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So much polynomial development on each half-space Re (z) ::::: c. in addition, Op(t) relies holomorphically on t for Re t > O. normal references for far of the cloth at the derivation of spectral features, asymptotic expansions and analytic houses of spectral services are [A-P-S] and [Sh], specifically bankruptcy 2. to check the spectral features and their relation to the geometry and topology of X, you'll, for instance, take the average linked parabolic challenge as a place to begin. that's, contemplate the 'heat equation': (%t + p) u(x, t) = zero { u(x,O) = Uo(x), tP that's solved by way of the (heat) semi staff V(t) = e- ; particularly, u(·, t) = V(t)uoU· Assuming that V(t) is of hint category (which is assured, for example, if P has a good central symbol), it has a Schwartz kernel okay E COO(X x X x Rt,E* ®E), in the neighborhood given by means of 00 K(x,y; t) = L>-IAk(~k ® 'Pk)(X,y), k=O for an entire set of orthonormal eigensections 'Pk E COO(E). Taking the hint, we then receive: 00 tA Op(t) = trace(V(t)) = 2::>- okay. k=O Now, utilizing, e. g. , the Dunford calculus formulation (where C is an appropriate curve round a(P)) as a kick off point and the traditional for­ malism of pseudodifferential operators, one simply derives asymptotic expansions for the spectral services, for that reason for Op.

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