$(m,infty)$-expansive and $(m,infty)$-contractive Commuting tuple of operators on a Banach space }

Abstract

For a $d$-tuple of commuting operators
${\bf \large{S}}:=(S_1,\cdots,S_d)\in {\mathcal B}[{\mathcal X}]^d $, $m\in \mathbb{N}$ and $p >0$ be a real number, we define
$${\mathcal Q}_m^{(p)}({{\bf \large S}};\;u):=\displaystyle\sum_{0\leq k \leq
m}(-1)^k\binom{m}{k}\bigg(\sum_{ \begin{array}{c}
\mu \in \mathbb{N}_0^d \\
|\mu|=k
\end{array}
}\frac{k!}{\mu}\|{\bf\large S}^{\mu}u\| ^p\bigg).$$
As a natural extension of the concepts of $(m,p)$-expansive and $(m,p)$-contractive for tuple of commuting operators, we introduce and studied the concepts of $(m,\infty)$-expansive tuple and $(m,\infty)$-contractive tuple of commuting operators acting on Banach space.
We say that ${{\bf \large S}}$ is $(m,\infty)$-expansive $d$-tuple \big(resp. $(m,p)$-contractive $d$-tuple \big) of operators if ${\mathcal Q}_m^{(p)}( {\bf \large{S}};\;u) \leq 0\;\;\forall\;u \in {\mathcal X}$ and $p\to \infty $ \big(resp.
${\mathcal Q}_m^{(p)}({\bf \large{S}};\;u)\geq 0 \;\;\forall\;u \in {\mathcal X} $ and $p\to \infty$ \big).
These concepts extend the definition of $(m,\infty)$-isometric tuple
of bounded linear operators
acting on Banach spaces was introduced and studied in \cite{HF}.