Various aspects of elementary operators have been characterized by many mathematicians. In this paper, we consider norm-attainability and orthogonality of these operators in Banach spaces. ...Characterizations and generalizations of norm-attainability and orthogonality are given in details. We first give necessary and sufficient conditions for norm-attainability of Hilbert space operators then we give results on orthogonality of the range and the kernel of elementary operators when they are implemented by norm-attainable operators in Banach spaces.
Various notions of orthogonality of elementary operators have been characterized by many mathematicians in different classes. In this paper, we characterize orthogonality of these operators in ...norm-attainable classes. We first give necessary and sufficient conditions for norm-attainability of Hilbert space operators then we give results on orthogonality of the range and the kernel of elementary operators when they are implemented by norm-attainable operators in norm-attainable classes.
Let X be a right Hilbert module over a C⁎-algebra A equipped with the canonical operator space structure. We define an elementary operator on X as a map ϕ:X→X for which there exists a finite number ...of elements ui in the C⁎-algebra B(X) of adjointable operators on X and vi in the multiplier algebra M(A) of A such that ϕ(x)=∑iuixvi for x∈X. If X=A this notion agrees with the standard notion of an elementary operator on A. In this paper we extend Mathieu's theorem for elementary operators on prime C⁎-algebras by showing that the completely bounded norm of each elementary operator on a non-zero Hilbert A-module X agrees with the Haagerup norm of its corresponding tensor in B(X)⊗M(A) if and only if A is a prime C⁎-algebra.
Let A be a prime unital C⁎-algebra, X a countably generated Hillbert A-module, B(X) the C⁎-algebra of adjointable operators on X and K(X) the C⁎-algebra of (generalised) compact operators on X. We ...characterise multiplication operators and elementary operators on B(X) in terms of the size of their images. To obtain these characterisations we introduce the concept of a uniformly approximable subset of a C⁎-algebra. We show that MA,B(B(X))⊆K(X) if and only if at least one of A or B belongs to K(X). We show that the set MA,B(B(X)1) is a uniformly approximable subset of K(X), (B(X)1 is the unit ball of B(X)), if and only if A,B∈K(X). If Φ is an elementary operator on B(X), we show that Φ(B(X))⊆K(X) (resp. is a uniformly approximable subset of K(X)) if and only if there exist {Ai}i=1k,{Bi}i=1k⊆B(X) such that at least one of Ai or Bi (resp. both) belong to K(X) for i=1,…,k and Φ=∑i=1kMAi,Bi.
Let R be a prime ring with center Z(R) and with extended centroid C. We give a complete characterization of Jordan derivations of R when charR=2 and dimCRC=4: An additive map δ:R→RC is a Jordan ...derivation if and only if there exist a derivation d:R→RC and an additive map μ:R→C such that δ=d+μ and μ(x2)=0 for all x∈R. As consequences, it is proved among other things: Any Z(R)-linear Jordan derivation of R is a derivation if dimCRC<∞. Moreover, if C is either a finite field or an algebraically closed field, where charC=2 and n≥2, then every Jordan derivation of Mn(C) is a derivation.
In this paper, motivated by perturbation theory of operators, we present some upper bounds for ⦀f(A)Xg(B)+X⦀ in terms of ⦀|AXB|+|X|⦀ and ⦀f(A)Xg(B)−X⦀ in terms of ⦀|AX|+|XB|⦀, where A,B are G1 ...operators, ⦀⋅⦀ is a unitarily invariant norm and f,g are certain analytic functions. Further, we find some new upper bounds for the Schatten 2-norm of f(A)X±Xg(B). Several special cases are discussed as well.
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
.
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