On strict isometric and strict symmetric commuting $d$-tuples of Banach space operators

Abstract

Given commuting $d$-tuples ${\mathbb{S}}_i$ and ${\mathbb{T}}_i$, $1\le i\le 2$, Banach space operators such that the tensor products pair $({\mathbb{S}}_1\otimes {\mathbb{S}}_2,{\mathbb{T}}_1\otimes {\mathbb{T}}_2)$ is strict $m$-isometric (resp., ${\mathbb{S}}_1$, ${\mathbb{S}}_2$ are invertible and $({\mathbb{S}}_1\otimes {\mathbb{S}}_2,{\mathbb{T}}_1\otimes {\mathbb{T}}_2)$ is strict $m$-symmetric), there exist integers $m_i>0$, and a non-zero scalar $c$, such that $m=m_1+m_2-1$, $({\mathbb{S}}_1,\frac{1}{c}{\mathbb{T}}_1)$ is strict $m_1$-isometric and $({\mathbb{S}}_2,c{\mathbb{T}}_2)$ is strict $m_2$-isometric (resp., there exist integers $m_i>0$, and a non-zero scalar $c$, such that $m=m_1+m_2-1$, $({\mathbb{S}}_1,\frac{1}{c}{\mathbb{T}}_1)$ is strict $m_1$-symmetric and $({\mathbb{S}}_2,c{\mathbb{T}}_2)$ is strict $m_2$-symmetric. However, $({\mathbb{S}}_i,{\mathbb{T}}_i)$ is strict $m_i$-isometric (resp., strict $m_i$-symmetric) for $1\le i\le 2$ implies only that $({\mathbb{S}}_1\otimes {\mathbb{S}}_2,{\mathbb{T}}_1\otimes {\mathbb{T}}_2)$ is $m$-isometric (resp., $({\mathbb{S}}_1\otimes {\mathbb{S}}_2,{\mathbb{T}}_1\otimes {\mathbb{T}}_2)$ is $m$-symmetric).

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