Regularized functional calculi, semigroups, and cosine functionsfor pseudodifferential operators

Abstract

Abstract. Let iA j (1 ≤ j ≤ n) be generators of commuting bounded strongly continuous groups, A ≡ (A 1 , A 2 , ..., A n ). We show that, when f has sufficiently many polynomially bounded derivatives, then there exist k, r > 0 such that f (A) has a (1+|A| 2 ) −r -regularized BC k (f (R n )) functional calculus. This immediately produces regularized semigroups and cosine functions with an explicit representation; in particular, when IntroductionIn finite dimensions, the Jordan canonical form for matrices guarantees that, although a linear operator may not be diagonalizable, which is equivalent to having a BC(C) functional calculus, it will be generalized scalar, that is, have a BC k (C) functional calculus, for some k; specifically, k may be chosen to be n − 1, where n is the order of the largest Jordan block.In infinite dimensions, even a bounded linear operator on a Hilbert space may fail to be generalized scalar; consider the left shift on 2 .Our favorite unbounded operators fail to be generalized scalar, on Banach spaces that are not Hilbert spaces. The operator i d dx , on L 2 (R), is selfadjoint and thus has a BC(R) functional calculus. However, on L p (R), p = 2, it does not have a BC m (R) functional calculus, for any nonnegative integer m; that is, it is not even generalized scalar (see [2, Lemma 5.3]).

Matrices of operators and regularized cosine functions

“…The wellposedness of systems (1.1) and (1.2) is discussed in Section 3. In the special case N = 1, our results improve the corresponding results in [2,12].…”

“…It thus seems to be important to study their application to (1.2). Up to now, only is the case N = 1 considered by some authors [2,12].…”

Spectral multiplier theorems via $$H^\infty $$ H ∞ calculus and R-bounds

“…Operators as in 7.6 are considered in [16], where they are called regularized semigroup, and in particular in [52,Sections 7.3,7.4.2], [49,Theorems 2,3] in connection with spectral multipliers. Moreover, the link with analytic semigroups on the right half plane is studied in [9, Theorems 2.2, 2.3].…”

“…They are well-defined operators at least on the domain D 0 = R(e −A ), (which is dense in X by the analyticity of the dual semigroup (e −tA ) ′ ), by the formula (1 + |s|A) −α e isA x = (1 + |s|A) −α e (is−1)A y for x = e −A y ∈ D 0 . Operators as in 7.6 are considered in [16], where they are called regularized semigroup, and in particular in [52,Sections 7.3,7.4.2], [49,Theorems 2,3] in connection with spectral multipliers. Moreover, the link with analytic semigroups on the right half plane is studied in [9, Theorems 2.2, 2.3].…”