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 $${1 \leq p \leq \infty}$$ .
Positivity – Springer Journals
Published: Jan 17, 2012
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