Fuglede–Putnam type theorems via the Aluthge transform

Fuglede–Putnam type theorems via the Aluthge transform Let A = U|A| and B = V|B| be the polar decompositions of $${A \in \mathbb{B}(\fancyscript{H}_1)}$$ and $${B\in \mathbb{B}(\fancyscript{H}_2)}$$ and let Com(A, B) stand for the set of operators $${X \in \mathbb{B}(\fancyscript{H}_2,\fancyscript{H}_1)}$$ such that AX =  XB. A pair (A, B) is said to have the FP-property if Com(A, B) $${\subseteq}$$ Com(A*, B*). Let $${\tilde{C}}$$ denote the Aluthge transform of a bounded linear operator C. We show that (i) if A and B are invertible and (A, B) has the FP-property, then so is $${(\tilde{A}, \tilde{B})}$$ ; (ii) if A and B are invertible, the spectrums of both U and V are contained in some open semicircle and $${(\tilde{A}, \tilde{B})}$$ has the FP-property, then so is (A, B); (iii) if (A, B) has the FP-property, then Com(A, B) $${\subseteq}$$ Com $${(\tilde{A},\tilde{B})}$$ , moreover, if A is invertible, then Com(A, B) = Com $${(\tilde{A}, \tilde{B})}$$ . Finally, if $${\mbox{Re}(U|A|^{1\over2})\geq a>0}$$ and $${\mbox{Re}(V|B|^{1\over2})\geq a>0}$$ and X is an operator such that U* X = XV, then we prove that $${\|\tilde{A}^* X-X\tilde{B}\|_p\geq 2a\|\,|B|^{1\over2}X-X|B|^{1\over2} \|_p}$$ for any $${1 \leq p \leq \infty}$$ . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

Fuglede–Putnam type theorems via the Aluthge transform

Positivity, Volume 17 (1) – Jan 17, 2012
12 pages

/lp/springer_journal/fuglede-putnam-type-theorems-via-the-aluthge-transform-hdC40gSyx6
Publisher
Springer Journals
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-011-0154-4
Publisher site
See Article on Publisher Site

Abstract

Let A = U|A| and B = V|B| be the polar decompositions of $${A \in \mathbb{B}(\fancyscript{H}_1)}$$ and $${B\in \mathbb{B}(\fancyscript{H}_2)}$$ and let Com(A, B) stand for the set of operators $${X \in \mathbb{B}(\fancyscript{H}_2,\fancyscript{H}_1)}$$ such that AX =  XB. A pair (A, B) is said to have the FP-property if Com(A, B) $${\subseteq}$$ Com(A*, B*). Let $${\tilde{C}}$$ denote the Aluthge transform of a bounded linear operator C. We show that (i) if A and B are invertible and (A, B) has the FP-property, then so is $${(\tilde{A}, \tilde{B})}$$ ; (ii) if A and B are invertible, the spectrums of both U and V are contained in some open semicircle and $${(\tilde{A}, \tilde{B})}$$ has the FP-property, then so is (A, B); (iii) if (A, B) has the FP-property, then Com(A, B) $${\subseteq}$$ Com $${(\tilde{A},\tilde{B})}$$ , moreover, if A is invertible, then Com(A, B) = Com $${(\tilde{A}, \tilde{B})}$$ . Finally, if $${\mbox{Re}(U|A|^{1\over2})\geq a>0}$$ and $${\mbox{Re}(V|B|^{1\over2})\geq a>0}$$ and X is an operator such that U* X = XV, then we prove that $${\|\tilde{A}^* X-X\tilde{B}\|_p\geq 2a\|\,|B|^{1\over2}X-X|B|^{1\over2} \|_p}$$ for any $${1 \leq p \leq \infty}$$ .

Journal

PositivitySpringer Journals

Published: Jan 17, 2012

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 Explore the DeepDyve Library 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 Freelancer DeepDyve Pro Price FREE$49/month
\$360/year

Save searches from
PubMed

Create lists to

Export lists, citations