TENSOR PRODUCTS OF ALUTHGE TRANSFORMS AND A-ADJOINTS OF m-ISOMETRIC OPERATORS

Abstract

Given an $m$-isometric Hilbert space operator $A\in \mathcal{ℬ}(\mathcal{\mathscr{H}})$, ${△}_{A^{\ast },A}^m(I)={\sum }_{j=0}^m{(-1)}^j\left(\genfrac{}{}{0.0pt}{}{m}{j}\right)A^{\ast j}{A}^j=0$, with polar decomposition $A=U\left|A\right|$, the Aluthge transform $Ã=\left|A{|}^{1∕2}U\right|A{|}^{1∕2}$ preserves almost all the spectral properties of $A$. However, the $m$-isometric property of an operator neither implies nor is implied by the $m$-isometric property of its Aluthge transform. The operator $A$ has an $\left|A\right|$-adjoint $\mathcal{𝒜}$, ${\mathcal{𝒜}}^{\ast }={[A]}^{\ast }=U^{\ast }\left|A\right|$. If $A_i$, $i=1,2$, doubly commute and ${Ã}_i$ (resp. ${\mathcal{𝒜}}_i$) is strict $m_i$-isometric, then $\widetilde{A_1{A}_2}$ (resp. ${\mathcal{𝒜}}_1{\mathcal{𝒜}}_2$) is strict $(m_1+m_2-1)$-isometric. The converse fails for products $A_1{A}_2$, ${Ã}_1{Ã}_2$ and ${\mathcal{𝒜}}_1{\mathcal{𝒜}}_2$, but has a positive answer for tensor products $A_1\otimes A_2$, ${Ã}_1\otimes {Ã}_2$, ${\mathcal{𝒜}}_1\otimes {\mathcal{𝒜}}_2$ (and their Hilbert–Schmidt class identifications with the elementary operators $L_{A_1}{R}_{A_2^{\ast }}$, $L_{{Ã}_1}{R}_{{Ã}_2^{\ast }}$ and $L_{{\mathcal{𝒜}}_1}{R}_{A_2^{\ast }}$); if $S\otimes T$, where $S\otimes T$ stands for either of the three tensor products above, is strict $m$-isometric, then there exist scalars $c$ and $d$, $\left|cd\right|=1$, and positive integers $m_1$ and $m_2$, $m=m_1+m_2-1$, such that $cS$ is strict $m_1$-isometric and $dT$ is strict $m_2$-isometric.