΘ|X∗ |,|Y∗ | cosΘ|X|,|Y| . Here ΘA,B denotes the angle between non-zero Hilbert–Schmidt operators A and B. This enables us to present some inequalities for the Hilbert–Schmidt norm. In particular, we prove that ∥∥∥X + Y∥∥∥ 2 ≤ √ √ 2 + 1 2 ∥∥∥ |X| + |Y| ∥∥∥ 2
DOI : 10.2298/FIL2330435Z
Classification : 47A63, 47A30, 47B10
Keywords: Hilbert–Schmidt norm, Operator inequalities, Angle 
Ali Zamani. A geometric approach to inequalities for the Hilbert–Schmidt norm. Filomat, Tome 37 (2023) no. 30, p. 10435 . doi: 10.2298/FIL2330435Z
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     title = {A geometric approach to inequalities for the {Hilbert{\textendash}Schmidt} norm},
     journal = {Filomat},
     pages = {10435 },
     year = {2023},
     volume = {37},
     number = {30},
     doi = {10.2298/FIL2330435Z},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.2298/FIL2330435Z/}
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