Let 𝓛 be a 𝒥-subspace lattice on a Banach space X and Alg 𝓛 the associated 𝒥-subspace lattice algebra. Assume that δ: Alg 𝓛 → Alg 𝓛 is an additive map. It is shown that δ satisfies δ(AB + BA) = δ(A)B + Aδ(B) + δ(B)A + Bδ(A) for any A,B ∈ Alg 𝓛 with AB + BA = 0 if and only if δ(A) = τ(A) + δ(I)A for all A, where τ is an additive derivation; if X is complex with dim X ≥ 3 and if δ is linear, then δ satisfies δ(AB + BA) = δ(A)B + Aδ(B) + δ(B)A + Bδ(A) for any A,B ∈ Alg 𝓛 with AB + BA = I if...

We study the mapping property of the commutator of Hardy-Littlewood maximal function on Triebel-Lizorkin spaces. Also, some new characterizations of the Lipschitz spaces are given.

In this paper we obtain some results concerning the set $ℳ=\cup \left\{\overline{R\left({\delta }_A\right)}\cap {\left\{A\right\}}^{\text{'}}A\in ℒ\left(H\right)\right\}$, where $\overline{R\left({\delta }_A\right)}$ is the closure in the norm topology of the range of the inner derivation ${\delta }_A$ defined by ${\delta }_A\left(X\right)=AX-XA.$ Here $\mathscr{H}$ stands for a Hilbert space and we prove that every compact operator in ${\overline{R\left({\delta }_A\right)}}^w\cap {\left\{A^{*}\right\}}^{\text{'}}$ is quasinilpotent if $A$ is dominant, where ${\overline{R\left({\delta }_A\right)}}^w$ is the closure of the range of ${\delta }_A$ in the weak topology.

Si $A$ est une fonction de classe ${𝒞}^1$ à support compacte et si $T$ est un opérateur pseudo-différentiel classique d’ordre 1, l’opérateur $f\to T\left(Af\right)-AT\left(f\right)$ est borné sur $L^2$. Ce résultat se généralise aux commutateurs d’ordre supérieur.

Let $N\subseteq M$ be an inclusion of $I{I}_1$ factors with finite Jones index. Then $M=(N^{\text{'}}\cap M)\oplus [N,M]$ as a vector space. Here $[N,M]$ denotes the vector space spanned by the commutators of the form $[a,b]$ where $a\in N,b\in M$.

We prove the Schatten-Lorentz ideal criteria for commutators of multiplications and projections based on the Calderón reproducing formula and the decomposition theorem for the space of symbols corresponding to commutators in the Schatten ideal.

The set of commutators in a Banach *-algebra A, with continuous involution, is examined. Applications are made to the case where A = B(ℓ₂), the algebra of all bounded linear operators on ℓ₂.

It is proved that the set Q of quasinilpotent elements in a Banach algebra is an ideal, i.e. equal to the Jacobson radical, if (and only if) the condition [Q,Q] ⊆ Q (or a similar condition concerning anticommutators) holds. In fact, if the inner derivation defined by a quasinilpotent element p maps Q into itself then p ∈ Rad A. Higher commutator conditions of quasinilpotents are also studied. It is shown that if a Banach algebra satisfies such a condition, then every quasinilpotent element has some...

Let $M_{\beta }$ be the fractional maximal function. The commutator generated by $M_{\beta }$ and a suitable function $b$ is defined by $[M_{\beta },b]f=M_{\beta }\left(bf\right)-b{M}_{\beta }\left(f\right)$. Denote by $𝒫\left({ℝ}^n\right)$ the set of all measurable functions $p(·):{ℝ}^n\to [1,\infty )$ such that $$1<p_{-}:=\underset{x\in {ℝ}^n}{\mathrm{ess}\inf }p\left(x\right)\text{and}{p}_{+}:=\underset{x\in {ℝ}^n}{\mathrm{ess}\sup }p\left(x\right)<\infty ,$$ and by $ℬ\left({ℝ}^n\right)$ the set of all $p(·)\in 𝒫\left({ℝ}^n\right)$ such that the Hardy-Littlewood maximal function $M$ is bounded on $L^{p(·)}\left({ℝ}^n\right)$. In this paper, the authors give some characterizations of $b$ for which $[M_{\beta },b]$ is bounded from $L^{p(·)}\left({ℝ}^n\right)$ into $L^{q(·)}\left({ℝ}^n\right)$, when $p(·)\in 𝒫\left({ℝ}^n\right)$, $0<\beta <n/p_{+}$ and $1/q(·)=1/p(·)-\beta /n$ with $q(·)(n-\beta )/n\in ℬ\left({ℝ}^n\right)$.

Let T be a bounded linear operator on $X={\left(\sum {\ell }_q\right)}_p$ with 1 ≤ q < ∞ and 1 < p < ∞. Then T is a commutator if and only if for all non-zero λ ∈ ℂ, the operator T - λI is not X-strictly singular.