Ann.
Acad.
Rom.
Sci.
Ser.
Math.
Appl.
ISSN
2066-6594
Vol.
18,
No.
3/2026
ON
UNIFORM
POLYNOMIAL
TRISPLITTING
OF
EVOLUTION
OPERATORS
∗
Claudia
Luminit
¸a
Mihit
¸
†
Ghiocel
Mot
¸
‡
Dedicated
to
the
memory
of
Professor
Mihail
Megan
DOI
10.56082/annalsarscimath.2026.3.145
Abstract
The
article
explores
the
concept
of
uniform
polynomial
trisplitting,
as
generalization
of
the
uniform
polynomial
trichotomy
for
the
case
of
evolution
operators
in
Banach
spaces.
We
prove
that
the
uniform
polynomial
trisplitting
in
continuous-time
is
equivalent
to
the
uniform
polynomial
trisplitting
in
discrete-time
(assuming
the
existence
of
the
uniform
polynomial
growth),
using
compatible
families
of
projections,
respectively
strongly
compatible
families
of
projections.
Also,
discrete
criteria
for
this
property
are
given
and
in
particular,
new
results
for
uniform
polynomial
trichotomy
are
obtained.
Keywords:
evolution
operators,
polynomial
trisplitting,
polynomial
tri-
chotomy.
MSC:
34D05,
34D09.
1
Introduction
The
property
of
trichotomy
played
a
central
role
in
the
qualitative
theory
of
dynamical
systems
and
was
intensively
approached
in
the
last
years
from
∗
Accepted
for
publication
on
July
28,
2026
†
claudia.mihit@uav.ro
,
Department
of
Mathematics
and
Computer
Science,
Aurel
Vlaicu
University
of
Arad,
Elena
Dr˘
agoi
Street
no.
2,
310330
Arad,
Romania
‡
ghiocel.mot@uav.ro
,
Department
of
Mathematics
and
Computer
Science,
Aurel
Vlaicu
University
of
Arad,
Elena
Dr˘
agoi
Street
no.
2,
310330
Arad,
Romania
145
On
uniform
polynomial
trisplitting
of
evolutions
operators
146
different
points
of
view
(see
[
1
,
4
,
25
,
27
]).
R.J.
Sacker
and
G.R.
Sell
[
24
]
in-
troduced
for
the
first
time
the
notion
of
an
exponential
trichotomy
for
skew-
product
semiflows
and
the
studies
were
further
developed
by
B.
Aulbach
[
2
]
in
the
case
of
differential
systems.
For
nonautonomous
dynamics,
notable
results
are
obtained
by
K.J.
Palmer
[
22
],
where
the
author
showed
that
the
hyperbolicity
properties
could
be
described
through
trichotomies.
This
trichotomy
concept
represents
a
direct
generalization
of
the
classical
expo-
nential
dichotomy
notion
-
we
refer
here
to
the
dichotomy
notion
explored
in
recent
years
via
different
methods
in
[
17
,
18
,
26
]
and
the
references
therein.
Nevertheless,
in
some
situations
the
concept
of
exponential
dichotomy
is
con-
sidered
restrictive
and
therefore
it
is
important
to
consider
other
asymptotic
behaviors,
such
as
those
of
polynomial
type.
The
nonuniform
polynomial
dichotomy
was
introduced
by
L.
Barreira
and
C.
Valls
in
[
5
]
and
an
interesting
study
in
the
nonuniform
case
was
made
in
[
6
].
A
slightly
modified
concept
of
nonuniform
polynomial
dichotomy
was
studied
in
[
8
],
where
the
authors
proved
characterizations
using
some
fami-
lies
of
norms
compatible
with
the
projection
families.
In
the
case
of
discrete
nonautonomous
systems,
we
remark
the
accomplishments
from
[
10
]
for
the
polynomial
dichotomy
with
respect
to
a
sequence
of
norms.
Regarding
the
polynomial
trichotomic
behavior,
two
concepts
of
uniform
polynomial
tri-
chotomy
were
treated
in
[
21
]
for
the
skew-evolution
semiflows
and
a
sufficient
criteria
for
the
uniform
polynomial
trichotomy
was
proved,
based
on
a
result
from
[
9
].
Another
step
in
the
trichotomies
topics
was
made
by
S.
Elaydi
and
K.
Janglajew
in
[
16
],
where
the
authors
approached
for
the
first
time
the
expo-
nential
trichotomy
in
discrete-time
for
difference
equations
x
(
n
+
1)
=
A
(
n
)
x
(
n
)
,
n
∈
Z
,
where
A
(
n
)
was
a
k
×
k
invertible
matrix
defined
on
Z
and
then
the
results
were
extended
to
nonlinear
difference
equations.
Previously,
interesting
con-
cepts
of
trichotomy
were
studied
by
S.
Elaydi
and
O.
H`
ajek
(see
[
14
,
15
])
for
linear,
respectively
nonlinear
differential
systems.
We
emphasize
that
in
[
14
–
16
],
the
authors
treated
versions
of
trichotomies
that
are
reduced
to
double
dichotomies
which
is
in
a
sharp
contrast
with
the
approaches
in
[
2
,
22
,
24
].
Recently,
important
results
for
the
property
of
the
robust-
ness
of
exponential
trichotomy
in
the
original
sense
of
Sacker
and
Sell
have
been
obtained
in
uniform
and
nonuniform
context,
for
nonautonomous
and
variational
systems
(see
D.
Dragiˇ
cevi´
c,
A.
Sasu
and
B.
Sasu
[
11
,
12
]).
The
exponential
splitting
property
was
introduced
by
B.
Aulbach,
J.
Kalkbrenner
in
[
3
]
for
difference
equations,
to
extend
the
framework
of
C.L.
Mihit
¸,
G.
Mot
¸
147
dichotomy
to
a
more
general
setting.
In
[
19
],
the
properties
of
exponen-
tial
splitting,
respectively
strong
exponential
splitting
are
treated
for
linear
discrete-time
systems,
as
generalization
of
exponential
dichotomy,
respec-
tively
strong
exponential
dichotomy.
In
a
natural
manner,
the
uniform
ex-
ponential
trichotomy
was
extended
to
uniform
exponential
trisplitting
in
[
7
]
for
discrete
skew-product
semiflows.
In
1984,
K.M.
Przyluski
and
S.
Rolewicz
[
23
]
proved
the
discrete-time
version
of
the
well-known
stability
theorem
of
R.
Datko
[
9
]:
a
linear
discrete-
time
system
x
(
n
+
1)
=
A
(
n
)
x
(
n
)
,
n
≥
n
0
,
on
a
Banach
space
X
is
uniformly
exponentially
stable
if
and
only
if
there
exists
p
∈
[1
,
+
∞
)
such
that:
sup
n
≥
n
0
+
∞
j
=
n
+1
j
−
1
i
=
n
A
i
x

p
<
+
∞
.
Various
methods,
which
led
to
the
generalizations
of
the
above
result,
have
been
presented
in
[
13
]
and
[
20
].
Furthermore,
as
shown
in
[
13
]
this
type
of
results
have
substantial
applications
in
exploring
robustness
for
stability
and
instability.
An
important
and
difficult
problem
in
the
asymptotic
theory
of
dynam-
ical
systems
is
the
connection
between
discrete-time
and
continuous
time
dynamics.
In
this
setting,
the
full
recovery
of
the
exponential
trichotomy
in
continuous-time
from
discrete-time
exponential
trichotomy
was
proved
for
the
first
time
in
A.
Sasu
and
B.
