Fuglede-Putnam Theorem has been proved for a considerably large number of
class of operators. In this paper by using the spectral theory, we obtain a
theoretical and general framework from which ...Fuglede-Putnam theorem may be
promptly established for many classes of operators.
nema
In this note we consider circular and strongly circular two-sided multiplications
acting on
or on minimal norm ideals of
. We prove that strong circularity of
implies circularity of A or of B. If A ...and B are irreducible and
is acting on some minimal norm ideal different from the Hilbert-Schmidt class, then
is strongly circular if and only if A or B is strongly circular.
The transfer property for the generalized Browder’s theorem both of the tensor product and of the left-right multiplication operator will be characterized in terms of the B-Weyl spectrum inclusion. ...In addition, the isolated points of these two classes of operators will be fully characterized.
In this article the notion of circular operator is extended to the Banach space setting. In particular, this property is considered for elementary operators of lengths one and two acting on minimal ...norm ideals of ℬ(ℋ). Necessary and sufficient conditions for the circularity of generalized derivations and Lüders operators are also obtained.
Let
be a division algebra finite-dimensional over its centre
and let
be an involution on the matrix algebra
. Suppose that
is of the second kind and
. We study elementary operators in the context of ...closed prime algebras with involution and then characterize all sesquilinear functionals on
in terms of reduced traces and a specific involution on elementary operators.
Let
A
,
B
∗
∈
B
(
H
)
be
w-hyponormal operators, and let
d
AB
∈
B
(
B
(
H
)
)
denote either the generalized derivation
δ
AB
(
X
)
=
AX
-
XB
or the length two elementary operator
▵
AB
(
X
)
=
AXB
-
X
.... We prove that
d
AB
has the single–valued extension property, and the quasinilpotent part
H
0
(
d
AB
-
λ
)
of
d
AB
at
λ
∈
iso
σ
(
d
AB
)
equals
(
d
AB
-
λ
)
-
1
(
0
)
. Let
H
(
σ
(
d
AB
)
)
denote the space of functions which are analytic on
σ
(
d
AB
)
, and let
H
c
(
σ
(
d
AB
)
)
denote the space of
f
∈
H
(
σ
(
d
AB
)
)
which are non-constant on every connected component of
σ
(
d
AB
)
. It is proved that, for every
h
∈
H
(
σ
(
d
AB
)
)
and
f
,
g
∈
H
c
(
σ
(
d
AB
)
)
, the complement of the Weyl spectrum
σ
w
(
h
(
d
f
(
A
)
g
(
B
)
)
)
of
h
(
d
f
(
A
)
g
(
B
)
)
in
σ
(
h
(
d
f
(
A
)
g
(
B
)
)
)
consists of isolated points in
σ
(
h
(
d
f
(
A
)
g
(
B
)
)
)
which are eigenvalues of finite multiplicity.
Let X be a finite-dimensional complex vector space. We give an explicit formula for the reflexivity defect of the kernel of an arbitrary elementary operator of length 2, i.e., an elementary operator ...of the form Δ(T)=A1TB1-A2TB2(T∈L(X)) where A1,A2 and B1,B2 are linearly independent.
In this paper, we present various notions and aspects of orthogonality in normed spaces. Characterizations and generalizations of orthogonality are also consid- ered. Results on orthogonality of the ...range and the kernel of elementary operators and the operators implementing them also are given.
Let
R
be a semiprime ring with the maximal right ring of quotients
Q
mr
. An additive map
d
:
R
→
Q
mr
is called a generalized skew derivation if there exists a ring endomorphism
σ
:
R
→
R
and a map
...such that
for all
x
,
y
∈
R
. If
σ
is surjective, we determine the structure of generalized skew derivations for which there exists a finite number of elements
a
i
,
b
i
∈
Q
mr
such that
d
(
x
) =
a
1
xb
1
+ ⋯ +
a
n
xb
n
for all
x
∈
R
.
Sharp upper estimates for the norm of the weighted elementary operator of the form
∑
n
=
1
∞
C
n
Z
n
A
n
⊗
B
n
W
n
D
n
, acting from one symmetrically normed ideal of compact Hilbert space operators ...to another, are given. Particularly, we relate the norm of
∑
n
=
1
∞
C
n
Z
n
A
n
⊗
A
n
∗
W
n
C
n
∗
with norms of
∑
n
=
1
∞
A
n
⊗
A
n
∗
and
∑
n
=
1
∞
C
n
⊗
C
n
∗
on the appropriate domains and co-domains.
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