In this paper, we introduce the class of
m
-complex symmetric operators and study various properties of this class. In particular, we show that if
T
is an
m
-complex symmetric operator, then
T
n
is ...also an
m
-complex symmetric operator for any
n
∈
N
. In addition, we prove that if
T
is an
m
-complex symmetric operator, then
σ
a
(
T
)
,
σ
SVEP
(
T
)
,
σ
β
(
T
)
, and
σ
(
β
)
ϵ
(
T
)
are symmetric about the real axis. Finally, we investigate the stability of an
m
-complex symmetric operator under perturbation by nilpotent operators commuting with
T
.
indestructible complex symmetric operators, in the sense that tensoring them with any operator yields a complex symmetric operator. In fact, we prove that this property characterizes nilpotents of ...order two among all nonzero bounded operators. Second, we establish that every nilpotent of order two is unitarily equivalent to a truncated Toeplitz operator.>
Let
B
(
X
)
be the algebra of all bounded linear operators on the Banach space
X, and let
N
(
X
)
be the set of nilpotent operators in
B
(
X
)
. Suppose
ϕ
:
B
(
X
)
→
B
(
X
)
is a surjective map such ...that
A
,
B
∈
B
(
X
)
satisfy
AB
∈
N
(
X
)
if and only if
ϕ
(
A
)
ϕ
(
B
)
∈
N
(
X
)
. If
X is infinite dimensional, then there exists a map
f
:
B
(
X
)
→
C
⧹
{
0
}
such that one of the following holds:
(a)
There is a bijective bounded linear or conjugate-linear operator
S
:
X
→
X
such that
ϕ has the form
A
↦
S
f
(
A
)
A
S
-
1
.
(b)
The space
X is reflexive, and there exists a bijective bounded linear or conjugate-linear operator
S
:
X′
→
X such that
ϕ has the form
A
↦
S
f(
A)
A′
S
−1.
If
X has dimension
n with 3
⩽
n
<
∞, and
B
(
X
)
is identified with the algebra
M
n
of
n
×
n complex matrices, then there exist a map
f
:
M
n
→
C
⧹
{
0
}
, a field automorphism
ξ
:
C
→
C
, and an invertible
S
∈
M
n
such that
ϕ has one of the following forms:
A
=
a
ij
↦
f
(
A
)
S
ξ
(
a
ij
)
S
-
1
or
A
=
a
ij
↦
f
(
A
)
S
ξ
(
a
ij
)
t
S
-
1
,
where
A
t denotes the transpose of
A.
The results are extended to the product of more than two operators and to other types of products on
B
(
X
)
including the Jordan triple product
A
∗
B
=
ABA. Furthermore, the results in the finite dimensional case are used to characterize surjective maps on matrices preserving the spectral radius of products of matrices.
The optimal control of the solutions of the initial-finish problem for Sobolev-type equations is studied. The abstract results obtained in the paper are applied to the linear Hoff model on graphs.
We show that (1) if A is a nonzero quasinilpotent operator with ran A
n
closed for some n ≥ 1, then its numerical range W(A) contains 0 in its interior and has a differentiable boundary, and (2) a ...noncircular elliptic disc can be the numerical range of a nilpotent operator with nilpotency 3 on an infinite-dimensional separable space. (1) is a generalization of the known result for nonzero nilpotent operators, and (2) is in contrast to the finite-dimensional case, where the only elliptic discs which are the numerical ranges of nilpotent finite matrices are the circular ones centred at the origin.
New growth conditions on the Cesàro means of higher order are investigated for Banach space operators with peripheral spectrum reduced to {1}. Certain consequences concerning the powers of such ...operators are derived. The uniform and strong convergence of the differences of consecutive Cesàro means are studied, and several examples are presented. These topics are related to the boundedness and convergence of Cesàro means of higher order, and also to Gelfand.Hille and Esterle-Katznelson-Tzafriri type theorems. In particular, if
V
denotes the classical Volterra operator, then our results provide a simultaneous conceptual proof showing that the operator
I-V
is Cesàro ergodic on
L
p
(0, 1) for 1 ≤
p
< ∞, completing the known cases
p
= 1 and
p
= 2. Even every power of the latter operator is Cesàro ergodic, though the operator itself is not power-bounded if
p
≠ 2. Analogous examples, with respect to uniform ergodicity, are given as well. We also obtain improvements on the general 1939 Lorch theorem, within the above spectral picture.
.
If
denotes the polar decomposition of a bounded linear operator
T
, then the
Aluthge transform
of
T
is defined to be the operator
. In this note we study the relationship between the Aluthge ...transform and the class of complex symmetric operators (
T
is
complex symmetric
if there exists a conjugate-linear, isometric involution
so that
T
=
CT
*
C
). In this note we prove that: (1) the Aluthge transform of a complex symmetric operator is complex symmetric, (2) if
T
is complex symmetric, then
and
are unitarily equivalent, (3) if
T
is complex symmetric, then
if and only if
T
is normal, (4)
if and only if
T
2
= 0, and (5) every operator which satisfies
T
2
= 0 is necessarily complex symmetric.
For any operator
A on a Hilbert space, let
W
(
A
)
,
w
(
A
)
and
w
0
(
A
)
denote its numerical range, numerical radius and the distance from the origin to the boundary of its numerical range, ...respectively. We prove that if
A
n
=
0
, then
w
(
A
)
⩽
(
n
-
1
)
w
0
(
A
)
, and, moreover, if
A attains its numerical radius, then the following are equivalent: (1)
w
(
A
)
=
(
n
-
1
)
w
0
(
A
)
, (2)
A is unitarily equivalent to an operator of the form
aA
n
⊕
A
′
, where
a is a scalar satisfying
|
a
|
=
2
w
0
(
A
)
,
A
n
is the
n-by-
n matrix
0
1
⋯
1
0
⋱
⋮
⋱
1
0
and
A
′
is some other operator, and (3)
W
(
A
)
=
bW
(
A
n
)
for some scalar
b.
.
The present paper deals with operators similar to partial isometries. We get some (necessary and) sufficient conditions for the similarity to (adjoints of) quasinormal partial isometries, or more ...general, to power partial isometries. We illustrate our results on the class of
n
-quasi-isometries, obtaining that a
n
-quasi-isometry is similar to a power partial isometry if and only if the ranges
are closed. In particular if
n
= 2, these conditions ensure the similarity to quasinormal partial isometries of Duggal and Aluthge transforms of 2-quasi-isometries. The case when a
n
-quasi-isometry is a partial isometry is also studied, and a structure theorem for
n
-quasi-isometries which are power partial isometries is given.
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