Sasu
[
25
],
through
admissibility
techniques.
In
a
distinct
direction,
the
analysis
of
continuous
and
discrete
exponential
trichotomy
for
skew-evolution
semiflows
is
made
in
[
28
]
and
discrete
charac-
terizations
were
given.
In
the
context
of
current
developments
and
new
trends
in
the
qualitative
theory
of
dynamical
systems,
we
approach
in
this
paper
the
property
of
uni-
form
polynomial
trisplitting
of
evolution
operators
in
terms
of
compatible
and
strongly
compatible
families
of
projections.
We
prove
the
equivalence
between
uniform
polynomial
trisplitting
in
continuous-time
and
the
uni-
form
polynomial
trisplitting
in
discrete-time
and
we
deduce
the
equivalence
between
uniform
polynomial
trichotomy
in
continuous-time
and
the
uni-
form
polynomial
trichotomy
in
discrete-time.
Discrete
results
of
Przyluski-
Rolewicz
type
are
established
and
also,
the
connection
between
uniform
polynomial
trisplitting
and
uniform
polynomial
trichotomy
are
obtained.
On
uniform
polynomial
trisplitting
of
evolutions
operators
148
2
Evolution
operators.
Families
of
projections
In
this
section,
we
introduce
the
main
concepts
of
evolution
operators,
pro-
jections
and
compatibility
properties
between
them.
We
consider
X
a
Ba-
nach
space,
B
(
X
)
will
denote
the
Banach
algebra
of
all
bounded
linear
op-
erators
on
X
and
the
norms
on
X
,
respectively
on
B
(
X
)
will
be
denoted
by
||
·
||
.
Further,
we
will
use
the
sets
∆
=
{
(
t,
s
)
∈
R
2
+
:
t
≥
s
}
,
∆
d
=
{
(
m,
n
)
∈
N
2
:
m
≥
n
}
,
T
=
{
(
t,
s,
t
0
)
∈
R
3
+
:
t
≥
s
≥
t
0
}
,
T
d
=
{
(
m,
n,
p
)
∈
N
3
:
m
≥
n
≥
p
}
.
Definition
1.
We
say
that
U
:
∆
→
B
(
X
)
is
an
evolution
operator
on
X
if
the
following
conditions
hold:
(
eo
1
)
U
(
s,
s
)
=
I
(the
identity
operator
on
X
),
for
every
s
≥
0;
(
eo
2
)
U
(
t,
s
)
U
(
s,
t
0
)
=
U
(
t,
t
0
)
,
for
all
(
t,
s,
t
0
)
∈
T.
Definition
2.
A
mapping
P
:
R
+
→
B
(
X
)
is
a
family
of
projections
if
P
(
s
)
2
=
P
(
s
)
,
for
every
s
≥
0
.
Let
U
:
∆
→
B
(
X
)
be
an
evolution
operator
and
let
P
1
,
P
2
,
P
3
:
R
+
→
B
(
X
)
be
three
families
of
projections.
Definition
3.
We
say
that
P
=
{
P
1
,
P
2
,
P
3
}
is
a
family
of
compatible
pro-
jections
with
U
if:
(
c
1
)
P
1
(
s
)
+
P
2
(
s
)
+
P
3
(
s
)
=
I,
for
all
s
≥
0;
(
c
2
)
P
i
(
s
)
P
j
(
s
)
=
0
,
for
all
i,
j
∈
1
,
3
,
i
=
j,
s
≥
0;
(
c
3
)
P
i
(
t
)
U
(
t,
s
)
=
U
(
t,
s
)
P
i
(
s
)
,
for
all
(
t,
s
)
∈
∆
,
i
∈
1
,
3
.
Definition
4.
The
family
of
projections
P
=
{
P
1
,
P
2
,
P
3
}
is
said
to
be
strongly
compatible
with
U
if
it
is
compatible
with
U
and
for
all
(
t,
s
)
∈
∆,
the
mapping
U
(
t,
s
)
is
an
isomorphism
from
Range
P
i
(
s
)
to
Range
P
i
(
t
)
,
i
=
2
,
3
.
Proposition
1.
If
P
=
{
P
1
,
P
2
,
P
3
}
is
strongly
compatible
with
U
,
then
there
exist
V
i
:
∆
→
B
(
X
)
,
such
that
V
i
(
t,
s
)
is
an
isomorphism
from
Range
P
i
(
t
)
to
Range
P
i
(
s
)
,
i
=
2
,
3
,
with
the
following
properties:
(
v
1
)
U
(
t,
s
)
V
i
(
t,
s
)
P
i
(
t
)
=
P
i
(
t
)
;
C.L.
Mihit
¸,
G.
Mot
¸
149
(
v
2
)
V
i
(
t,
s
)
U
(
t,
s
)
P
i
(
s
)
=
P
i
(
s
)
;
(
v
3
)
V
i
(
t,
s
)
P
i
(
t
)
=
P
i
(
s
)
V
i
(
t,
s
)
P
i
(
t
);
(
v
4
)
P
i
(
t
)
=
V
i
(
t,
t
)
P
i
(
t
)
=
P
i
(
t
)
V
i
(
t,
t
)
P
i
(
t
);
(
v
5
)
V
i
(
t,
t
0
)
P
i
(
t
)
=
V
i
(
s,
t
0
)
V
i
(
t,
s
)
P
i
(
t
)
,
for
all
(
t,
s,
t
0
)
∈
T.
Proof.
See
[
18
].
In
what
follows,
we
will
denote
by
U
i
(
t,
s
)
=
U
(
t,
s
)
P
i
(
s
)
,
for
all
(
t,
s
)
∈
∆
,
i
=
1
,
3
,
respectively
V
j
j
(
t,
s
)
=
V
j
(
t,
s
)
P
j
(
t
)
,
for
all
(
t,
s
)
∈
∆
,
j
=
2
,
3
.
3
Uniform
polynomial
trisplitting
with
compatible
families
of
projections
In
this
section,
we
prove
the
equivalence
between
the
uniform
polynomial
trisplitting
in
continuous-time
and
the
uniform
polynomial
trisplitting
in
discrete-time,
using
the
uniform
polynomial
growth
property
and
compatible
families
of
projections.
Next,
we
give
Przyluski-Rolewicz
type
criteria
for
the
uniform
polynomial
trisplitting
and
as
consequences,
we
obtain
the
results
for
the
uniform
polynomial
trichotomy.
We
consider
U
:
∆
→
B
(
X
)
an
evolution
operator
and
P
=
{
P
1
,
P
2
,
P
3
}
a
compatible
family
of
projections
with
U.
Definition
5.
We
say
that
the
pair
(
U,
P
)
has
an
uniform
polynomial
trisplitting
if
there
exist
some
real
constants
N
≥
1
and
α
<
β,
γ
<
δ
such
that
the
following
relations
are
satisfied:
(
ups
1
)
(
s
+
1)
α
||
U
1
(
t,
t
0
)
x
0
||
≤
N
(
t
+
1)
α
||
U
1
(
s,
t
0
)
x
0
||
;
(
ups
2
)
(
t
+
1)
β
||
U
2
(
s,
t
0
)
x
0
||
≤
N
(
s
+
1)
β
||
U
2
(
t,
t
0
)
x
0
||
;
(
ups
3
)
(
t
+
1)
γ
||
U
3
(
t,
t
0
)
x
0
||
≤
N
(
s
+
1)
γ
||
U
3
(
s,
t
0
)
x
0
||
;
(
ups
4
)
(
s
+
1)
δ
||
U
3
(
s,
t
0
)
x
0
||
≤
N
(
t
+
1)
δ
||
U
3
(
t,
t
0
)
x
0
||
,
for
all
(
t,
s,
t
0
,
x
0
)
∈
T
×
X.
