In this paper, we study the behaviours of the commutators $[\vec b,\,T]$ generated by multilinear Calderón–Zygmund operators $T$ with $\vec b=(b_1,\,\ldots,\,b_m)\in L_{\rm loc}(\mathbb {R}^n)$ on weighted Hardy spaces. We show that for some $p_i\in (0,\,1]$ with $1/p=1/p_1+\cdots +1/p_m$, $\omega \in A_\infty$ and $b_i\in \mathcal {BMO}_{\omega,p_i}$ ($1\le i\le m$), which are a class of non-trivial subspaces of ${\rm BMO}$, the commutators $[\vec b,\,T]$ are bounded from $H^{p_1}(\omega )\times \cdots \times H^{p_m}(\omega )$ to $L^p(\omega )$. Meanwhile, we also establish the corresponding results for a class of maximal truncated multilinear commutators $T_{\vec b}^*$.

Let $C_{\||.\||}$ be an ideal of compact operators with symmetric norm $\||.\||$ . In this paper, we extend the van Hemmen–Ando norm inequality for arbitrary bounded operators as follows: if f is an operator monotone function on $[0,\infty)$ and S and T are bounded operators in $\mathbb{B}(\mathscr{H}\;\,)$ such that ${\rm{sp}}(S),{\rm{sp}}(T) \subseteq \Gamma_a=\{z\in \mathbb{C} \ | \ {\rm{re}}(z)\geq a\}$ , then

\begin{equation*}\||f(S)X-Xf(T)\|| \leq\;f'(a) \ \||SX-XT\||,\end{equation*}

$X\in C_{\||.\||}$

${\rm{sp}}(S), {\rm{sp}}(T) \subseteq \Gamma_a$

\begin{equation*}\||S^r X-XT^r\|| \leq r a^{r-1} \ \||SX-XT\||,\end{equation*}

$X\in C_{\||.\||}$

$0\leq r\leq 1$

This paper obtains new characterizations of weighted Hardy spaces and certain weighted $BMO$ type spaces via the boundedness of variation operators associated with approximate identities and their commutators, respectively.

Let $\mathcal {X}$ be a Banach space over the complex field $\mathbb {C}$ and $\mathcal {B(X)}$ be the algebra of all bounded linear operators on $\mathcal {X}$ . Let $\mathcal {N}$ be a nontrivial nest on $\mathcal {X}$ , $\text {Alg}\mathcal {N}$ be the nest algebra associated with $\mathcal {N}$ , and $L\colon \text {Alg}\mathcal {N}\longrightarrow \mathcal {B(X)}$ be a linear mapping. Suppose that $p_n(x_1,x_2,\ldots ,x_n)$ is an $(n-1)\,$ th commutator defined by n indeterminates $x_1, x_2, \ldots , x_n$ . It is shown that L satisfies the rule

$$ \begin{align*}L(p_n(A_1, A_2, \ldots, A_n))=\sum_{k=1}^{n}p_n(A_1, \ldots, A_{k-1}, L(A_k), A_{k+1}, \ldots, A_n) \end{align*} $$

for all $A_1, A_2, \ldots , A_n\in \text {Alg}\mathcal {N}$ if and only if there exist a linear derivation $D\colon \text {Alg}\mathcal {N}\longrightarrow \mathcal {B(X)}$ and a linear mapping $H\colon \text {Alg}\mathcal {N}\longrightarrow \mathbb {C}I$ vanishing on each $(n-1)\,$ th commutator $p_n(A_1,A_2,\ldots , A_n)$ for all $A_1, A_2, \ldots , A_n\in \text {Alg}\mathcal {N}$ such that $L(A)=D(A)+H(A)$ for all $A\in \text {Alg}\mathcal {N}$ . We also propose some related topics for future research.

Let $S$ be a discrete inverse semigroup, $l^{1}(S)$ the Banach semigroup algebra on $S$ and $\mathbb{X}$ a Banach $l^{1}(S)$-bimodule which is an $L$-embedded Banach space. We show that under some mild conditions ${\mathcal{H}}^{1}(l^{1}(S),\mathbb{X})=0$. We also provide an application of the main result.

The purpose of this note is to prove the following. Suppose $\mathfrak{R}$ is a semiprime unity ring having an idempotent element $e$ ($e\neq 0,~e\neq 1$) which satisfies mild conditions. It is shown that every additive generalized Jordan derivation on $\mathfrak{R}$ is a generalized derivation.

