Spectral theory for fractal pseudodifferential operators

Abstract

The paper deals with the distribution of eigenvalues of the compact fractal pseudodifferential operator $T^\mu_\tau$, \[ \big( T^\mu_\tau f\big)(x) = \int_{\mathbb{R}^n} e^{-ix\xi} \, \tau(x,\xi) \, \big( f\mu \big)^\vee (\xi) \, \mathrm{d} \xi, \qquad x\in \mathbb{R}^n, \] in suitable special Besov spaces $B^s_p (\mathbb{R}^n) = B^s_{p,p} (\mathbb{R}^n)$, $s>0$, $1<p<\infty$. Here $\tau(x,\xi)$ are the symbols of (smooth) pseudodifferential operators belonging to appropriate H\"{o}rmander classes $\Psi^\sigma_{1, \varrho} (\mathbb{R}^n)$, $\sigma <0$, $0 \le \varrho \le 1$ (including the exotic case $\varrho =1$) whereas $\mu$ is the Hausdorff measure of a compact $d$-set $\Gamma$ in $\mathbb{R}^n$, $0<d<n$. This extends previous assertions for the positive-definite selfadjoint fractal differential operator $(\mathrm{id} - \Delta)^{\sigma/2} \mu$ based on Hilbert space arguments in the context of suitable Sobolev spaces $H^s (\mathbb{R}^n) = B^s_2 (\mathbb{R}^n)$. We collect the outcome in the {Main Theorem} below. Proofs are based on estimates for the entropy numbers of the compact trace operator \[ \mathrm{tr}_\mu: \quad B^s_p (\mathbb{R}^n) \hookrightarrow L_p (\Gamma, \mu), \quad s>0, \quad 1<p<\infty. \] 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.