On
uniform
polynomial
trisplitting
of
evolutions
operators
150
As
a
particular
case,
if
α
<
0
<
β,
γ
<
0
<
δ
,
then
we
say
that
(
U,
P
)
is
uniformly
polynomially
trichotomic
.
Remark
1.
The
pair
(
U,
P
)
is
uniformly
polynomially
trichotomic
if
and
only
if
there
exist
some
constants
N
≥
1
,
ν
1
,
ν
2
>
0
such
that:
(
upt
1
)
(
t
+
1)
ν
1
||
U
1
(
t,
t
0
)
x
0
||
≤
N
(
s
+
1)
ν
1
||
U
1
(
s,
t
0
)
x
0
||
;
(
upt
2
)
(
t
+
1)
ν
1
||
U
2
(
s,
t
0
)
x
0
||
≤
N
(
s
+
1)
ν
1
||
U
2
(
t,
t
0
)
x
0
||
;
(
upt
3
)
(
s
+
1)
ν
2
||
U
3
(
t,
t
0
)
x
0
||
≤
N
(
t
+
1)
ν
2
||
U
3
(
s,
t
0
)
x
0
||
;
(
upt
4
)
(
s
+
1)
ν
2
||
U
3
(
s,
t
0
)
x
0
||
≤
N
(
t
+
1)
ν
2
||
U
3
(
t,
t
0
)
x
0
||
,
for
all
(
t,
s,
t
0
,
x
0
)
∈
T
×
X.
In
what
follows,
we
show
that
the
concepts
of
uniform
polynomial
tri-
chotomy
and
uniform
polynomial
trisplitting
are
distinct,
more
precisely
uniform
polynomial
trichotomy
implies
uniform
polynomial
trisplitting,
but,
for
all
that,
the
converse
implication
is
false.
Example
1.
On
X
=
R
3
,
endowed
with
the
norm
||
(
x
1
,
x
2
,
x
3
)
||
=
|
x
1
|
+
|
x
2
|
+
|
x
3
|
,
x
=
(
x
1
,
x
2
,
x
3
)
∈
X,
we
consider
the
evolution
operator
U
:
∆
→
B
(
X
)
,
defined
by
U
(
t,
t
0
)
x
=
t
+
1
t
0
+
1
−
e
2
x
1
,
t
+
1
t
0
+
1
−
e
x
2
,
t
0
+
1
t
+
1
e
x
3
and
P
i
:
R
+
→
B
(
X
)
,
i
=
1
,
3
the
canonical
families
of
projections.
We
immediately
see
that
(
U,
P
)
has
a
uniform
polynomial
trisplitting
with
N
=
1
,
α
=
−
e
2
,
β
=
−
e,
γ
=
e
and
δ
=
2
e.
Let
us
suppose
that
(
U,
P
)
is
uniformly
polynomially
trichotomic.
It
follows
that
there
are
N
≥
1
,
ν
1
>
0
with
(
t
+1)
ν
1
||
U
2
(
s,
t
0
)
x
0
||
≤
N
(
s
+1)
ν
1
||
U
2
(
t,
t
0
)
x
0
||
,
for
all
(
t,
s,
t
0
,
x
0
)
∈
T
×
X.
Thus,
we
obtain
t
+
1
s
+
1
ν
1
+
e
≤
N,
for
all
(
t,
s
)
∈
∆
and
for
s
=
e
−
1,
t
→
+
∞
,
it
yields
a
contradiction.
In
conclusion,
(
U,
P
)
is
not
uniformly
polynomially
trichotomic.
C.L.
Mihit
¸,
G.
Mot
¸
151
Definition
6.
We
say
that
(
U,
P
)
has
uniform
polynomial
growth
if
there
exist
M
≥
1
,
ω
1
,
ω
2
>
0
with:
(
upg
1
)
(
s
+
1)
ω
1
||
U
1
(
t,
t
0
)
x
0
||
≤
M
(
t
+
1)
ω
1
||
U
1
(
s,
t
0
)
x
0
||
;
(
upg
2
)
(
s
+
1)
ω
1
||
U
2
(
s,
t
0
)
x
0
||
≤
M
(
t
+
1)
ω
1
||
U
2
(
t,
t
0
)
x
0
||
;
(
upg
3
)
(
s
+
1)
ω
2
||
U
3
(
t,
t
0
)
x
0
||
≤
M
(
t
+
1)
ω
2
||
U
3
(
s,
t
0
)
x
0
||
;
(
upg
4
)
(
s
+
1)
ω
2
||
U
3
(
s,
t
0
)
x
0
||
≤
M
(
t
+
1)
ω
2
||
U
3
(
t,
t
0
)
x
0
||
,
for
all
(
t,
s,
t
0
,
x
0
)
∈
T
×
X.
Remark
2.
If
(
U,
P
)
has
a
uniform
polynomial
trisplitting,
then
(
U,
P
)
admits
uniform
polynomial
growth.
Theorem
1.
The
pair
(
U,
P
)
admits
a
uniform
polynomial
trisplitting
if
and
only
if
(
U,
P
)
has
uniform
polynomial
growth
and
there
exist
some
real
constants
S
≥
1
,
a
<
b,
c
<
d
such
that:
(
dups
1
)
(
n
+
1)
a
||
U
1
(
m,
p
)
x
0
||
≤
S
(
m
+
1)
a
||
U
1
(
n,
p
)
x
0
||
;
(
dups
2
)
(
m
+
1)
b
||
U
2
(
n,
p
)
x
0
||
≤
S
(
n
+
1)
b
||
U
2
(
m,
p
)
x
0
||
;
(
dups
3
)
(
m
+
1)
c
||
U
3
(
m,
p
)
x
0
||
≤
S
(
n
+
1)
c
||
U
3
(
n,
p
)
x
0
||
;
(
dups
4
)
(
n
+
1)
d
||
U
3
(
n,
p
)
x
0
||
≤
S
(
m
+
1)
d
||
U
3
(
m,
p
)
x
0
||
,
for
all
(
m,
n,
p,
x
0
)
∈
T
d
×
X.
Proof.
Necessity.
It
is
obvious.
Sufficiency.
We
prove
that
the
inequalities
(
ups
i
)
,
i
=
1
,
4
from
Definition
5
are
satisfied.
(
ups
1
)
Let
(
t,
s,
t
0
)
∈
T,
x
0
∈
X
and
we
denote
m
=
[
t
]
,
n
=
[
s
]
,
p
=
[
t
0
]
.
Case
1.
If
m
≥
n
+
1
≥
p
+
1
,
then
we
have
||
U
1
(
t,
t
0
)
x
0
||
≤
M
t
+
1
m
+
1
ω
1
||
U
1
(
m,
n
+
1)
U
1
(
n
+
1
,
t
0
)
x
0
||
≤
MS
2
ω
1
m
+
1
n
+
2
a
||
U
1
(
n
+
1
,
s
)
U
1
(
s,
t
0
)
x
0
||
≤
M
2
S
2
ω
1
m
+
1
n
+
2
a
n
+
2
s
+
1
ω
1
||
U
1
(
s,
t
0
)
x
0
||
≤
M
2
S
2
2
ω
1
m
+
1
n
+
2
a
||
U
1
(
s,
t
0
)
x
0
||
On
uniform
polynomial
trisplitting
of
evolutions
operators
152
≤
M
2
S
2
2
ω
1
max
{
1
,
2
−
2
a
}
t
+
1
s
+
1
a
||
U
1
(
s,
t
0
)
x
0
||
≤
N
t
+
1
s
+
1
a
||
U
1
(
s,
t
0
)
x
0
||
,
where
N
=
M
2
S
2
2(
ω
1
+
ω
2
)
max
{
1
,
2
−
2
a
,
2
2
b
,
2
2
c
,
2
−
2
d
}
.
