Given two n-tuples $a=(a_1,...,a_n)$ and $b=(b_1,...,b_n)$ of bounded linear operators on a Hilbert space the question of when there exists an elementary operator E such that $E{a}_j=b_j$ for all j =1,...,n, is studied. The analogous question for left multiplications (instead of elementary operators) is answered in any C*-algebra A, as a consequence of the characterization of closed left A-submodules in $A^n$.

Let T be a bounded linear operator on a complex Hilbert space H. In this paper we introduce a new class, denoted *, of operators satisfying $T^{*k}\left(\right|T²|-|T*|²)T^k\ge 0$ where k is a natural number, and we prove basic structural properties of these operators. Using these results, we also show that if E is the Riesz idempotent for a non-zero isolated point μ of the spectrum of T ∈ *, then E is self-adjoint and EH = ker(T-μ) = ker(T-μ)*. Some spectral properties are also presented.

Suppose ℒ₁ and ℒ₂ are subspace lattices on complex separable Banach spaces X and Y, respectively. We prove that under certain lattice-theoretic conditions every isomorphism from algℒ₁ to algℒ₂ is quasi-spatial; in particular, if a subspace lattice ℒ of a complex separable Banach space X contains a sequence $E_i$ such that $\left(E_i\right)₋\ne X$, $E_i\subseteq E_{i+1}$, and ${\bigvee }_{i=1}^{\infty }{E}_i=X$ then every automorphism of algℒ is quasi-spatial.

Motivated by Problem 2 in [2], Jordan *-derivation pairs and n-Jordan *-mappings are studied. From the results on these mappings, an affirmative answer to Problem 2 in [2] is given when E = F in (1) or when 𝓐 is unital. For the general case, we prove that every Jordan *-derivation pair is automatically real-linear. Furthermore, a characterization of a non-normal prime *-ring under some mild assumptions and a representation theorem for quasi-quadratic functionals are provided.

The commutator of a singular integral operator with homogeneous kernel Ω(x)/|x|ⁿ is studied, where Ω is homogeneous of degree zero and has mean value zero on the unit sphere. It is proved that $\Omega \in L{\left(logL\right)}^{k+1}\left(S^{n-1}\right)$ is a sufficient condition for the kth order commutator to be bounded on $L^p\left(ℝⁿ\right)$ for all 1 < p < ∞. The corresponding maximal operator is also considered.

It is proved that if $J_i$ is a Jordan operator on a Hilbert space with the Jordan decomposition $J_i=N_i+Q_i$, where $N_i$ is normal and $Q_i$ is compact and quasinilpotent, i = 1,2, and the Lie algebra generated by J₁,J₂ is an Engel Lie algebra, then the Banach algebra generated by J₁,J₂ is an Engel algebra. Some results for normal operators and Jordan operators on Banach spaces are given.

It is proved that every locally inner derivation on a symmetric norm ideal of operators is an inner derivation.

A linear map T from a Banach algebra A into another B preserves zero products if T(a)T(b) = 0 whenever a,b ∈ A are such that ab = 0. This paper is mainly concerned with the question of whether every continuous linear surjective map T: A → B that preserves zero products is a weighted homomorphism. We show that this is indeed the case for a large class of Banach algebras which includes group algebras. Our method involves continuous bilinear maps ϕ: A × A → X (for some Banach space X) with the property...

Under the assumption that m is a non-doubling measure on Rd, the authors obtain the (Lp,Lq)-boundedness and the weak type endpoint estimate for the multilinear commutators generated by fractional integrals with RBMO (m) functions of Tolsa or with Osc exp Lr(m) functions for r greater than or equal to 1, where Osc exp Lr(m) is a space of Orlicz type satisfying that Osc exp Lr(m)=RBMO(m) if r=1 and Osc exp Lr(m) is a subset of RBMO(m) if r&gt;1.

A classification of weakly compact multiplication operators on $L\left(L_p\right),$1<p<$,isgiven.ThisanswersaquestionraisedbySaksmanandTylliin1992.Theclassificationinvolvestheconceptof$p$-strictlysingularoperators,andwealsoinvestigatethestructureofgeneral$p$-strictlysingularoperatorson$Lp$.Themainresultisthatifanoperator$T$on$Lp$,$1<p<2$,is$p$-strictlysingularand$T|X$isanisomorphismforsomesubspace$X$of$Lp$,then$X$embedsinto$Lr$forall$r<2$,but$X$neednotbeisomorphictoaHilbertspace.$It is also shown that if $T$ is convolution by a biased coin on $L_p$ of the Cantor group, $1\le p<2$, and $T_{|X}$ is an isomorphism for some reflexive subspace $X$ of $L_p$, then $X$ is isomorphic to a Hilbert space. The case $p=1$ answers a question asked by Rosenthal in 1976.

Let $𝒳$ be a Banach space of dimension $n>1$ and $𝔄\subset ℬ\left(𝒳\right)$ be a standard operator algebra. In the present paper it is shown that if a mapping $d:𝔄\to 𝔄$ (not necessarily linear) satisfies $$d\left(\right[[U,V],W\left]\right)=\left[\right[d\left(U\right),V],W]+\left[\right[U,d\left(V\right)],W]+\left[\right[U,V],d(W\left)\right]$$ for all $U,V,W\in 𝔄$, then $d=\psi +\tau$, where $\psi$ is an additive derivation of $𝔄$ and $\tau :𝔄\to 𝔽I$ vanishes at second commutator $\left[\right[U,V],W]$ for all $U,V,W\in 𝔄$. Moreover, if $d$ is linear and satisfies the above relation, then there exists an operator $S\in 𝔄$ and a linear mapping $\tau$ from $𝔄$ into $𝔽I$ satisfying $\tau \left(\right[[U,V],W\left]\right)=0$ for all $U,V,W\in 𝔄$, such that $d\left(U\right)=SU-US+\tau \left(U\right)$ for all $U\in 𝔄$.

Let $\mathscr{H}$ be an infinite-dimensional complex Hilbert space and $𝔄$ be a standard operator algebra on $\mathscr{H}$ which is closed under the adjoint operation. It is shown that every nonlinear $*$-Lie higher derivation $𝒟={\left\{{\delta }_n\right\}}_{n\in \mathbb{N}}$ of $𝔄$ is automatically an additive higher derivation on $𝔄$. Moreover, $𝒟={\left\{{\delta }_n\right\}}_{n\in \mathbb{N}}$ is an inner $*$-higher derivation.