# 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

, Volume OnlineFirst – Jun 5, 2018
11 pages

/lp/springer_journal/on-the-absolute-value-of-the-product-and-the-sum-of-linear-operators-Xxp5PRKmBd
Publisher
Springer Milan
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

## You’re reading a free preview. Subscribe to read the entire article.

### DeepDyve is your personal research library

It’s your single place to instantly
that matters to you.

over 18 million articles from more than
15,000 peer-reviewed journals.

All for just \$49/month

### Search

Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly

### Organize

Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place.

### Access

Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals.

### Your journals are on DeepDyve

Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more.

All the latest content is available, no embargo periods.

DeepDyve

DeepDyve

### Pro

Price

FREE

\$49/month
\$360/year

Save searches from
PubMed

Create lists to

Export lists, citations