Affiliation:
1. Faculty of Sciences and Mathematics, University of Niš , P. O. Box 224 , 18000 Niš , Serbia
2. Department of Mathematics, Incheon National University , Incheon , 22012 , Korea
Abstract
Abstract
Given Banach space operators
S
i
{S}_{i}
and
T
i
{T}_{i}
,
i
=
1
,
2
i=1,2
, we use elementary properties of the left and right multiplication operators to prove, that if the tensor products pair
(
S
1
⊗
S
2
,
T
1
⊗
T
2
)
\left({S}_{1}\otimes {S}_{2},{T}_{1}\otimes {T}_{2})
is strictly
m
m
-isometric, i.e.,
Δ
S
1
⊗
S
2
,
T
1
⊗
T
2
m
(
I
⊗
I
)
=
∑
j
=
0
m
(
−
1
)
j
m
j
(
S
1
⊗
S
2
)
m
−
j
(
T
1
⊗
T
2
)
m
−
j
=
0
{\Delta }_{{S}_{1}\otimes {S}_{2},{T}_{1}\otimes {T}_{2}}^{m}\left(I\otimes I)={\sum }_{j=0}^{m}{\left(-1)}^{j}\left(\begin{array}{c}m\\ j\end{array}\right){\left({S}_{1}\otimes {S}_{2})}^{m-j}{\left({T}_{1}\otimes {T}_{2})}^{m-j}=0
, then there exist a non-zero scalar
c
c
and positive integers
m
1
,
m
2
≤
m
{m}_{1},{m}_{2}\le m
such that
m
=
m
1
+
m
2
−
1
m={m}_{1}+{m}_{2}-1
,
(
S
1
,
c
T
1
)
\left({S}_{1},c{T}_{1})
is strict-
m
1
{m}_{1}
-isometric and
S
2
,
1
c
T
2
\left({S}_{2},\frac{1}{c}{T}_{2}\right)
is strict
m
2
{m}_{2}
-isometric.
Subject
Applied Mathematics,Analysis