Ann.
Acad.
Rom.
Sci.
Ser.
Math.
Appl.
ISSN
2066-6594
Vol.
18,
No.
3/2026
LOCAL
ERGODIC
PROPERTIES
FOR
POWERS
OF
OPERATORS
IN
BANACH
SPACES
∗
Laurian
Suciu
†
Dedicated
to
the
memory
of
Professor
Mihail
Megan
for
his
scientific
contributions
and
enduring
enthusiasm
for
mathematics
DOI
10.56082/annalsarscimath.2026.3.129
Abstract
In
this
paper
we
deal
with
the
local
convergence
of
a
sequence
{
T
n
}
in
the
convex
hull
of
the
powers
of
an
operator
T
on
a
Banach
space
X
.
We
refer
to
the
case
when
the
sequence
{
T
n
(
T
−
I
)
}
is
bounded,
and
to
the
subspaces
of
X
on
which
these
two
sequences
converge
to
zero,
respectively.
Also,
we
study
the
behavior
of
f
(
T
n
x
)
or
f
(
T
n
(
T
−
I
)
x
)
for
x
∈
X
and
f
in
the
dual
space
X
∗
of
X
,
in
particular
when
X
is
a
Banach
algebra
and
f
a
character
of
X
.
Some
consequences
concerning
the
Ces`
aro
means
of
higher
order
are
also
derived.
Keywords:
Ces`
aro
mean,
ergodicity,
ascent
of
an
operator.
MSC:
47A35,
47A10,
47A16.
1
Introduction
and
preliminaries
Let
X
,
Y
be
Banach
spaces
and
B
(
X
,
Y
)
be
the
Banach
space
of
all
bounded
linear
operators
T
:
X
→
Y
.
As
usual
denote
B
(
X
)
=
B
(
X
,
X
),
which
is
a
Banach
algebra
with
the
identity
operator
I
∈
B
(
X
).
The
kernel,
the
range
and
the
spectrum
of
T
∈
B
(
X
)
will
be
denoted
by
N
(
T
)
,
R
(
T
)
and
σ
(
T
),
∗
Accepted
for
publication
on
June
17,
2026
†
laurians2002@yahoo.com
,
Lucian
Blaga
University
of
Sibiu,
Department
of
Mathe-
matics
and
Informatics,
Sibiu,
Romania
129
L.
Suciu
135
We
say
that
a
sequence
{
T
n
}
⊂
co
(
T
)
is
(
uniformly
)
regular
if
it
is
bounded
and
there
exists
an
integer
n
0
≥
0
such
that
{
TT
n
−
T
n
+
n
0
}
strongly
(respectively,
uniformly)
converges
to
zero.
In
this
case,
{
T
n
}
is
called
(uni-
formly)
regular
of
order
n
0
.
Clearly,
the
ergodicity
means
regularity
of
order
n
0
=
0.
But
{
T
n
}
can
be
regular
of
an
order
n
0
≥
1
without
being
ergodic.
Ces`
aro
averages
of
order
p
form
a
regular
sequence
of
order
1
(see
relation
(
4
)
below)
but
these
means
are
not
necessary
ergodic
sequences,
in
general.
Other
examples
can
be
provided
as
follows
Example
1.
Let
T
∈
B
(
X
)
be
Ces`
aro
bounded
such
that
{
1
n
T
n
}
does
not
strongly
converge
to
zero.
(Such
operators
are
provided
in
[
6
],
for
instance;
or
on
C
2
can
be
considered
the
Assani’s
example
T
=
−
1
2
0
−
1
).
Let
{
T
n
}
∈
co
(
T
)
given
by
T
n
=
1
2
(
M
n
(
T
)
+
M
n
+1
(
T
))
,
n
∈
N
.
Then
{
T
n
}
is
bounded
and
by
the
relation
(
4
)
we
have
TT
n
−
T
n
+1
=
1
2
(
TM
n
(
T
)
−
M
n
+1
(
T
)
+
TM
n
+1
(
T
)
−
M
n
+2
(
T
))
→
0
,
hence
{
T
n
}
is
regular
of
order
1
.
Clearly,
by
recurrence
we
can
find
such
a
sequence
for
any
order
n
0
>
1
.
On
the
other
hand
one
has
T
n
(
T
−
I
)
=
1
2
T
n
+1
n
+
1
(
T
+
I
)
−
T
n
+2
(
n
+
1)(
n
+
2)
−
2
n
+
3
(
n
+
1)(
n
+
2)
I
.
As
{
M
n
(
T
)
}
is
bounded,
the
second
term
in
the
right
side
uniformly
con-
verges
to
zero.
Thus,
assuming
that
R
(
T
+
I
)
=
X
,
under
the
above
assumption
on
T
it
follows
from
this
relation
that
{
T
n
(
T
−
I
)
}
does
not
strongly
converge
to
zero,
that
is
{
T
n
}
is
not
ergodic.
As
another
case,
we
can
consider
T
an
invertible
isometry
on
X
such
that
1
,
−
1
/
∈
σ
(
T
)
and
T
n
=
1
2
(
T
n
+
T
n
+1
)
,
n
∈
N
.
Here
TT
n
−
T
n
+1
=
0
therefore
{
T
n
}
is
regular
of
order
1,
but
T
n
(
T
−
I
)
=
1
2
T
n
(
T
2
−
I
)
does
not
converge
strongly
to
zero,
hence
it
is
not
ergodic.
Notice
that
in
[
17
]
the
regularity
condition
was
used
to
derive
certain
er-
godic
properties
for
the
Ces`
aro
means
of
higher
order,
as
well
as
to
obtain
two
extensions
of
the
Esterle-Katznelson-Tzafriri
theorem
for
some
sequences
in
the
convex
hull
of
an
operator.
Now
we
present
results
on
local
convergence
for
some
sequences
of
oper-
ators
which
satisfy
local
regularity
conditions.