On the spectrum of multipliers in Bessel potential spaces. (English) Zbl 0578.46022

By definition, a multiplier in a function space S is such a function that multiplication by it maps S into itself. The space of multipliers in S is denoted by MS. A description of the pointwise \(\sigma_ p(\gamma)\), residual \(\sigma_ r(\gamma)\) and continuous \(\sigma_ c(\gamma)\) spectra of a multiplier \(\gamma\) in the Bessel potential space \(H_ p^{\ell}(R^ n)\) and its dual \(H_{p'}^{\ell}(R^ n)\), \(p\in (1,\infty)\), \(\ell >0\), is obtained. For example, let \(\gamma \in MH_{p'}^{-\ell}\) and \(\lambda\) be a point of the spectrum of \(\gamma\). Let \(z_{\gamma}=\{x:\gamma (x)=\lambda \}\) and let \(cap(e,H_ p^{\ell})\) be the capacity of a set \(e\subset R^ n\) generated by the norm in \(H_ p^{\ell}\). Then
1) \(\lambda \in \sigma_ p(\gamma)\) if and only if \(cap(Z_{\gamma},H_ p^{\ell})>0;\)
2) \(\lambda \in \sigma_ c(\gamma)\) if and only if \(cap(Z_{\lambda},H_ p^{\ell})=0.\)
This result is applied to a characterization of the same spectra of a convolution operator in a weighted \(L_ 2\)-space.


46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
44A35 Convolution as an integral transform
47A10 Spectrum, resolvent
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