Case
2.
For
p
+
1
≤
m
<
n
+
1
,
or
m
<
p
+
1
≤
n
+
1
,
we
obtain
||
U
1
(
t,
t
0
)
x
0
||
≤
M
t
+
1
s
+
1
ω
1
||
U
1
(
s,
t
0
)
x
0
||
≤
M
2
2
ω
1
max
{
1
,
2
−
2
a
}
t
+
1
s
+
1
a
||
U
1
(
s,
t
0
)
x
0
||
≤
N
t
+
1
s
+
1
a
||
U
1
(
s,
t
0
)
x
0
||
.
It
yields
that
(
s
+
1)
a
||
U
1
(
t,
t
0
)
x
0
||
≤
N
(
t
+
1)
a
||
U
1
(
s,
t
0
)
x
0
||
,
for
all
(
t,
s,
t
0
,
x
0
)
∈
T
×
X.
(
ups
2
)
We
consider
(
t,
s,
t
0
,
x
0
)
∈
T
×
X
and
m
=
[
t
]
,
n
=
[
s
]
,
p
=
[
t
0
]
.
Case
1.
For
m
≥
n
+
1
≥
p
+
1
,
we
deduce
||
U
2
(
t,
t
0
)
x
0
||
≥
1
M
m
+
1
t
+
1
ω
1
||
U
2
(
m,
n
+
1)
U
2
(
n
+
1
,
t
0
)
x
0
||
≥
1
MS
2
−
ω
1
m
+
1
n
+
2
b
||
U
2
(
n
+
1
,
t
0
)
x
0
||
≥
1
M
2
S
2
−
ω
1
m
+
1
n
+
2
b
n
+
2
s
+
1
−
ω
1
||
U
2
(
s,
t
0
)
x
0
||
≥
1
M
2
S
2
−
2
ω
1
m
+
1
n
+
2
b
||
U
2
(
s,
t
0
)
x
0
||
≥
1
M
2
S
2
−
2
ω
1
1
max
{
1
,
2
2
b
}
t
+
1
s
+
1
b
||
U
2
(
s,
t
0
)
x
0
||
,
which
implies
that
(
t
+
1)
b
||
U
2
(
s,
t
0
)
x
0
||
≤
N
(
s
+
1)
b
||
U
2
(
t,
t
0
)
x
0
||
.
Case
2.
If
p
+
1
≤
m
<
n
+
1
,
or
m
<
p
+
1
≤
n
+
1
,
then
||
U
2
(
t,
t
0
)
x
0
||
≥
1
M
t
+
1
s
+
1
−
ω
1
||
U
2
(
s,
t
0
)
x
0
||
C.L.
Mihit
¸,
G.
Mot
¸
153
≥
1
M
2
−
2
ω
1
1
max
{
1
,
2
2
b
}
t
+
1
s
+
1
b
||
U
2
(
s,
t
0
)
x
0
||
.
Thus,
(
t
+
1)
b
||
U
2
(
s,
t
0
)
x
0
||
≤
N
(
s
+
1)
b
||
U
2
(
t,
t
0
)
x
0
||
,
for
all
(
t,
s,
t
0
,
x
0
)
∈
T
×
X.
(
ups
3
)
Let
(
t,
s,
t
0
)
∈
T,
x
0
∈
X
and
we
denote
m
=
[
t
]
,
n
=
[
s
]
,
p
=
[
t
0
]
.
Case
1.
If
m
≥
n
+
1
≥
p
+
1
,
then
||
U
3
(
t,
t
0
)
x
0
||
≤
M
t
+
1
m
+
1
ω
2
||
U
3
(
m,
n
+
1)
U
3
(
n
+
1
,
t
0
)
x
0
||
≤
M
2
S
2
ω
2
n
+
2
m
+
1
c
n
+
2
s
+
1
ω
2
||
U
3
(
s,
t
0
)
x
0
||
≤
M
2
S
2
2
ω
2
max
{
1
,
2
2
c
}
s
+
1
t
+
1
c
||
U
3
(
s,
t
0
)
x
0
||
≤
N
s
+
1
t
+
1
c
||
U
3
(
s,
t
0
)
x
0
||
.
Case
2.
For
p
+
1
≤
m
<
n
+
1
,
or
m
<
p
+
1
≤
n
+
1
,
it
follows
that
||
U
3
(
t,
t
0
)
x
0
||
≤
M
t
+
1
s
+
1
ω
2
||
U
3
(
s,
t
0
)
x
0
||
≤
M
2
2
ω
2
max
{
1
,
2
2
c
}
s
+
1
t
+
1
c
||
U
3
(
s,
t
0
)
x
0
||
≤
N
s
+
1
t
+
1
c
||
U
3
(
s,
t
0
)
x
0
||
.
We
obtain
(
t
+
1)
c
||
U
3
(
t,
t
0
)
x
0
||
≤
N
(
s
+
1)
c
||
U
3
(
s,
t
0
)
x
0
||
,
for
all
(
t,
s,
t
0
,
x
0
)
∈
T
×
X.
(
ups
4
)
Based
on
an
analogous
argument
as
in
(
ups
2
)
,
we
deduce
(
s
+
1)
d
||
U
3
(
s,
t
0
)
x
0
||
≤
M
2
S
2
2
ω
2
max
{
1
,
2
−
2
d
}
(
t
+
1)
d
||
U
3
(
t,
t
0
)
x
0
||
and
then
(
s
+
1)
d
||
U
3
(
s,
t
0
)
x
0
||
≤
N
(
t
+
1)
d
||
U
3
(
t,
t
0
)
x
0
||
,
for
all
(
t,
s,
t
0
,
x
0
)
∈
T
×
X.
We
conclude
that
(
U,
P
)
has
a
uniform
polynomial
trisplitting.
On
uniform
polynomial
trisplitting
of
evolutions
operators
154
Corollary
1.
The
pair
(
U,
P
)
is
uniformly
polynomially
trichotomic
if
and
only
if
(
U,
P
)
has
uniform
polynomial
growth
and
there
are
some
constants
T
≥
1
,
a
<
0
<
b,
c
<
0
<
d
with:
(
dupt
1
)
(
n
+
1)
a
||
U
1
(
m,
p
)
x
0
||
≤
T
(
m
+
1)
a
||
U
1
(
n,
p
)
x
0
||
;
(
dupt
2
)
(
m
+
1)
b
||
U
2
(
n,
p
)
x
0
||
≤
T
(
n
+
1)
b
||
U
2
(
m,
p
)
x
0
||
;
(
dupt
3
)
(
m
+
1)
c
||
U
3
(
m,
p
)
x
0
||
≤
T
(
n
+
1)
c
||
U
3
(
n,
p
)
x
0
||
;
(
dupt
4
)
(
n
+
1)
d
||
U
3
(
n,
p
)
x
0
||
≤
T
(
m
+
1)
d
||
U
3
(
m,
p
)
x
0
||
,
for
all
(
m,
n,
p,
x
0
)
∈
T
d
×
X.
Proof.
