It is shown that the Schatten–von Neumann p-norm of a Volterra integral operator of the form

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is sandwiched, when 2 [les ] p < ∞, by an integral expression similar to the well-known Hilbert–Schmidt norm.

The autonomous composition operator is the nonlinear map which takes a pair of functions into its composite function. The composition operator often appears in problems of nonlinear analysis and to analyse such problems it is often important to know whether the composition operator is continuous or differentiable. A fairly large number of papers in the literature have been devoted to the study of composition operators. For fullscale references, we refer the reader to the extensive monographs of Appell and Zabrejko [1] and Runst and Sickel [8]. To exemplify a typical situation, we consider the semilinear Dirichlet boundary value problem

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where Ω denotes a sufficiently regular bounded open subset of ℝN, and h0 a map of ℝ to ℝ, and where u is the unknown of the problem. We assume that we know that a certain function u0 belonging to a certain function space [Xscr ] solves (1.1). Then if we wish to know whether by perturbing h0 in a certain function space, say [Yscr ], the solutions u depend on h continuously, with differentiability, with analyticity or bifurcate, we could set G[h, u] ≡ Δu+h ∘ u, recast problem (1.1) into the abstract form

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and study the solution set of equation (1.2) around the pair (h0, u0) by means of the implicit function theorem or by local bifurcation theorems in a Banach space setting.

Let T be the compact real torus, and TC its complexification. Fix an integral weight α, and consider the α-weighted T-action on TC. If ω is a T-invariant Kähler form on TC, it corresponds to a pre-quantum line bundle L over TC. Let Hω be the square-integrable holomorphic sections of L. The weighted T-action lifts to a unitary T-representation on the Hilbert space Hω, and the multiplicity of its irreducible sub-representations is considered. It is shown that this is controlled by the image of the moment map, as well as the principle that ‘quantization commutes with reduction’.

A full characterization is given of those compact Lie groups G with the property that every G-map X→X on a finite-dimensional G-complex X of finite orbit type, XG = Ø, is (non-equivariantly) essential. For arbitrary G, conditions are given on the G-space X which guarantee this property. Finally, conditions are given for the non-existence of a G-map X→Y inducing a homotopy equivalence XG≃YG on the fixed point sets. These results have applications to critical point theory of almost G-invariant functionals.