A spectral multiplier theorem for non-self-adjoint operators

Ouhabaz, El Maati

doi:10.1090/S0002-9947-09-04754-0

A spectral multiplier theorem for non-self-adjoint operators
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Trans. Amer. Math. Soc. 361 (2009), 6567-6582 Request permission

Abstract:

We prove a spectral multiplier theorem for non-self-adjoint operators. More precisely, we consider non-self-adjoint operators $A: D(A) \subset L^2 \to L^2$ having numerical range in a sector $\Sigma (w)$ of angle $w,$ and whose heat kernel satisfies a Gaussian upper bound. We prove that for every bounded holomorphic function $f$ on $\Sigma (w),$ $f(A)$ acts on $L^p$ with $L^p-$norm estimated by the behavior of a finite number of derivatives of $f$ on the boundary of $\Sigma (w).$

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