In this paper the author considers the spectrum of drift operators \(\mathcal L = \sum_{i,j=1}^n b_{ij}x_j D_i\) and the Ornstein–Uhlenbeck operators \({\mathcal A}= \sum_{i,j=1}^n q_{ij} D_{ij}+ \mathcal L\) in \(L^p(\mathbb R^n)\) and \(BUC(\mathbb R^n)\). Let \(B\) be the (nonzero real) matrix \((b_{ij})\), and let \(Q\) be the real symmetric positive-definite matrix \((q_{ij})\). For a linear operator \(\mathcal B\) let the spectral bound of \(\mathcal B\) be \(s(\mathcal B) = \{ \sup \text{Re}\, \mu \, : \,\mu \in \sigma (\mathcal B) \}\). Next, let the boundary spectrum of \(\mathcal B\) be \(\sigma(\mathcal B) \cap \{ \mu \in \mathbb C:\text{Re} \, \mu = s(\mathcal B)\}\). Finally, set \(D_p(\mathcal B)\) be the set of all \(u\in L^p\) such that \(\mathcal B u \in L^p\). In Section 2 of the paper, the author characterises the \(L^p\) spectrum of the drift operator \(\mathcal L\), \(\sigma_p(\mathcal L)\), under different conditions on the matrix \(B\). It turns out that the spectrum is \(p\)-dependent; a typical result is Theorem 2.3. which shows that if \(\text{tr}(B) \neq 0\), then \(\sigma_p(\mathcal L) = - \text{tr}(B)/p + i \mathbb R\).
The main result of Section 3 is Theorem 3.3, which states that the boundary spectrum of \((\mathcal A, \; D_p(\mathcal A))\) contains the spectrum of the drift operator \((\mathcal L, \; D_p(\mathcal L))\).
Section 4, the central section of the paper, concerns the \(L^p\) spectrum of Ornstein-Uhlenbeck operators. Precise results here depend on \(p\), and on the location of the spectrum of \(B\). The results in this section assume that either \(\sigma(\mathcal B) \subset \mathbb C_+\) or \(\sigma(\mathcal B) \subset \mathbb C_-\). As a typical result we quote Theorem 4.12: If \(1 \leq p \leq \infty\) and \(\sigma (B) \subset \mathbb C_-\), then \[ \sigma_p (\mathcal A) = \{ \mu \in \mathbb C \, : \, \hbox{Re} \, \mu \leq -\text{tr}(B)/p \}. \] Finally, in Section 5 the author considers \(L^p\) spectra of Ornstein-Uhlenbeck operators under less restrictive conditions on the spectrum of \(B\), but under the additional assumptions that \(B\) is symmetric and that \(B\) and \(Q\) commute. A result similar to Theorem 4.12, Theorem 5.1, holds in that case.
In Section 6 the author proves results about spectra in \(BUC(\mathbb R^n)\). For example (see Theorem 6.2), if \(\sigma(B) \subset \mathcal C_-\), the spectrum of \((\mathcal A, \, \mathcal D( \mathcal A))\) is the left half-plane; here the domain of \(\mathcal A\) is defined as \[ \{ u \in BUC (\mathbb R^n) \cap W^{2,p}_{\text{loc}} (\mathbb R^n) \; \forall \, p > n \, : \, u \in BUC(\mathbb R^n) \}. \] The proofs use an attractive mixture of results from semigroup theory, Fourier methods and results of Arendt.