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 ≥
Rendiconti del Circolo Matematico di Palermo – Springer Journals
Published: Jun 5, 2018
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