Spectral radius formulas for the dilation-convolution integral operator
Let \(M\) be an \(s\times s\) nonsingular matrix with the property \(\lim_{n\to \infty} M^{- n}= 0\), \[ (Tf)(x)= \int_{\mathbb{R}^s} a(x- My) f(y) dy,\quad x\in \mathbb{R}^s, \] \(a\) be a compactly supported bounded function. The spectral radius of the operator \(T\) on \(L^p(\mathbb{R}^s)\) is studied. It is shown that there is a compact set \(\Omega\subset \mathbb{R}^s\) such that the spectral radius \(\sigma_\infty(T_\Omega)\) of the compact operator \[ (T_\Omega f)(x)= \int_\Omega a(x- My) f(y) dy,\quad x\in \mathbb{R}^s, \] is related to the spectral radius \(\sigma_p(T)\) of \(T\) by the formula \[ \sigma_p(T)= |\text{det } M|^{1/p} \sigma_\infty(T_\Omega). \] Moreover, it is proved that \[ \sigma_p(T)= \lim_{n\to \infty} |a^n|^{1/n}_p= \lim_{n\to \infty} |\widehat{a^n}|^{1/n}_q, \] where \(p^{- 1}+ q^{- 1}= 1\) and \(\widehat{a^n}\) stands for the Fourier transform of \(a^n\).
scientific article; zbMATH DE number 853044