It
is
immediate
from
Theorem
1
.
A
necessary
condition
for
the
uniform
polynomial
trisplitting
is
given
by:
Theorem
2.
If
the
pair
(
U,
P
)
has
a
uniform
polynomial
trisplitting,
then
there
exist
some
real
constants
D
≥
1
,
d
1
<
d
2
,
d
3
<
d
4
such
that:
(
Dps
1
)
+
∞
j
=
n
(
j
+
1)
−
d
1
−
1
||
U
1
(
j,
p
)
x
0
||
≤
D
(
n
+
1)
−
d
1
||
U
1
(
n,
p
)
x
0
||
;
(
Dps
2
)
m
j
=
n
(
j
+
1)
−
d
2
||
U
2
(
j,
p
)
x
0
||
≤
D
(
m
+
1)
1
−
d
2
||
U
2
(
m,
p
)
x
0
||
;
(
Dps
3
)
+
∞
j
=
n
(
j
+
1)
d
3
−
1
||
U
3
(
j,
p
)
x
0
||
≤
D
(
n
+
1)
d
3
||
U
3
(
n,
p
)
x
0
||
;
(
Dps
4
)
m
j
=
n
(
j
+
1)
d
4
||
U
3
(
j,
p
)
x
0
||
≤
D
(
m
+
1)
d
4
+1
||
U
3
(
m,
p
)
x
0
||
,
for
all
(
m,
n,
p,
x
0
)
∈
T
d
×
X.
Proof.
Let
α
<
d
1
<
d
2
<
β,
d
3
<
γ
<
δ
<
d
4
and
D
=
N
max
d
1
−
α
+
1
d
1
−
α
,
γ
−
d
3
+
1
γ
−
d
3
.
Thus,
for
all
(
m,
n,
p,
x
0
)
∈
T
d
×
X
the
following
relations
hold:
(
Dps
1
)
+
∞
j
=
n
(
j
+
1)
−
d
1
−
1
||
U
1
(
j,
p
)
x
0
||
≤
N
(
n
+
1)
−
α
+
∞
j
=
n
(
j
+
1)
α
−
d
1
−
1
||
U
1
(
n,
p
)
x
0
||
C.L.
Mihit
¸,
G.
Mot
¸
155
≤
N
(
n
+
1)
−
α
(
n
+
1)
α
−
d
1
−
1
+
1
d
1
−
α
(
n
+
1)
α
−
d
1
||
U
1
(
n,
p
)
x
0
||
≤
N
·
d
1
−
α
+
1
d
1
−
α
(
n
+
1)
−
d
1
||
U
1
(
n,
p
)
x
0
||
≤
D
(
n
+
1)
−
d
1
||
U
1
(
n,
p
)
x
0
||
;
(
Dps
2
)
m
j
=
n
(
j
+
1)
−
d
2
||
U
2
(
j,
p
)
x
0
||
≤
N
(
m
+
1)
−
β
m
j
=
n
(
j
+
1)
β
−
d
2
||
U
2
(
m,
p
)
x
0
||
≤
N
(
m
+
1)
1
−
d
2
||
U
2
(
m,
p
)
x
0
||
≤
D
(
m
+
1)
1
−
d
2
||
U
2
(
m,
p
)
x
0
||
;
(
Dps
3
)
+
∞
j
=
n
(
j
+
1)
d
3
−
1
||
U
3
(
j,
p
)
x
0
||
≤
N
(
n
+
1)
γ
+
∞
j
=
n
(
j
+
1)
d
3
−
γ
−
1
||
U
3
(
n,
p
)
x
0
||
≤
N
(
n
+
1)
γ
(
n
+
1)
d
3
−
γ
−
1
+
1
γ
−
d
3
(
n
+
1)
d
3
−
γ
||
U
3
(
n,
p
)
x
0
||
≤
D
(
n
+
1)
d
3
||
U
3
(
n,
p
)
x
0
||
;
(
Dps
4
)
m
j
=
n
(
j
+
1)
d
4
||
U
3
(
j,
p
)
x
0
||
≤
N
(
m
+
1)
δ
m
j
=
n
(
j
+
1)
d
4
−
δ
||
U
3
(
m,
p
)
x
0
||
≤
N
(
m
+
1)
d
4
+1
||
U
3
(
m,
p
)
x
0
||
≤
D
(
m
+
1)
d
4
+1
||
U
3
(
m,
p
)
x
0
||
.
So,
the
inequalities
are
verified.
Corollary
2.
If
(
U,
P
)
is
uniformly
polynomially
trichotomic,
then
there
are
some
constants
D
≥
1
,
d
1
<
0
<
d
2
,
d
3
<
0
<
d
4
with:
(
Dpt
1
)
+
∞
j
=
n
(
j
+
1)
−
d
1
−
1
||
U
1
(
j,
p
)
x
0
||
≤
D
(
n
+
1)
−
d
1
||
U
1
(
n,
p
)
x
0
||
;
(
Dpt
2
)
m
j
=
n
(
j
+
1)
−
d
2
||
U
2
(
j,
p
)
x
0
||
≤
D
(
m
+
1)
1
−
d
2
||
U
2
(
m,
p
)
x
0
||
;
(
Dpt
3
)
+
∞
j
=
n
(
j
+
1)
d
3
−
1
||
U
3
(
j,
p
)
x
0
||
≤
D
(
n
+
1)
d
3
||
U
3
(
n,
p
)
x
0
||
;
(
Dpt
4
)
m
j
=
n
(
j
+
1)
d
4
||
U
3
(
j,
p
)
x
0
||
≤
D
(
m
+
1)
d
4
+1
||
U
3
(
m,
p
)
x
0
||
,
On
uniform
polynomial
trisplitting
of
evolutions
operators
156
for
all
(
m,
n,
p,
x
0
)
∈
T
d
×
X.
Proof.
It
is
a
consequence
of
Theorem
2
.
In
what
follows,
we
present
a
sufficient
condition
for
the
uniform
poly-
nomial
trisplitting.
Theorem
3.
Let
(
U,
P
)
be
a
pair
with
uniform
polynomial
growth.
If
there
are
some
real
constants
D
≥
1
,
d
1
<
d
2
,
d
3
<
d
4
such
that:
(
Dps
1
)
+
∞
j
=
n
(
j
+
1)
−
d
1
||
U
1
(
j,
p
)
x
0
||
≤
D
(
n
+
1)
−
d
1
||
U
1
(
n,
p
)
x
0
||
;
(
Dps
2
)
m
j
=
n
(
j
+
1)
−
d
2
||
U
2
(
j,
p
)
x
0
||
≤
D
(
m
+
1)
−
d
2
||
U
2
(
m,
p
)
x
0
||
;
(
Dps
3
)
+
∞
j
=
n
(
j
+
1)
d
3
||
U
3
(
j,
p
)
x
0
||
≤
D
(
n
+
1)
d
3
||
U
3
(
n,
p
)
x
0
||
;
(
Dps
4
)
m
j
=
n
(
j
+
1)
d
4
||
U
3
(
j,
p
)
x
0
||
≤
D
(
m
+
1)
d
4
||
U
3
(
m,
p
)
x
0
||
,
for
all
(
m,
n,
p,
x
0
)
∈
T
d
×
X,
then
(
U,
P
)
admits
a
uniform
polynomial
trisplitting.
Proof.
