On L^p -spectra and essential spectra of second-order elliptic operators

Weighted norm estimates and Lp-spectral independence of linear operators

“…In this subsection we present a result of [15] on L p -spectral independence for closed operators acting in L p (Ω) for some open set Ω ⊂ R N . Choosing an appropriate semi-metric d on R N we show that this result is also a consequence of our main theorem.…”

“…This corresponds to the semi-metric d(x, y) := |ϕ(x)−ϕ(y)| on R N . In [15], L p -spectral independence was proved for closed (not necessarily selfadjoint) operators assuming a weighted norm estimate for a single resolvent. In Subsection 2.2 we discuss properties of L 1 -regular functions and show that Theorem 1 of the present paper extends [15,Thm.…”

“…In [15], this theorem was applied to (not necessarily selfadjoint) strictly elliptic operators in divergence form with lower order terms. We show that, for the semi-metric on Ω ′ := R N defined by d(x, y) := |ϕ(x)−ϕ(y)| ∞ and for µ := |·| the Lebesgue measure, assumptions (1) The following lemma in particular shows that (2) and (3) hold.…”

“…One method that has been widely used in the context of the L p -theory of elliptic operators relies on the exploitation of certain bounds, especially of Gaussian type, that hold for the kernels of associated integral operators, namely semigroup or resolvent operators ( [9], [1], [20], [7], [10], [13], [14]). If these operators are not integral operators, which is in general the case if p > 1 and q < ∞, then r-independence of the spectrum still holds under the assumption of certain weighted norm estimates ( [15]- [18]). It is the latter kind of assumptions, i.e., weighted norm estimates, that we shall use in this paper.…”

Trace estimates and invariance of the essential spectrum

“…Various sufficient conditions exist under which the resolvent difference (H + 1) −1 − (H + 1) −1 is compact and, subsequently, H andH have the same essential spectrum. See [6,7,8,5] and references therein. These conditions typically involve some decay of the differencẽ a ij − a ij of the respective coefficients near infinity.…”