Given a DT-operator Z whose Brown measure is radially symmetric and has a certain concentration property, it is shown that Z is not spectral in the sense of Dunford. This is accomplished by showing that the angles between certain complementary Haagerup–Schultz projections of Z approach zero. New estimates on norms and traces of powers of algebra-valued circular operators over commutative C*-algebras are also proved.

Let Γ<G be a discrete subgroup of a locally compact unimodular group G. Let m∈C _b(G) be a p-multiplier on G with 1≤p<∞ and let T _m:L _p(G^)→L _p(G^) be the corresponding Fourier multiplier. Similarly, let Tm| _Γ:L _p(Γ^)→L _p(Γ^) be the Fourier multiplier associated to the restriction m| _Γ of m to Γ. We show that (Formula presented.) for a specific constant 0≤c(U)≤1 that is defined for every U⊆Γ. The function c quantifies the failure of G to admit small almost Γ-invariant neighbourhoods and can be determined explicitly in concrete cases. In particular, c(Γ)=1 when G has small almost Γ-invariant neighbourhoods. Our result thus extends the de Leeuw restriction theorem from Caspers et al. (Forum Math Sigma 3(e21):59, 2015) as well as de Leeuw’s classical theorem (Ann Math 81(2):364–379, 1965). For real reductive Lie groups G we provide an explicit lower bound for c in terms of the maximal dimension d of a nilpotent orbit in the adjoint representation. We show that c(B _ρ ^G)≥ρ ^-d/4 where B _ρ ^G is the ball of g∈G with ‖Ad _g‖<ρ. We further prove several results for multilinear Fourier multipliers. Most significantly, we prove a multilinear de Leeuw restriction theorem for pairs Γ<G with c(Γ)=1. We also obtain multilinear versions of the lattice approximation theorem, the compactification theorem and the periodization theorem. Consequently, we are able to provide the first examples of bilinear multipliers on nonabelian groups.

Let G be a locally compact unimodular group, and let φ be some function of n variables on G. To such a φ, one can associate a multilinear Fourier multiplier, which acts on some n-fold product of the noncommutative L _p-spaces of the group von Neumann algebra. One may also define an associated Schur multiplier, which acts on an n-fold product of Schatten classes S _p(L ₂(G). We generalize well-known transference results from the linear case to the multilinear case. In particular, we show that the so-called multiplicatively bounded (p1,....,pn)-norm"of a multilinear Schur multiplier is bounded above by the corresponding multiplicatively bounded norm of the Fourier multiplier, with equality whenever the group is amenable. Furthermore, we prove that the bilinear Hilbert transform is not bounded as a vector-valued map L _p1(R,S _p1) × L _p2 (R,S _p2), → L ₁(R,S ₁), whenever p1 and p2 are such that {equation presented}. A similar result holds for certain Calderón-Zygmund-type operators. This is in contrast to the nonvector-valued Euclidean case.