Considering
j
=
m
in
(
Dps
1
),
respectively
(
Dps
3
),
we
deduce:
(
n
+
1)
d
1
||
U
1
(
m,
p
)
x
0
||
≤
D
(
m
+
1)
d
1
||
U
1
(
n,
p
)
x
0
||
,
respectively
(
m
+
1)
d
3
||
U
3
(
m,
p
)
x
0
||
≤
D
(
n
+
1)
d
3
||
U
3
(
n,
p
)
x
0
||
,
for
all
(
m,
n,
p,
x
0
)
∈
T
d
×
X.
Similarly,
taking
j
=
n
in
(
Dps
2
),
respectively
(
Dps
4
),
we
obtain:
(
m
+
1)
d
2
||
U
2
(
n,
p
)
x
0
||
≤
D
(
n
+
1)
d
2
||
U
2
(
m,
p
)
x
0
||
,
respectively,
(
n
+
1)
d
4
||
U
3
(
n,
p
)
x
0
||
≤
D
(
m
+
1)
d
4
||
U
3
(
m,
p
)
x
0
||
,
for
all
(
m,
n,
p,
x
0
)
∈
T
d
×
X.
From
Theorem
1
,
we
conclude
that
(
U,
P
)
has
a
uniform
polynomial
trisplit-
ting.
C.L.
Mihit
¸,
G.
Mot
¸
157
In
particular,
we
obtain:
Corollary
3.
Let
(
U,
P
)
be
a
pair
with
uniform
polynomial
growth.
If
there
exist
D
≥
1
,
d
1
<
0
<
d
2
,
d
3
<
0
<
d
4
with:
(
Dpt
1
)
+
∞
j
=
n
(
j
+
1)
−
d
1
||
U
1
(
j,
p
)
x
0
||
≤
D
(
n
+
1)
−
d
1
||
U
1
(
n,
p
)
x
0
||
;
(
Dpt
2
)
m
j
=
n
(
j
+
1)
−
d
2
||
U
2
(
j,
p
)
x
0
||
≤
D
(
m
+
1)
−
d
2
||
U
2
(
m,
p
)
x
0
||
;
(
Dpt
3
)
+
∞
j
=
n
(
j
+
1)
d
3
||
U
3
(
j,
p
)
x
0
||
≤
D
(
n
+
1)
d
3
||
U
3
(
n,
p
)
x
0
||
;
(
Dpt
4
)
m
j
=
n
(
j
+
1)
d
4
||
U
3
(
j,
p
)
x
0
||
≤
D
(
m
+
1)
d
4
||
U
3
(
m,
p
)
x
0
||
,
for
all
(
m,
n,
p,
x
0
)
∈
T
d
×
X,
then
(
U,
P
)
is
uniformly
polynomially
tri-
chotomic.
4
Uniform
polynomial
trisplitting
with
strongly
compatible
families
of
projections
In
what
follows,
we
demonstrate
the
equivalence
between
the
uniform
poly-
nomial
trisplitting
in
continuous-time
and
the
uniform
polynomial
trisplit-
ting
in
discrete-time,
using
the
uniform
polynomial
growth,
strongly
com-
patible
families
of
projections
and
the
properties
from
Proposition
1
.
First,
we
prove
two
auxiliary
results
for
the
uniform
polynomial
trisplitting,
respec-
tively
uniform
polynomial
growth
in
terms
of
strongly
compatible
families
of
projections
(Proposition
2
,
respectively
Proposition
3
).
Next,
we
obtain
Przyluski-Rolewicz
type
conditions
for
the
uniform
polynomial
trisplitting
and
in
particular,
Przyluski-Rolewicz
type
criteria
for
the
uniform
polyno-
mial
trichotomy.
Let
U
:
∆
→
B
(
X
)
be
an
evolution
operator
and
let
P
=
{
P
1
,
P
2
,
P
3
}
be
a
strongly
compatible
family
of
projections
with
U.
A
first
characterization
for
the
uniform
polynomial
trisplitting,
using
strongly
compatible
families
of
projections
is
given
by:
On
uniform
polynomial
trisplitting
of
evolutions
operators
158
Proposition
2.
The
pair
(
U,
P
)
admits
a
uniform
polynomial
trisplitting
if
and
only
if
there
are
some
real
constants
N
≥
1
,
α
<
β
and
γ
<
δ
such
that:
(
ups
1
)
(
s
+
1)
α
||
U
1
(
t,
t
0
)
x
0
||
≤
N
(
t
+
1)
α
||
U
1
(
s,
t
0
)
x
0
||
;
(
sups
2
)
(
s
+
1)
β
||
V
2
2
(
t,
t
0
)
x
0
||
≤
N
(
t
0
+
1)
β
||
V
2
2
(
t,
s
)
x
0
||
;
(
ups
3
)
(
t
+
1)
γ
||
U
3
(
t,
t
0
)
x
0
||
≤
N
(
s
+
1)
γ
||
U
3
(
s,
t
0
)
x
0
||
;
(
sups
4
)
(
t
0
+
1)
δ
||
V
3
3
(
t,
t
0
)
x
0
||
≤
N
(
s
+
1)
δ
||
V
3
3
(
t,
s
)
x
0
||
,
for
all
(
t,
s,
t
0
,
x
0
)
∈
T
×
X.
Proof.
We
demonstrate
(
sups
2
)
⇔
(
ups
2
)
and
(
sups
4
)
⇔
(
ups
4
).
For
the
implication
(
sups
2
)
⇒
(
ups
2
),
we
observe
that:
(
t
+
1)
β
||
U
2
(
s,
t
0
)
x
0
||
=
(
t
+
1)
β
||
V
2
2
(
t,
s
)
V
2
2
(
t,
t
)
U
2
(
t,
s
)
U
2
(
s,
t
0
)
x
0
||
≤
N
(
s
+
1)
β
||
V
2
2
(
t,
t
)
U
2
(
t,
t
0
)
x
0
||
=
N
(
s
+
1)
β
||
U
2
(
t,
t
0
)
x
0
||
,
for
all
(
t,
s,
t
0
,
x
0
)
∈
T
×
X.
For
the
converse
implication
(
ups
2
)
⇒
(
sups
2
),
we
have:
(
s
+
1)
β
||
V
2
2
(
t,
t
0
)
x
0
||
=
(
s
+
1)
β
||
U
2
(
t,
t
0
)
V
2
2
(
t,
t
0
)
x
0
||
≤
N
(
t
0
+
1)
β
||
U
2
(
s,
t
0
)
V
2
2
(
s,
t
0
)
V
2
2
(
t,
s
)
x
0
||
=
N
(
t
0
+
1)
β
||
V
2
2
(
t,
s
)
x
0
||
,
for
all
(
t,
s,
t
0
,
x
0
)
∈
T
×
X.
Similarly,
it
yields
that
(
sups
4
)
⇔
(
ups
4
).
Concerning
the
uniform
polynomial
growth
with
strongly
compatible
families
of
projections,
we
obtain:
Proposition
3.
The
pair
(
U,
P
)
admits
a
uniform
polynomial
growth
if
and
only
if
there
exist
M
≥
1
,
ω
1
,
ω
2
>
0
with:
(
upg
1
)
(
s
+
1)
ω
1
||
U
1
(
t,
t
0
)
x
0
||
≤
M
(
t
+
1)
ω
1
||
U
1
(
s,
t
0
)
x
0
||
;
(
supg
2
)
(
t
0
+
1)
ω
1
||
V
2
2
(
t,
t
0
)
x
0
||
≤
M
(
s
+
1)
ω
1
||
V
2
2
(
t,
s
)
x
0
||
;
(
upg
3
)
(
s
+
1)
ω
2
||
U
3
(
t,
t
0
)
x
0
||
≤
M
(
t
+
1)
ω
2
||
U
3
(
s,
t
0
)
x
0
||
;
(
supg
4
)
(
t
0
+
1)
ω
2
||
V
3
3
(
t,
t
0
)
x
0
||
≤
M
(
s
+
1)
ω
2
||
V
3
3
(
t,
s
)
x
0
||
,
for
all
(
t,
s,
t
0
,
x
0
)
∈
T
×
X.
