## Existence of functional calculi over some algebras of pseudo-differential operators and related topics |

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2nitk According to lemma according to theorem amplitudes assert asymptotic expansion Banach algebra Banach space basic calculus belongs Bochner integral Borel measurable bounded sets C(A+X calculus is continuous commuting self-adjoint operators compact support complete functional calculus complete space concludes the proof consider the algebra continuous derivatives continuous operator converges deduce definition denotes derivatives of orders differential operators e-2nitk e-2nixc entire function erator exists a unique extended to continuous fact FjCy.u follows functions f Given f e Hilbert space hypothesis of regularity integral Lf(x linear modulo Moreover n-tuple n+N+l-j norm operator from H order to prove order zero P(BX proof of theorem pseudo-differential operators R.S. Strichartz readily seen real function regularizing operator seminorms spectrum suffices sup S(t sup x,y supp suppose symbol tain term of order theorem 1.1 tinuous topology tuple vanishes variables verifying the adequate write