Second order elliptic operators with essential spectrum(0,∞)on lp

On Lp -spectra and essential spectra of second-order elliptic operators

“…Then by [33] the semigroup extends to a C 0semigroup on L p , for p 0 < p < q 0 p 0 < p < 1 if q 0 1. Now representing the resolvent via the semigroup, by (21) we show that the condition (ii) of Theorem 1 is ful®lled for 1=p À 1=q < 2m=d , with p 0 < p < q < q 0 . Therefore by Theorem 1 we conclude that the spectrum of T p is p-independent.…”

“…Moreover, (21) holds in a certain interval around 2, for higher-order superelliptic operators for which the L p -theory is developed in [10, § 7], in the absence of pointwise Gaussian estimates.…”

“…Theorem 3 is a direct generalization of the main result in [21] and of Example 1 from [13]. In the latter paper only self-adjoint operators are studied, while in the former one there is a requirement of ultracontractivity and the conditions on the coef®cients of the operator are much more restrictive.…”

“…This problem was addressed by M. S. Birman [7], M. Reed and B. Simon [23], M. Schechter [24], D. E. Edmunds and W. D. Evans [11], R. Hempel [13], E. M. Ouhabaz [21], and in [22] in a more general setting.…”

“…Applying their results to our operator leads to additional restrictions. In both [21] and [22] unbounded coef®cients are treated by an approximation method which requires strict ellipticity of the matrix, while the approach described here avoids this. In the case of Schro Èdinger operators our result is not optimal, but we did not pursue generalizations in order to preserve simplicity.…”