Let A be a ring and
a ring endomorphism. A generalized skew (or σ-)derivation of A is an additive map
for which there exists a map
such that
for all
. If A is a prime
-algebra and σ is surjective, we ...determine the structure of generalized σ-derivations of A that belong to the cb-norm closure of elementary operators
on A; all such maps are of the form
for suitable elements a,b,c of the multiplier algebra
. As a consequence, if an epimorphism
lies in the cb-norm closure of
, then σ must be an inner automorphism. We also show that these results cannot be extended even to relatively well-behaved non-prime
-algebras like
.
Kernels of elementary operators of length two give rise to many interesting classes of operators. In this paper we study bounded reflexivity of these kernels. For general Banach spaces, a sufficient ...condition is given for the kernels to be boundedly reflexive, in particular this assures that the commutant of any Banach space operator is boundedly reflexive. We also show that the bounded reflexive cover of the set of all Toeplitz matrices in Mm×n is the entire Mm×n. For finite-dimensional Banach spaces, using Kronecker canonical form for matrix pencils, we obtain some necessary and sufficient conditions for the kernels to be boundedly reflexive. These conditions enable us to easily determine bounded reflexivity by simply computing rank or nullity of matrices.
The study of m-isometry elementary operator of length one is initiated by Botelho and Jamison. Several authors have contributed to obtain this result: if the two operator coefficients involved are ...m-isometries then the elementary operator is m-isometry on the Hilbert–Schmidt operator ideal. In this paper we show that the converse of this result is also true. We also give a necessary and sufficient condition for the generalized derivation to be an m-isometry. The sufficiency follows from results on the sum of m-isometries and nilpotent operators in two recent papers.
The aim of this paper is to investigate the Daugavet equation for a Jordan elementary operator. More precisely, we study the equation
‖
I
+
U
J
,
A
,
B
‖
=
1
+
2
‖
A
‖
‖
B
‖
,
where
I
stands for the ...identity operator,
A
and
B
are two bounded operators acting on a complex Hilbert space
H
,
J
is a norm ideal of operators on
H
, and
U
J
,
A
,
B
is the restriction of the Jordan operator
U
A
,
B
to
J
. In the particular case where
J
=
C
2
(
H
)
is the ideal of Hilbert–Schmidt operators, we give necessary and sufficient conditions under which the above equation holds.
We discuss some necessary and some sufficient conditions for an elementary operator x↦∑i=1naixbi on a Banach algebra A to be spectrally bounded. In the case of length three, we obtain a complete ...characterisation when A acts irreducibly on a Banach space of dimension greater than three.
We apply the inequality |〈x,y〉| ≤ ⩽x⩾ 〈y,y〉1/2 to give an easy and elementary proof of many operator inequalities for elementary operators and inner type product integral transformers obtained during ...last two decades, which also generalizes many of them.
Let D be a division algebra finite-dimensional over its center C, Ω:=Mm(D), the m×m matrix algebra over D, and V be a vector space over C. We characterize all n-linear forms on Ω in terms of reduced ...traces and elementary operators. For m>1, it is proved that a bilinear form B:Ω×Ω→V vanishes on zero products of xy and yx if and only if there exist linear maps g,h:Ω→V such that B(x,y)=g(xy)+h(yx) for all x,y∈Ω. As an application, a bilinear form B is completely characterized if B(x,y)=0 whenever x,y∈Ω satisfy xy+ξyx=0, where ξ is a fixed nonzero element in C.
The characterization of the points in
, the Von Neuman-Schatten
-classes, that are orthogonal to the range of elementary operators has been done for certain kinds of elementary operators. In this ...paper, we shall study this problem of characterization on an abstract reflexive, smooth and strictly convex Banach space for arbitrary operator. As an application, we consider other kinds of elementary operators defined on the spaces
, and finally, we give a counterexample to Mecheri’s result given in this context.
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