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.
Given unital Banach algebras A and B and elements a ∈ A and b ∈ B, the Drazin spectrum of
will be fully characterized, where
is a Banach algebra that is the completion of A ⊗ B with respect to a ...uniform crossnorm. To this end, however, first the isolated points of the spectrum of
need to be characterized. On the other hand, given Banach spaces X and Y and Banach space operators S ∈ L(X) and T ∈ L(Y), using similar arguments the Drazin spectrum of τ
ST
∈ L(L(Y, X)), the elementary operator defined by S and T, will be fully characterized.
Let
D
i
be a strongly double triangle subspace lattice on a Banach space
X
i
, where
i
=
1
,
2
. If operator pair
(
M
,
M
∗
)
is a surjective elementary operator on
Alg
D
1
⊗
Alg
D
2
, then there ...exist closed, densely defined linear operators
T
¯
from the domain
D
(
T
¯
)
of
T
¯
in
X
1
into
X
2
and
S
¯
from the domain
D
(
S
¯
)
of
S
¯
in
X
2
into
X
1
satisfying
A
1
R
(
S
¯
)
⊆
D
(
T
¯
)
and
A
2
R
(
T
¯
)
⊆
D
(
S
¯
)
for all
A
1
∈
Alg
D
1
,
A
2
∈
Alg
D
2
such that
M
(
A
1
)
y
=
T
¯
A
1
S
¯
y
,
y
∈
D
(
S
¯
)
and
M
∗
(
A
2
)
x
=
S
¯
A
2
T
¯
x
,
x
∈
D
(
T
¯
)
.
On weakly central C ∗-algebras Magajna, Bojan
Journal of mathematical analysis and applications,
06/2008, Letnik:
342, Številka:
2
Journal Article
Recenzirano
Odprti dostop
A unital C
∗-algebra
A is weakly central if and only if for every
x
∈
A
there exists a sequence of elementary unital completely positive maps
α
n
on
A such that the sequence
(
α
n
(
x
)
)
converges ...to a central element.
A Hilbert space operator
T
belongs to class A if
|
T
2
|
−
|
T
|
2
≥
0
. Let
d
A
B
denote either
δ
A
B
or
△
A
B
, where
δ
A
B
and
△
A
B
denote the generalized derivation and the elementary operator ...on a Banach space
B
(
H
)
defined by
δ
A
B
X
=
A
X
−
X
B
and
△
A
B
X
=
A
X
B
−
X
respectively. If
A
and
B
∗
are class A operators, we show that
d
A
B
is polaroid and generalized Weyl’s theorem holds for
f
(
d
A
B
)
, generalized
a
-Weyl’s theorem holds for
f
(
(
d
A
B
)
∗
)
for every
f
∈
H
(
σ
(
d
A
B
)
)
and
f
is not constant on each connected component of the open set
U
containing
σ
(
d
A
B
)
, where
H
(
σ
(
d
A
B
)
)
denotes the set of all analytic functions in a neighborhood of
σ
(
d
A
B
)
.
MSC:
47B20, 47A63.
We study s-functions of elementary operators acting on C*-algebras. The main results are the following: If τ is any tensor norm and A, B ∈ B(𝓗) are such that the sequences s(A), s(B) of their ...singular numbers belong to a tensor stable Calkin space 𝔦 then the sequence of approximation numbers of A ⊗τ B belongs to 𝔦. If 𝒜 is a C*-algebra, 𝔦 is a tensor stable Calkin space, s is an s-number function, and ai, bi ∈ 𝒜, i = 1, 2,..., m are such that s(π(ai)), s(π(bi)) ∈ 𝔦, i = 1, 2,..., m for some faithful representation π of 𝒜 then $\mathrm{s}\left(\sum _{\mathrm{i}=1}^{\mathrm{m}}{\mathrm{M}}_{{\mathrm{a}}_{\mathrm{i}},{\mathrm{b}}_{\mathrm{i}}}\right)\in \mathfrak{i}$. The converse implication holds if and only if the ideal of compact elements of 𝒜 has finite spectrum. We also prove a quantitative version of a result of Ylinen.
The question of the existence of non-trivial ideals of Lie algebras of compact operators is considered from different points of view. One of the approaches is based on the concept of a tractable Lie ...algebra, which can be of interest independently of the main theme of the paper. Among other results it is shown that an infinite-dimensional closed Lie or Jordan algebra of compact operators cannot be simple. Several partial answers to Wojtyński’s problem on the topological simplicity of Lie algebras of compact quasinilpotent operators are also given.
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