MSC 2010: Primary: 447B37; Secondary: 47B38, 47A15

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.

The relationship between the joint spectrum γ(A) of an n-tuple $A=(A_1,...,A_n)$ of selfadjoint operators and the support of the corresponding Weyl calculus T(A) : f ↦ f(A) is discussed. It is shown that one always has γ(A) ⊂ supp (T(A)). Moreover, when the operators are compact, equality occurs if and only if the operators $A_j$ mutually commute. In the non-commuting case the equality fails badly: While γ(A) is countable, supp(T(A)) has to be an uncountable set. An example is given showing that, for non-compact operators,...

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.

We prove that two Toeplitz operators acting on the pluriharmonic Bergman space with radial symbol and pluriharmonic symbol respectively commute only in an obvious case.