In this paper, we show that the spectrum, Weyl spectrum, and Browder spectrum are continuous on the set of all 2-quasi-2-isometric operators. Further, we show that k-quasi-2-isometric operator ...satisfies Bishop's property β. Moreover, we prove Weyl type theorems for f(dTS), where dTS denote the generalized derivation or the elementary operator with k-quasi-2-isometric operator entries T and S* and f ∈ H(σ(dTS)), the set of analytic functions which are defined on an open neighborhood of σ(dTS).
Let
H
be a complex Hilbert space and let
B
(
H
)
denote the algebra of all bounded linear operators on
H
. For
A
,
B
∈
B
(
H
)
, the Jordan elementary operator
U
A
,
B
is defined by
U
A
,
B
(
X
)
=
A
...X
B
+
B
X
A
,
∀
X
∈
B
(
H
)
. In this short note, we discuss the norm of
U
A
,
B
. We show that if
dim
H
=
2
and
‖
U
A
,
B
‖
=
‖
A
‖
‖
B
‖
, then either
A
B
∗
or
B
∗
A
is 0. We give some examples of Jordan elementary operators
U
A
,
B
such that
‖
U
A
,
B
‖
=
‖
A
‖
‖
B
‖
but
A
B
∗
≠
0
and
B
∗
A
≠
0
, which answer negatively a question posed by M. Boumazgour in M. Boumazgour, Norm inequalities for sums of two basic elementary operators, J. Math. Anal. Appl. 342 (2008) 386–393.
A Hilbert space operator
A
∈
B
(
H
)
is
p-hyponormal,
A
∈
(
p
-
H
)
, if
|
A
∗
|
2
p
⩽
|
A
|
2
p
; an invertible operator
A
∈
B
(
H
)
is log-hyponormal,
A
∈
(
ℓ
-
H
)
, if
log
(
TT
∗
)
⩽
log
(
T
∗
T
...)
. Let
d
AB
=
δ
AB
or
▵
AB
, where
δ
AB
∈
B
(
B
(
H
)
)
is the generalised derivation
δ
AB
(
X
)
=
AX
-
XB
and
▵
AB
∈
B
(
B
(
H
)
)
is the elementary operator
▵
AB
(
X
)
=
AXB
-
X
. It is proved that if
A
,
B
∗
∈
(
ℓ
-
H
)
∪
(
p
-
H
)
, then, for all complex
λ
,
(
d
AB
-
λ
)
-
1
(
0
)
⊆
(
d
A
∗
B
∗
-
λ
¯
)
-
1
(
0
)
, the ascent of
(
d
AB
-
λ
)
⩽
1
, and
d
AB
satisfies the range-kernel orthogonality inequality
‖
X
‖
⩽
‖
X
-
(
d
AB
-
λ
)
Y
‖
for all
X
∈
(
d
AB
-
λ
)
-
1
(
0
)
and
Y
∈
B
(
H
)
. Furthermore, isolated points of
σ
(
d
AB
)
are simple poles of the resolvent of
d
AB
. A version of the elementary operator
E
(
X
)
=
A
1
XA
2
-
B
1
XB
2
and perturbations of
d
AB
by quasi–nilpotent operators are considered, and Weyl’s theorem is proved for
d
AB
.
Let
H be a Hilbert space and let
A
and
B
be standard ∗-operator algebras on
H. Denote by
A
s
and
B
s
the set of all self-adjoint operators in
A
and
B
, respectively. Assume that
M
:
A
s
→
B
s
and
M
∗
...:
B
s
→
A
s
are surjective maps such that
M
(
A
M
∗
(
B
)
A
)
=
M
(
A
)
B
M
(
A
)
and
M
∗
(
B
M
(
A
)
B
)
=
M
∗
(
B
)
A
M
∗
(
B
)
for every pair
A
∈
A
s
,
B
∈
B
s
. Then there exist an invertible bounded linear or conjugate-linear operator
T
:
H
→
H
and a constant
c
∈
{
−
1
,
1
}
such that
M
(
A
)
=
c
T
A
T
∗
,
A
∈
A
s
, and
M
∗
(
B
)
=
c
T
∗
B
T
,
B
∈
B
s
.
We determine kernels of similarity-preserving bounded linear maps on B(H) and give characterizations for elementary operators of length 1 to be similarity-preserving.
Let
H
H
be an infinite dimensional separable Hilbert space over the complex field. Structure characterizations are given for some elementary operators on
B
(
H
)
B(H)
which preserve point spectrum.
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