There are a number of spectra studied in the literature which do not fit into the axiomatic theory of Żelazko. This paper is an attempt to give an axiomatic theory for these spectra, which, apart from the usual types of spectra, like one-sided, approximate point or essential spectra, include also the local spectra, the Browder spectrum and various versions of the Apostol spectrum (studied under various names, e.g. regular, semiregular or essentially semiregular).

We give a survey of results concerning various classes of bounded linear operators in a Banach space defined by means of kernels and ranges. We show that many of these classes define a spectrum that satisfies the spectral mapping property.

Let X and Y be locally compact Hausdorff spaces, let E and F be Banach spaces, and let T be a linear isometry from C₀(X,E) into C₀(Y,F). We provide three new answers to the Banach-Stone problem: (1) T can always be written as a generalized weighted composition operator if and only if F is strictly convex; (2) if T is onto then T can be written as a weighted composition operator in a weak sense; and (3) if T is onto and F does not contain a copy of $\ell {₂}^{\infty }$ then T can be written as a weighted composition...

We obtain asymptotic formulas for eigenvalues and eigenfunctions of the operator generated by a system of ordinary differential equations with summable coefficients and periodic or antiperiodic boundary conditions. Then using these asymptotic formulas, we find necessary and sufficient conditions on the coefficients for which the system of eigenfunctions and associated functions of the operator under consideration forms a Riesz basis.

We characterize the dynamics of the finite time blow-up solutions with minimal mass for the focusing mass-critical Hartree equation with H¹(ℝ⁴) data and L²(ℝ⁴) data, where we make use of the refined Gagliardo-Nirenberg inequality of convolution type and the profile decomposition. Moreover, we analyze the mass concentration phenomenon of such blow-up solutions.

We give necessary and sufficient conditions on the weights v and w such that the differentiation operator D: Hv(Ω) → Hw(Ω) between two weighted spaces of holomorphic functions is bounded and onto. Here Ω = ℂ or Ω = 𝔻. In particular we characterize all weights v such that D: Hv(Ω) → Hw(Ω) is bounded and onto where w(r) = v(r)(1-r) if Ω = 𝔻 and w = v if Ω = ℂ. This leads to a new description of normal weights.

Per funzioni opportune $f,g$ si ottiene una formula di Parseval ${⟨{𝐑}^Q,𝒬f𝒬g⟩}_{\lambda }=⟨{\mathrm{\Delta }}_Q^{-1/2}f,{\mathrm{\Delta }}_Q^{-1/2}g⟩$ per operatori differenziali singolari di tipo dell'operatore radiale di Laplace-Beltrami. ${𝐑}^Q$ è una funzione spettrale generalizzata di tipo Marčenko e può essere rappresentata per mezzo di un certo nucleo della trasmutazione.

We consider one-parameter (C₀)-semigroups of operators in the space ${}^{\text{'}}(ℝⁿ;{ℂ}^m)$ with infinitesimal generator of the form ${(G*)|}_{{}^{\text{'}}(ℝⁿ;{ℂ}^m)}$ where G is an $M_{m\times m}$-valued rapidly decreasing distribution on ℝⁿ. It is proved that the Petrovskiĭ condition for forward evolution ensures not only the existence and uniqueness of the above semigroup but also its nice behaviour after restriction to whichever of the function spaces $(ℝⁿ;{ℂ}^m)$, ${}_{L^p}(ℝⁿ;{ℂ}^m)$, p ∈ [1,∞], ${(}_a)(ℝⁿ;{ℂ}^m)$, a ∈ ]0,∞[, or the spaces ${}_{L^q}^{\text{'}}(ℝⁿ;{ℂ}^m)$, q ∈ ]1,∞], of bounded distributions.