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

Abstract

Given commuting $d$-tuples $\mathbb{S}_i$ and $\mathbb{T}_i$, $1\leq i\leq 2$, 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 $1\leq i\leq 2$ 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).