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

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
,
∞
)
-expansive tuple and
(
m
,
∞
)
-contractive tuple of commuting operators acting on Banach space. We say that
S
is
(
m
,
∞
)
-expansive d-tuple \big(resp. (m,p)-contractive d-tuple \big) of operators if
Q
m
(
p
)
(
S
;
  
u
)
≤
0
  
  
∀
  
u
∈
X
and
p
→
∞
\big(resp.
Q
m
(
p
)
(
S
;
  
u
)
≥
0
  
  
∀
  
u
∈
X
and
p
→
∞
\big). These concepts extend the definition of
(
m
,
∞
)
-isometric tuple of bounded linear operators acting on Banach spaces was introduced and studied in \cite{HF}.