Proof.
It
follows
based
on
an
analogous
technique
as
in
the
proof
of
Propo-
sition
2
.
C.L.
Mihit
¸,
G.
Mot
¸
159
Theorem
4.
The
pair
(
U,
P
)
has
a
uniform
polynomial
trisplitting
if
and
only
if
(
U,
P
)
has
uniform
polynomial
growth
and
there
exist
some
real
con-
stants
S
≥
1
,
a
<
b,
c
<
d
such
that:
(
dups
1
)
(
n
+
1)
a
||
U
1
(
m,
p
)
x
0
||
≤
S
(
m
+
1)
a
||
U
1
(
n,
p
)
x
0
||
;
(
sdups
2
)
(
n
+
1)
b
||
V
2
2
(
m,
p
)
x
0
||
≤
S
(
p
+
1)
b
||
V
2
2
(
m,
n
)
x
0
||
;
(
dups
3
)
(
m
+
1)
c
||
U
3
(
m,
p
)
x
0
||
≤
S
(
n
+
1)
c
||
U
3
(
n,
p
)
x
0
||
;
(
sdups
4
)
(
p
+
1)
d
||
V
3
3
(
m,
p
)
x
0
||
≤
S
(
n
+
1)
d
||
V
3
3
(
m,
n
)
x
0
||
,
for
all
(
m,
n,
p,
x
0
)
∈
T
d
×
X.
Proof.
Necessity.
It
is
obvious
from
Proposition
2
.
Sufficiency.
It
is
sufficient
to
prove
the
implications
(
sdups
2
)
⇒
(
sups
2
)
and
(
sdups
4
)
⇒
(
sups
4
),
where
(
sups
2
)
and
(
sups
4
)
are
the
inequalities
from
Proposition
2
.
(
sdups
2
)
⇒
(
sups
2
)
.
Let
(
t,
s,
t
0
,
x
0
)
∈
T
×
X
and
we
denote
m
=
[
t
]
,
n
=
[
s
]
,
p
=
[
t
0
]
.
Case
1.
If
m
≥
n
≥
p
+
1
,
then
||
V
2
2
(
t,
t
0
)
x
0
||
=
||
V
2
2
(
n,
t
0
)
V
2
2
(
t,
n
)
x
0
||
≤
M
p
+
2
t
0
+
1
ω
1
||
V
2
2
(
n,
p
+
1)
V
2
2
(
t,
n
)
x
0
||
≤
MS
2
ω
1
p
+
2
n
+
1
b
||
V
2
2
(
s,
n
)
V
2
2
(
t,
s
)
x
0
||
≤
M
2
S
2
2
ω
1
p
+
2
n
+
1
b
||
V
2
2
(
t,
s
)
x
0
||
≤
M
2
S
2
2
ω
1
max
{
1
,
2
2
b
}
t
0
+
1
s
+
1
b
||
V
2
2
(
t,
s
)
x
0
||
,
which
implies
that
(
s
+
1)
b
||
V
2
2
(
t,
t
0
)
x
0
||
≤
N
(
t
0
+
1)
b
||
V
2
2
(
t,
s
)
x
0
||
,
where
N
=
M
2
S
2
2(
ω
1
+
ω
2
)
max
{
1
,
2
−
2
a
,
2
2
b
,
2
2
c
,
2
−
2
d
}
.
Case
2.
If
n
≤
m
<
p
+
1
,
we
deduce
||
V
2
2
(
t,
t
0
)
x
0
||
≤
M
s
+
1
t
0
+
1
ω
1
||
V
2
2
(
t,
s
)
x
0
||
On
uniform
polynomial
trisplitting
of
evolutions
operators
160
≤
M
2
2
ω
1
max
{
1
,
2
2
b
}
t
0
+
1
s
+
1
b
||
V
2
2
(
t,
s
)
x
0
||
and
then
(
s
+
1)
b
||
V
2
2
(
t,
t
0
)
x
0
||
≤
N
(
t
0
+
1)
b
||
V
2
2
(
t,
s
)
x
0
||
,
for
all
(
t,
s,
t
0
,
x
0
)
∈
T
×
X.
In
a
similar
manner
we
obtain
(
sdups
4
)
⇒
(
sups
4
)
,
more
precisely,
for
all
(
t,
s,
t
0
,
x
0
)
∈
T
×
X
it
yields
that
(
t
0
+
1)
d
||
V
3
3
(
t,
t
0
)
x
0
||
≤
M
2
S
2
2
ω
2
max
{
1
,
2
−
2
d
}
(
s
+
1)
d
||
V
3
3
(
t,
s
)
x
0
||
and
then
(
t
0
+
1)
d
||
V
3
3
(
t,
t
0
)
x
0
||
≤
N
(
s
+
1)
d
||
V
3
3
(
t,
s
)
x
0
||
.
Using
Proposition
2
,
we
deduce
that
(
U,
P
)
has
a
uniform
polynomial
trisplit-
ting.
Corollary
4.
The
pair
(
U,
P
)
has
a
uniform
polynomial
trichotomy
if
and
only
if
(
U,
P
)
has
uniform
polynomial
growth
and
there
exist
some
constants
T
≥
1
,
a
<
0
<
b,
c
<
0
<
d
with
(
dupt
1
)
(
n
+
1)
a
||
U
1
(
m,
p
)
x
0
||
≤
T
(
m
+
1)
a
||
U
1
(
n,
p
)
x
0
||
;
(
sdupt
2
)
(
n
+
1)
b
||
V
2
2
(
m,
p
)
x
0
||
≤
T
(
p
+
1)
b
||
V
2
2
(
m,
n
)
x
0
||
;
(
dupt
3
)
(
m
+
1)
c
||
U
3
(
m,
p
)
x
0
||
≤
T
(
n
+
1)
c
||
U
3
(
n,
p
)
x
0
||
;
(
sdupt
4
)
(
p
+
1)
d
||
V
3
3
(
m,
p
)
x
0
||
≤
T
(
n
+
1)
d
||
V
3
3
(
m,
n
)
x
0
||
,
for
all
(
m,
n,
p,
x
0
)
∈
T
d
×
X.
Proof.
It
follows
from
Theorem
4
.
Next,
we
give
a
necessary
criterion
for
the
uniform
polynomial
trispli-
tting.
Theorem
5.
If
the
pair
(
U,
P
)
has
a
uniform
polynomial
trisplitting,
then
there
are
some
real
constants
D
≥
1
,
d
1
<
d
2
,
d
3
<
d
4
with:
(
Dps
1
)
+
∞
j
=
n
(
j
+
1)
−
d
1
−
1
||
U
1
(
j,
p
)
x
0
||
≤
D
(
n
+
1)
−
d
1
||
U
1
(
n,
p
)
x
0
||
;
(
sDps
2
)
n
j
=
p
(
j
+
1)
−
d
2
||
V
2
2
(
m,
j
)
x
0
||
≤
D
(
n
+
1)
1
−
d
2
||
V
2
2
(
m,
n
)
x
0
||
;
C.L.
Mihit
¸,
G.
