Spectral properties of matrix continuous refinement operators
For \(1\leq p\leq\infty\), let \(L^p_r(\mathbb{R}^d)\) be the Banach space of vector-valued functions \((f_1(x),\dots,f_r(x))\), \(x\in\mathbb{R}^d\), satisfying: \[ \sum_{1\leq i\leq r} \Bigl\{\int|f_i(x)|^p dx\Bigr\}^{1/p}<\infty. \] Let \(M\) be a \(d\times d\) matrix with real entries and with eigenvalues of modulus greater than 1; and let \(C\) be an \(r\times r\) matrix with entries \(c_{ij}\in L^1(\mathbb{R}^d)\). Consider the operator \(W_C: L^p_r\to L^p_r\) defined by: \[ (W_C f)(x)=|\text{det}(M)|\int C(Mx-y)f(y)dy, \quad x\in\mathbb{R}^d. \] If \(c_{ij}\) are compactly supported, then there exists a subspace \(L^p_{r,K}\) of \(L^p_r\) which is invariant under \(W_C\), where \(K\subseteq\mathbb{R}^d\) is a compact set which is invariant with respect to \(M\). In this paper, the authors study the spectrum of the restriction of the operator \(W_C\) to \(L^p_{r,K}\). Among other results, the author prove that: When \(M=\text{diag}(1/\lambda_1,\dots,1/\lambda_d)\), the equation \((W_Cf)(x)=|\lambda_1\dots\lambda_d|^{-1} \int C(Mx-y)f(y)dy\), has a compactly supported solution in \(L^p_r\) if and only if \(C^\land(0)\) has an eigenvalue \(1/\lambda^\beta\) for some \(\beta\in\mathbb{Z}^d_+\), where \(C^\land\) denotes the Fourier transform of \(C\).
scientific article; zbMATH DE number 1083932