We introduce the concept of strong property $(\mathbb{B})$ with a constant for Banach algebras and, by applying a certain analysis on the Fourier algebra of the unit circle, we show that all C*-algebras and group algebras have the strong property $(\mathbb{B})$ with a constant given by $288\unicode[STIX]{x1D70B}(1+\sqrt{2})$. We then use this result to find a concrete upper bound for the hyperreflexivity constant of ${\mathcal{Z}}^{n}(A,X)$, the space of bounded $n$-cocycles from $A$ into $X$, where $A$ is a C*-algebra or the group algebra of a group with an open subgroup of polynomial growth and $X$ is a Banach $A$-bimodule for which ${\mathcal{H}}^{n+1}(A,X)$ is a Banach space. As another application, we show that for a locally compact amenable group $G$ and $1<p<\infty$, the space $CV_{P}(G)$ of convolution operators on $L^{p}(G)$ is hyperreflexive with a constant given by $384\unicode[STIX]{x1D70B}^{2}(1+\sqrt{2})$. This is the generalization of a well-known result of Christensen [‘Extensions of derivations. II’, Math. Scand. 50(1) (1982), 111–122] for $p=2$.

For a von Neumann subalgebra $A \, \subseteq \, {\cal B}({\cal H})$ and any two elements a, b ∈ A with a normal, such that the corresponding derivations da and db satisfy the condition ‖db(x)‖ ≤ ‖da(x)‖ for all x ∈ A, there exist completely bounded (a)ʹ-bimodule map $\varphi : {\cal B}({\cal H}) \rightarrow {\cal B}({\cal H})$ such that db|A = φ da|A=daφ|A. (In particular db(A) ⊆ da(A).) Moreover, if A is a factor, then φ can be taken to be normal and these equalities hold on ${\cal B}({\cal H})$ instead of just on A. This result is not true for general (even primitive) C*-algebras ${\cal A}$.

Extending the notion of parallelism we introduce the concept of approximate parallelism in normed spaces and then substantially restrict ourselves to the setting of Hilbert space operators endowed with the operator norm. We present several characterizations of the exact and approximate operator parallelism in the algebra $\mathbb{B}\left( H \right)$ of bounded linear operators acting on a Hilbert space $H$. Among other things, we investigate the relationship between the approximate parallelism and norm of inner derivations on $\mathbb{B}\left( H \right)$. We also characterize the parallel elements of a ${{C}^{*}}$-algebra by using states. Finally we utilize the linking algebra to give some equivalent assertions regarding parallel elements in a Hilbert ${{C}^{*}}$-module.

Let ${{H}^{2}}\left( \Omega \right)$ be the Hardy space on a strictly pseudoconvex domain $\Omega \,\subset \,{{\mathbb{C}}^{n}}$, and let $A\,\subset \,{{L}^{\infty }}\left( \partial \Omega \right)$ denote the subalgebra of all ${{L}^{\infty }}$-functions $f$ with compact Hankel operator ${{H}_{f}}$. Given any closed subalgebra $B\,\subset \,A$ containing $C\left( \partial \Omega \right)$, we describe the first Hochschild cohomology group of the corresponding Toeplitz algebra $\mathcal{T}\left( B \right)\,\subset \,B\left( {{H}^{2}}\left( \Omega \right) \right)$. In particular, we show that every derivation on $\mathcal{T}\left( A \right)$ is inner. These results are new even for $n\,=\,1$, where it follows that every derivation on $\mathcal{T}\left( {{H}^{\infty }}\,+\,C \right)$ is inner, while there are non-inner derivations on $\mathcal{T}\left( {{H}^{\infty }}\,+\,C\left( \partial {{\mathbb{B}}_{n}} \right) \right)$ over the unit ball ${{\mathbb{B}}_{n}}$ in dimension $n\,>\,1$.

We introduce generalized triple homomorphisms between Jordan–Banach triple systems as a concept that extends the notion of generalized homomorphisms between Banach algebras given by K. Jarosz and B. E. Johnson in 1985 and 1987, respectively. We prove that every generalized triple homomorphism between $\text{J}{{\text{B}}^{*}}$-triples is automatically continuous. When particularized to ${{C}^{*}}$-algebras, we rediscover one of the main theorems established by Johnson. We will also consider generalized triple derivations from a Jordan–Banach triple $E$ into a Jordan–Banach triple $E$-module, proving that every generalized triple derivation from a $\text{J}{{\text{B}}^{*}}$-triple $E$ into itself or into ${{E}^{*}}$is automatically continuous.

Let A be a unital C*-algebra with the canonical (H) C*-bundle over the maximal ideal space of the centre of A, and let E(A) be the set of all elementary operators on A. We consider derivations on A which lie in the completely bounded norm closure of E(A), and show that such derivations are necessarily inner in the case when each fibre of is a prime C*-algebra. We also consider separable C*-algebras A for which is an (F) bundle. For these C*-algebras we show that the following conditions are equivalent: E(A) is closed in the operator norm; A as a Banach module over its centre is topologically finitely generated; fibres of have uniformly finite dimensions, and each restriction bundle of over a set where its fibres are of constant dimension is of finite type as a vector bundle.

Under the assumption that $\mu $ is a nondoubling measure, we study certain commutators generated by the Lipschitz function and the Marcinkiewicz integral whose kernel satisfies a Hörmander-type condition. We establish the boundedness of these commutators on the Lebesgue spaces, Lipschitz spaces, and Hardy spaces. Our results are extensions of known theorems in the doubling case.

