M-Isometric tensor products

Abstract

Abstract Given Banach space operators S i {S}_{i} and T i {T}_{i} , i = 1 , 2 i=1,2 , we use elementary properties of the left and right multiplication operators to prove, that if the tensor products pair ( S 1 ⊗ S 2 , T 1 ⊗ T 2 ) \left({S}_{1}\otimes {S}_{2},{T}_{1}\otimes {T}_{2}) is strictly m m -isometric, i.e., Δ S 1 ⊗ S 2 , T 1 ⊗ T 2 m ( I ⊗ I ) = ∑ j = 0 m ( − 1 ) j m j ( S 1 ⊗ S 2 ) m − j ( T 1 ⊗ T 2 ) 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 c c and positive integers m 1 , m 2 ≤ m {m}_{1},{m}_{2}\le m such that m = m 1 + m 2 − 1 m={m}_{1}+{m}_{2}-1 , ( S 1 , c T 1 ) \left({S}_{1},c{T}_{1}) is strict- m 1 {m}_{1} -isometric and S 2 , 1 c T 2 \left({S}_{2},\frac{1}{c}{T}_{2}\right) is strict m 2 {m}_{2} -isometric.