We consider operators of the form $\left(\Omega f\right)\left(y\right)={ʃ}_{-\infty }^{\infty }\Omega (y,u)f\left(u\right)du$ with Ω(y,u) = K(y,u)h(y-u), where K is a Calderón-Zygmund kernel and $h\in L^{\infty }$ (see (0.1) and (0.2)). We give necessary and sufficient conditions for such operators to map the Besov space ${Ḃ}_1^{0,1}$ (= B) into itself. In particular, all operators with $h\left(y\right)=e^{{i\left|y\right|}^a}$, a > 0, a ≠ 1, map B into itself.

It is proved that a linear surjection $\Phi 𝒜\to ℬ$, which preserves noninvertibility between semisimple, unital, complex Banach algebras, is automatically injective.

We consider the so-called Jordan triple automorphisms of some important sets of self-adjoint operators without the assumption of linearity. These transformations are bijective maps which satisfy the equality ϕ(ABA) = ϕ(A)ϕ(B)ϕ(A) on their domains. We determine the general forms of these maps (under the assumption of continuity) on the sets of all invertible positive operators, of all positive operators, and of all invertible self-adjoint operators.

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.

Motivated by the powerful and elegant works of Miers (1971, 1973, 1978) we mainly study nonlinear Lie-type derivations of von Neumann algebras. Let 𝓐 be a von Neumann algebra without abelian central summands of type I₁. It is shown that every nonlinear Lie n-derivation of 𝓐 has the standard form, that is, can be expressed as a sum of an additive derivation and a central-valued mapping which annihilates each (n-1)th commutator of 𝓐. Several potential research topics related to our work are also...

We prove that if F is a Lipschitz map from the set of all complex n × n matrices into itself with F(0) = 0 such that given any x and y we know that F(x) - F(y) and x-y have at least one common eigenvalue, then either $F\left(x\right)=ux{u}^{-1}$ or $F\left(x\right)=u{x}^t{u}^{-1}$ for all x, for some invertible n × n matrix u. We arrive at the same conclusion by supposing F to be of class ¹ on a domain in ℳₙ containing the null matrix, instead of Lipschitz. We also prove that if F is of class ¹ on a domain containing the null matrix satisfying F(0) = 0...

In this paper we prove that every bijection preserving Lie products from a triangular algebra onto a normal triangular algebra is additive modulo centre. As an application, we described the form of bijections preserving Lie products on nest algebras and block upper triangular matrix algebras.

In this work we introduce a nonparametric recursive aggregation process called Multilayer Aggregation (MLA). The name refers to the fact that at each step the results from the previous one are aggregated and thus, before the final result is derived, the initial values are subjected to several layers of aggregation. Most of the conventional aggregation operators, as for instance weighted mean, combine numerical values according to a vector of weights (parameters). Alternatively, the MLA operators...

Necessary and sufficient conditions are given for a (complete) commutative algebra that is regular in the sense of von Neumann to have a non-zero derivation. In particular, it is shown that there exist non-zero derivations on the algebra L(M) of all measurable operators affiliated with a commutative von Neumann algebra M, whose Boolean algebra of projections is not atomic. Such derivations are not continuous with respect to measure convergence. In the classical setting of the algebra S[0,1] of all...

Let A and B be uniform algebras. Suppose that α ≠ 0 and A 1 ⊂ A. Let ρ, τ: A 1 → A and S, T: A 1 → B be mappings. Suppose that ρ(A 1), τ(A 1) and S(A 1), T(A 1) are closed under multiplications and contain expA and expB, respectively. If ‖S(f)T(g) − α‖∞ = ‖ρ(f)τ(g) − α‖∞ for all f, g ∈ A 1, S(e 1)−1 ∈ S(A 1) and S(e 1) ∈ T(A 1) for some e 1 ∈ A 1 with ρ(e 1) = 1, then there exists a real-algebra isomorphism $$\tilde{S}$$ : A → B such that $$\tilde{S}$$ (ρ(f)) = S(e 1)−1 S(f) for every f ∈ A 1. We also give some applications...

We show that a positive semigroup $T_t$ on $L_p(\Omega ,\nu )$ with generator A and ||R(α + i β)|| → 0 as |β| → ∞ for some α ∈ ℝ is continuous in the operator norm for t>0. The proof is based on a criterion for norm continuity in terms of “smoothing properties” of certain convolution operators on general Banach spaces and an extrapolation result for the $L_p$-scale, which may be of independent interest.

Let X be a closed subspace of $L^p\left(\mu \right)$, where μ is an arbitrary measure and 1 < p < ∞. Let U be an invertible operator on X such that $su{p}_{n\in \mathbb{Z}}\left|\right|Uⁿ\left|\right|<\infty$. Motivated by applications in ergodic theory, we obtain (optimal) conditions for the convergence of series like ${\sum }_{n\ge 1}\left(Uⁿf\right)/n^{1-\alpha }$, 0 ≤ α < 1, in terms of $\left|\right|f+\cdots +U^{n-1}f{\left|\right|}_p$, generalizing results for unitary (or normal) operators in L²(μ). The proofs make use of the spectral integration initiated by Berkson and Gillespie and, more particularly, of results from a paper by Berkson-Bourgain-Gillespie....