Nilpotent perturbations of m-isometric and m-symmetric tensor products of commuting d-tuples of operators

Abstract

If T1{{\mathbb{T}}}_{1} and T2{{\mathbb{T}}}_{2} are commuting dd-tuples of Hilbert space operators in B(ℋ)dB{\left({\mathcal{ {\mathcal H} }})}^{d} such that (T1*⊗I+I⊗T2*,T1⊗I+I⊗T2)\left({{\mathbb{T}}}_{1}^{* }\otimes I+I\otimes {{\mathbb{T}}}_{2}^{* },{{\mathbb{T}}}_{1}\otimes I+I\otimes {{\mathbb{T}}}_{2}) is strictly mm-isometric (resp., mm-symmetric) for some positive integer mm, then there exist a scalar dd-tuple λ\lambda and positive integers mi{m}_{i}, 1≤i≤21\le i\le 2, such that m=m1+2m2−2m={m}_{1}+2{m}_{2}-2, (T1*+λ¯,T1+λ)\left({{\mathbb{T}}}_{1}^{* }+\overline{\lambda },{{\mathbb{T}}}_{1}+\lambda ) is m1{m}_{1} isometric, and T2−λ{{\mathbb{T}}}_{2}-\lambda is m2{m}_{2}-nilpotent (resp., m=m1+m2−1m={m}_{1}+{m}_{2}-1, (T1*+λ¯,T1+λ)\left({{\mathbb{T}}}_{1}^{* }+\overline{\lambda },{{\mathbb{T}}}_{1}+\lambda ) is m1{m}_{1}-symmetric and (T2*−λ¯,T2−λ)\left({{\mathbb{T}}}_{2}^{* }-\overline{\lambda },{{\mathbb{T}}}_{2}-\lambda ) is m2{m}_{2} symmetric).

Keywords

• banach space
• left/right multiplication operator
• m-isometric and m-symmetric commuting d-tuples of operators
• tensor product of operators
• 47a05
• 47a55
• 47a11
• 47b47