On the absolute value of the product and the sum of linear operators

On the absolute value of the product and the sum of linear operators Rend. Circ. Mat. Palermo, II. Ser https://doi.org/10.1007/s12215-018-0356-8 On the absolute value of the product and the sum of linear operators 1,2 Mohammed Hichem Mortad Received: 7 May 2018 / Accepted: 26 May 2018 © Springer-Verlag Italia S.r.l., part of Springer Nature 2018 Abstract Let A, B ∈ B(H ). In the present paper, we establish simple and interesting facts on when we have | A||B|=|B|| A|, |AB|=| A||B|, | A± B|≤| A|+|B|, || A|−|B|| ≤ | A± B| and | A|−|B| ≤  A ± B,where |·| denotes the absolute value (or modulus) of an operator. Some of these results are known but the proofs here are Theorem-Spectral-free proofs. Some other interesting consequences are also established. Keywords Absolute value · Triangle inequality · Normal, Hyponormal, Self-adjoint and positive operators · Commutativity · Fuglede theorem Mathematics Subject Classification Primary 47A63; Secondary 47A62 · 47B15 · 47B20 1 Introduction Let H be a complex Hilbert space and let A, B ∈ B(H ).Wesay that A is positive, and we write A ≥ 0, if < Ax , x >≥ 0for all x ∈ H.Since H is a complex Hilbert space, a positive operator is clearly self-adjoint. We say that A ≥ http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Rendiconti del Circolo Matematico di Palermo Springer Journals

On the absolute value of the product and the sum of linear operators

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Publisher
Springer Journals
Copyright
Copyright © 2018 by Springer-Verlag Italia S.r.l., part of Springer Nature
Subject
Mathematics; Mathematics, general; Algebra; Geometry; Analysis; Applications of Mathematics
ISSN
0009-725X
eISSN
1973-4409
D.O.I.
10.1007/s12215-018-0356-8
Publisher site
See Article on Publisher Site

Abstract

Rend. Circ. Mat. Palermo, II. Ser https://doi.org/10.1007/s12215-018-0356-8 On the absolute value of the product and the sum of linear operators 1,2 Mohammed Hichem Mortad Received: 7 May 2018 / Accepted: 26 May 2018 © Springer-Verlag Italia S.r.l., part of Springer Nature 2018 Abstract Let A, B ∈ B(H ). In the present paper, we establish simple and interesting facts on when we have | A||B|=|B|| A|, |AB|=| A||B|, | A± B|≤| A|+|B|, || A|−|B|| ≤ | A± B| and | A|−|B| ≤  A ± B,where |·| denotes the absolute value (or modulus) of an operator. Some of these results are known but the proofs here are Theorem-Spectral-free proofs. Some other interesting consequences are also established. Keywords Absolute value · Triangle inequality · Normal, Hyponormal, Self-adjoint and positive operators · Commutativity · Fuglede theorem Mathematics Subject Classification Primary 47A63; Secondary 47A62 · 47B15 · 47B20 1 Introduction Let H be a complex Hilbert space and let A, B ∈ B(H ).Wesay that A is positive, and we write A ≥ 0, if < Ax , x >≥ 0for all x ∈ H.Since H is a complex Hilbert space, a positive operator is clearly self-adjoint. We say that A ≥

Journal

Rendiconti del Circolo Matematico di PalermoSpringer Journals

Published: Jun 5, 2018

References

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