m-Isometric tensor products

Abstract

Given Banach space operators Si{S}_{i} and Ti{T}_{i}, i=1,2i=1,2, we use elementary properties of the left and right multiplication operators to prove, that if the tensor products pair (S1⊗S2,T1⊗T2)\left({S}_{1}\otimes {S}_{2},{T}_{1}\otimes {T}_{2}) is strictly mm-isometric, i.e., ΔS1⊗S2,T1⊗T2m(I⊗I)=∑j=0m(−1)jmj(S1⊗S2)m−j(T1⊗T2)m−j=0{\Delta }_{{S}_{1}\otimes {S}_{2},{T}_{1}\otimes {T}_{2}}^{m}\left(I\otimes I)={\sum }_{j=0}^{m}{\left(-1)}^{j}\left(\begin{array}{c}m\\ j\end{array}\right){\left({S}_{1}\otimes {S}_{2})}^{m-j}{\left({T}_{1}\otimes {T}_{2})}^{m-j}=0, then there exist a non-zero scalar cc and positive integers m1,m2≤m{m}_{1},{m}_{2}\le m such that m=m1+m2−1m={m}_{1}+{m}_{2}-1, (S1,cT1)\left({S}_{1},c{T}_{1}) is strict-m1{m}_{1}-isometric and S2,1cT2\left({S}_{2},\frac{1}{c}{T}_{2}\right) is strict m2{m}_{2}-isometric.

Keywords

• banach space
• left/right multiplication operator
• strict-m-isometric operator
• tensor product
• primary 47b47
• 47a80
• secondary 47a80
• 47b10