We study the range-kernel weak orthogonality of certain elementary operators induced by non-normal operators, with respect to usual operator norm and the Von Newmann-Schatten
-norm
We identify concrete examples of hypercyclic generalised derivations acting on separable ideals of operators and establish some necessary conditions for their hypercyclicity. We also consider the ...dynamics of elementary operators acting on particular Banach algebras, which reveals surprising hypercyclic behaviour on the space of bounded linear operators on the Banach space constructed by Argyros and Haydon.
Given Hilbert space operators A,B∈B(H), define δA,B and △A,B in B(B(H)) by δA,B(X)=AX−XB and △A,B(X)=AXB−X for each X∈B(H). An operator A∈B(H) satisfies the Putnam–Fuglede properties δ, respectively ...△ (notation: A∈PF(δ), respectively A∈PF(△)), if for every isometry V∈B(H) for which the equation δA,V⁎(X)=0, respectively △A,V⁎(X)=0, has a non-trivial solution X∈B(H), the solution X also satisfies δA⁎,V(X)=0, respectively △A⁎,V(X)=0. We prove that an operator A∈B(H) is in PF(△) if and only if it is in PF(δ).
Let 𝓝 be a nest on a Hilbert space H and Alg 𝓝 the corresponding nest algebra. We obtain a characterization of the compact and weakly compact multiplication operators defined on nest algebras. This ...characterization leads to a description of the closed ideal generated by the compact elements of Alg 𝓝. We also show that there is no non-zero weakly compact multiplication operator on Alg 𝓝/Alg 𝓝 ∩ 𝒦(H).
Given Banach space operators Ai,Bi∈B(X), 1⩽i⩽2, let ΦAB∈B(B(X)) denote the elementary operator ΦAB(X)=A1XB1−A2XB2. Then ΦAB has finite ascent ⩽1 for a number of fairly general choices of the ...operators Ai and Bi. This information is applied to prove some necessary and sufficient conditions for the range of ΦAB to be closed and in deciding conditions on the tuples (A1,A2) and (B1,B2) so that ΦABn(X) compact for some integer n⩾1 and operator X implies ΦAB(X) compact. This generalizes some well known results of Anderson and Foiaş 4, and Yosun 25. Also considered is the question: What is a necessary and sufficient condition (on the tuples (A1,A2), (B1,B2) and ΦAB) for ΦABn to be compact for some integer n⩾1?
Let
ℳ
be a von Neumann algebra of operators on a Hilbert space
ℋ
,
τ
be a faithful normal semifinite trace on
ℳ
. We obtain some new inequalities for rearrangements of
τ
-measurable operators ...products. We also establish some sufficient
τ
-compactness conditions for products of selfadjoint
τ
-measurable operators. Next we obtain a
τ
-compactness criterion for product of a nonnegative
τ
-measurable operator with an arbitrary
τ
-measurable operator. We construct an example that shows importance of nonnegativity for one of the factors. The similar results are obtained also for elementary operators from
ℳ
. We apply our results to symmetric spaces on
(
ℳ
,
τ
)
. The results are new even for the *-algebra
ℬ
(
ℋ
)
of all linear bounded operators on
ℋ
endowed with the canonical trace
τ
= tr.
Let
M
be the von Neumann algebra of operators in a Hilbert space
H
and
τ
be an exact normal semi-finite trace on
M
. We obtain inequalities for permutations of products of
τ
-measurable operators. We ...apply these inequalities to obtain new submajorizations (in the sense of Hardy, Littlewood, and Pólya) of products of
τ
-measurable operators and a sufficient condition of orthogonality of certain nonnegative
τ
-measurable operators. We state sufficient conditions of the
τ
–compactness of products of self-adjoint
τ
-measurable operators and obtain a criterion of the
τ
-compactness of the product of a nonnegative
τ
-measurable operator and an arbitrary
τ
-measurable operator. We present an example that shows that the nonnegativity of one of the factors is substantial. We also state a criterion of the elementary nature of the product of nonnegative operators from
M
. All results are new for the *-algebra
B
(
H
) of all bounded linear operators in
H
endowed with the canonical trace
τ
= tr.
In this paper, we study the
k
-quasi-
M
-hyponormal operator and mainly prove that if
T
is a
k
-quasi-
M
-hyponormal operator, then
σ
j
a
(
T
)
\
{
0
}
=
σ
a
(
T
)
\
{
0
}
, and the spectrum is ...continuous on the class of all
k
-quasi-
M
-hyponormal operators; let
d
A
B
∈
B
(
B
(
H
)
)
denote either the generalized derivation
δ
A
B
=
L
A
-
R
B
or the elementary operator
Δ
A
B
=
L
A
R
B
-
I
, we show that if
A
and
B
∗
are
k
-quasi-
M
-hyponormal operators, then
d
A
B
is polaroid and generalized Weyl’s theorem holds for
f
(
d
A
B
)
, where
f
is an analytic function on
σ
(
d
A
B
)
and
f
is not constant on each connected component of the open set
U
containing
σ
(
d
A
B
)
. In additon, we discuss the hyperinvariant subspace problem for
k
-quasi-
M
-hyponormal operators.
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