Mot
¸
161
(
Dps
3
)
+
∞
j
=
n
(
j
+
1)
d
3
−
1
||
U
3
(
j,
p
)
x
0
||
≤
D
(
n
+
1)
d
3
||
U
3
(
n,
p
)
x
0
||
;
(
sDps
4
)
n
j
=
p
(
j
+
1)
d
4
||
V
3
3
(
m,
j
)
x
0
||
≤
D
(
n
+
1)
d
4
+1
||
V
3
3
(
m,
n
)
x
0
||
,
for
all
(
m,
n,
p,
x
0
)
∈
T
d
×
X.
Proof.
Let
α
<
d
1
<
d
2
<
β,
d
3
<
γ
<
δ
<
d
4
and
D
=
N
max
d
1
−
α
+
1
d
1
−
α
,
γ
−
d
3
+
1
γ
−
d
3
.
As
in
Theorem
2
,
we
obtain
that
(
Dps
1
)
and
(
Dps
3
)
hold.
For
all
(
m,
n,
p,
x
0
)
∈
T
d
×
X
,
we
deduce:
(
sDps
2
)
n
j
=
p
(
j
+
1)
−
d
2
||
V
2
2
(
m,
j
)
x
0
||
≤
N
(
n
+
1)
−
β
n
j
=
p
(
j
+
1)
β
−
d
2
||
V
2
2
(
m,
n
)
x
0
||
≤
N
(
n
+
1)
1
−
d
2
||
V
2
2
(
m,
n
)
x
0
||
≤
D
(
n
+
1)
1
−
d
2
||
V
2
2
(
m,
n
)
x
0
||
;
(
sDps
4
)
n
j
=
p
(
j
+
1)
d
4
||
V
3
3
(
m,
j
)
x
0
||
≤
N
(
n
+
1)
δ
n
j
=
p
(
j
+
1)
d
4
−
δ
||
V
3
3
(
m,
n
)
x
0
||
≤
N
(
n
+
1)
d
4
+1
||
V
3
3
(
m,
n
)
x
0
||
≤
D
(
n
+
1)
d
4
+1
||
V
3
3
(
m,
n
)
x
0
||
.
As
a
consequence,
we
deduce:
Corollary
5.
If
(
U,
P
)
is
uniformly
polynomially
trichotomic,
then
there
exist
some
constants
D
≥
1
,
d
1
<
0
<
d
2
,
d
3
<
0
<
d
4
such
that:
(
Dpt
1
)
+
∞
j
=
n
(
j
+
1)
−
d
1
−
1
||
U
1
(
j,
p
)
x
0
||
≤
D
(
n
+
1)
−
d
1
||
U
1
(
n,
p
)
x
0
||
;
(
sDpt
2
)
n
j
=
p
(
j
+
1)
−
d
2
||
V
2
2
(
m,
j
)
x
0
||
≤
D
(
n
+
1)
1
−
d
2
||
V
2
2
(
m,
n
)
x
0
||
;
On
uniform
polynomial
trisplitting
of
evolutions
operators
162
(
Dpt
3
)
+
∞
j
=
n
(
j
+
1)
d
3
−
1
||
U
3
(
j,
p
)
x
0
||
≤
D
(
n
+
1)
d
3
||
U
3
(
n,
p
)
x
0
||
;
(
sDpt
4
)
n
j
=
p
(
j
+
1)
d
4
||
V
3
3
(
m,
j
)
x
0
||
≤
D
(
n
+
1)
d
4
+1
||
V
3
3
(
m,
n
)
x
0
||
,
for
all
(
m,
n,
p,
x
0
)
∈
T
d
×
X.
Proof.
It
follows
from
Theorem
5
.
A
sufficient
condition
for
the
uniform
polynomial
trisplitting
is
the
fol-
lowing:
Theorem
6.
Let
(
U,
P
)
be
a
pair
with
uniform
polynomial
growth.
If
there
are
some
real
constants
D
≥
1
,
d
1
<
d
2
,
d
3
<
d
4
such
that:
(
Dps
1
)
+
∞
j
=
n
(
j
+
1)
−
d
1
||
U
1
(
j,
p
)
x
0
||
≤
D
(
n
+
1)
−
d
1
||
U
1
(
n,
p
)
x
0
||
;
(
sDps
2
)
n
j
=
p
(
j
+
1)
−
d
2
||
V
2
2
(
m,
j
)
x
0
||
≤
D
(
n
+
1)
−
d
2
||
V
2
2
(
m,
n
)
x
0
||
;
(
Dps
3
)
+
∞
j
=
n
(
j
+
1)
d
3
||
U
3
(
j,
p
)
x
0
||
≤
D
(
n
+
1)
d
3
||
U
3
(
n,
p
)
x
0
||
;
(
sDps
4
)
n
j
=
p
(
j
+
1)
d
4
||
V
3
3
(
m,
j
)
x
0
||
≤
D
(
n
+
1)
d
4
||
V
3
3
(
m,
n
)
x
0
||
,
for
all
(
m,
n,
p,
x
0
)
∈
T
d
×
X,
then
(
U,
P
)
has
a
uniform
polynomial
trisplit-
ting.
Proof.
For
j
=
m
in
(
Dps
1
),
respectively
(
Dps
3
),
we
obtain
the
conditions
(
dups
1
),
(
dups
3
)
from
Theorem
4
.
Similarly,
taking
j
=
p
in
(
sDps
2
),
respectively
(
sDps
4
),
we
deduce
that
(
sdups
2
)
and
(
sdups
4
)
from
Theorem
4
hold.
Thus,
(
U,
P
)
has
a
uniform
polynomial
trisplitting.
Corollary
6.
Let
(
U,
P
)
be
with
uniform
polynomial
growth.
If
there
are
D
≥
1
,
d
1
<
0
<
d
2
,
d
3
<
0
<
d
4
with:
C.L.
Mihit
¸,
G.
Mot
¸
163
(
Dpt
1
)
+
∞
j
=
n
(
j
+
1)
−
d
1
||
U
1
(
j,
p
)
x
0
||
≤
D
(
n
+
1)
−
d
1
||
U
1
(
n,
p
)
x
0
||
;
(
sDpt
2
)
n
j
=
p
(
j
+
1)
−
d
2
||
V
2
2
(
m,
j
)
x
0
||
≤
D
(
n
+
1)
−
d
2
||
V
2
2
(
m,
n
)
x
0
||
;
(
Dpt
3
)
+
∞
j
=
n
(
j
+
1)
d
3
||
U
3
(
j,
p
)
x
0
||
≤
D
(
n
+
1)
d
3
||
U
3
(
n,
p
)
x
0
||
;
(
sDpt
4
)
n
j
=
p
(
j
+
1)
d
4
||
V
3
3
(
m,
j
)
x
0
||
≤
D
(
n
+
1)
d
4
||
V
3
3
(
m,
n
)
x
0
||
,
for
all
(
m,
n,
p,
x
0
)
∈
T
d
×
X,
then
(
U,
P
)
is
uniformly
polynomially
tri-
chotomic.
Proof.
It
is
a
consequence
of
Theorem
6
.
Acknowledgements.
We
are
grateful
to
the
reviewer
for
the
careful
eval-
uation
and
detailed
reading
of
our
work.
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Alonso,
J.
Hong
and
R.
Obaya,
Exponential
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chotomy
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Comput.
Math.
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38
(1999),
41-
49.
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B.
Aulbach,
Invariant
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Anal.
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B.
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Kalkbrenner,
Exponential
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L.
Barreira,
D.
Dragiˇ
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c
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L.
Barreira
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