Spectral theory for fractal pseudodifferential operators

The paper deals with the distribution of eigenvalues of the compact fractal pseudodifferential operator T^μ_τ, \[ ( T^μ_τ f)(x) = ∫R^n e-ixξ \, τ(x,ξ) \, ( fμ )^ (ξ) \, d ξ, x∈ R^n, \] in suitable special Besov spaces Bˢₚ (Rⁿ) = Bˢp,p (Rⁿ), $s>0$, 1<p<∞. Here τ(x,ξ) are the symbols of (smooth) pseudodifferential operators belonging to appropriate H\"{o}rmander classes Ψ^σ1, (Rⁿ), σ <0, 0 ≤ ≤ 1 (including the exotic case =1) whereas μ is the Hausdorff measure of a compact d-set Γ in Rⁿ, $0<d<n$. This extends previous assertions for the positive-definite selfadjoint fractal differential operator (id - Δ)σ/2 μ based on Hilbert space arguments in the context of suitable Sobolev spaces Hˢ (Rⁿ) = Bˢ₂ (Rⁿ). We collect the outcome in the {Main Theorem} below. Proofs are based on estimates for the entropy numbers of the compact trace operator \[ tr_μ: B^s_p (R^n) L_p (Γ, μ), s>0, 1<p<∞. \] We add at the end of the paper a few personal reminiscences illuminating the role of Pietsch in connection with the creation of approximation numbers and entropy numbers.