Boundary spectral estimates for semiclassical Gevrey operators

We obtain the spectral and resolvent estimates for semiclassical pseudodifferential operators with symbol of Gevrey- s regularity, near the boundary of the range of the principal symbol. We prove that the boundary spectrum free region is of size {\mathcal{O}}(h^{1-\frac{1}{s}}) where the resolvent is at most fractional exponentially large in h , as the semiclassical parameter h\to 0^{+} . This is a natural Gevrey analogue of a result by N. Dencker, J. Sjöstrand, and M. Zworski in the C^{\infty} and analytic cases.