# 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

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

/lp/springer_journal/fuglede-putnam-type-theorems-via-the-aluthge-transform-hdC40gSyx6
Publisher
SP Birkhäuser Verlag Basel
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

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