Let $H$ be a separable, infinite-dimensional, complex Hilbert space and let $A,\,B\,\in \,\mathcal{L}\left( H \right)$, where $\mathcal{L}(H)$ is the algebra of all bounded linear operators on $H$. Let ${{\delta }_{AB}}\,:\mathcal{L}\left( H \right)\to \mathcal{L}\left( H \right)$ denote the generalized derivation ${{\delta }_{AB}}\left( X \right)\,=\,AX\,-\,XB$. This note will initiate a study on the class of pairs $\left( A,\,B \right)$ such that $\overline{R\left( {{\delta }_{AB}} \right)}\,=\,\overline{R\left( {{\delta }_{A*\,B*}} \right)}$.

Suppose that σ:𝔐→𝔐 is an ultraweakly continuous surjective *-linear mapping and d:𝔐→𝔐 is an ultraweakly continuous *-σ-derivation such that d(I) is a central element of 𝔐. We provide a Kadison–Sakai-type theorem by proving that ℌ can be decomposed into and d can be factored as the form , where δ:𝔐→𝔐 is an inner *-σ𝔎-derivation, Z is a central element, 2τ=2σ𝔏 is a *-homomorphism, and σ𝔎 and σ𝔏 stand for compressions of σ to 𝔎 and 𝔏 , respectively.

We consider the problem of determining for which square integrable functions $f$ and $g$ on the polydisk the densely defined Hankel product ${{H}_{f}}\,H_{g}^{*}$ is bounded on the Bergman space of the polydisk. Furthermore, we obtain similar results for the mixed Haplitz products ${{H}_{g}}\,{{T}_{{\bar{f}}}}$ and ${{T}_{f}}\,H_{g}^{*}$, where $f$ and $g$ are square integrable on the polydisk and $f$ is analytic.

The principal results in this paper are concerned with the description of domains of infinitesimal generators of strongly continuous groups of isometries in non-commutative operator spaces $E(\mathcal{M},\tau)$, which are induced by $\mathbb{R}$-flows on $\mathcal{M}$. In particular, we are concerned with the description of operator functions which leave the domain of such generators invariant in all symmetric operator spaces, associated with a semi-finite von Neumann algebra $\mathcal{M}$ and a separable function space $E$ on $(0,\infty)$. Furthermore, we apply our results to the study of operator functions for which $[D,x]\in E(\mathcal{M},\tau)$ implies that $[D,f(x)]\in E(\mathcal{M},\tau)$, where $D$ is an unbounded self-adjoint operator. Our methods are partly based on the recently developed theory of double operator integrals in symmetric operator spaces and the theory of adjoint $C_{0}$-semigroups.

Let $B(H)$ denote the algebra of all bounded linear operators on a separable, infinite-dimensional, complex Hilbert space $H$. Let $I$ be a two-sided ideal in $B(H)$. For operators $A, B$ and $X \in B(H)$, we say that $X$intertwines$A$and$B$modulo$I$ if $AX - XB \in I$. It is easy to see that if $X$ intertwines $A$ and $B$ modulo $I$, then it intertwines $A^{n}$ and $B^{n}$ modulo $I$ for every integer $n > $1. However, the converse is not true. In this paper, sufficient conditions on the operators $A$ and $B$ are given so that any operator $X$ which intertwines certain powers of $A$ and $B$ modulo $I$ also intertwines $A$ and $B$ modulo $J$ for some two-sided ideal $J \supseteq I$.

Let $L(X)$ be the space of bounded linear operators on the Banach space $X$. We study the strict singularity and cosingularity of the two-sided multiplication operators $S\,\mapsto \,ASB$ on $L(X)$, where $A,\,B\,\in \,L(X)$ are fixed bounded operators and $X$ is a classical Banach space. Let $1\,<\,p\,<\,\infty $ and $p\,\ne \,2$. Our main result establishes that the multiplication $S\,\mapsto \,ASB$ is strictly singular on $L\left( {{L}^{p}}\left( 0,\,1 \right) \right)$ if and only if the non-zero operators $A,\,B\,\in \,L\left( {{L}^{p}}\left( 0,\,1 \right) \right)$ are strictly singular. We also discuss the case where $X$ is a ${\mathcal{L}^{1}}-$ or a ${{\mathcal{L}}^{\infty }}-$space, as well as several other relevant examples.

Certain linear operators from a Banach algebra $A$ into a Banach $A$-bimodule $X$, which are called approximately local derivations, are studied. It is shown that when $A$ is a ${\rm C^*}$-algebra, a Banach algebra generated by idempotents, a semisimple annihilator Banach algebra, or the group algebra of a SIN or a totally disconnected group, bounded approximately local derivations from $A$ into $X$ are derivations. This, in particular, extends a result of B. E. Johnson that ‘local derivations on ${\rm C^*}$-algebras are derivations’ and provides an alternative proof of it.