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).
Categories: math.FA