\(L^p\)-spectrum of degenerate hypoelliptic Ornstein-Uhlenbeck operators. (English) Zbl 1454.35239
Summary: We describe the spectrum of degenerate hypoelliptic Ornstein-Uhlenbeck operators \(\mathcal{A} = \sum\nolimits_{i, j = 1}^n q_{ij} D_{ij} + \sum_{i, j = 1}^n b_{ij} x_j D_i\) in \(L^p(\mathbb{R}^n), 1 \leq p < + \infty\), and in \(C_0 (\mathbb{R}^n)\). We show that the spectrum of \(\mathcal{A}\) is the sum of \((- \infty, 0]\) and the spectrum of the drift term. Our result gives a complete picture of the spectral properties of Ornstein-Uhlenbeck operators in \(L^p\) spaces.
MSC:
| 35P05 | General topics in linear spectral theory for PDEs |
| 35J70 | Degenerate elliptic equations |
| 35K65 | Degenerate parabolic equations |
| 47D06 | One-parameter semigroups and linear evolution equations |
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