Spectral distributions and generalization of Stone's theorem

Fractional powers of momentum of a spectral distribution

“…Theorem 2.6 (Stone's generalized theorem, see [5]). Let A be a linear densely defined operator on a Banach space X.…”

“…A spectral distribution is a ^-functional calculus of the momentum B (see below), and in [5] we have shown how to define f(B) for any function / e W°°(V), where F is a neighborhood of o(B). This allowed us to define the ath power of (B + e) (a e C, o(B) c [0, +oo[, and e > 0) as the image of the function t .-► (t + e)a and finally Ba = lime^0(B + e)a • In [5] the authors introduced spectral distributions and gave a generalization of Stone's theorem to a Banach space which connects the momentum of a spectral distribution (see [5] and [16]) to the generator of a temperate integrated group (see [1]) and to the generator of a smooth distribution group (see [3] and [4]).…”

“…This allowed us to define the ath power of (B + e) (a e C, o(B) c [0, +oo[, and e > 0) as the image of the function t .-► (t + e)a and finally Ba = lime^0(B + e)a • In [5] the authors introduced spectral distributions and gave a generalization of Stone's theorem to a Banach space which connects the momentum of a spectral distribution (see [5] and [16]) to the generator of a temperate integrated group (see [1]) and to the generator of a smooth distribution group (see [3] and [4]). An extensive number of applications of this theorem to the fundamental Cauchy problem in Lp(Rn) is given in [14] and [16].…”

A functional calculus for unbounded generalized scalar operators on Banach spaces

“…A more general result was obtained by Andersson and Berndtsson [2002]. The case of a single generator of a bounded strongly continuous group was treated in [Balabane et al 1993]. For more details and references, see [deLaubenfels 1994].…”

Funkcionalni racuni za n-torke komutirajucih neogranicenih operatora

“…Rezultate koji su vezani za generator ograniqene jako neprekidne grupe dobili su Balabane i grupa autora u [5]. Vixe detaƩa vezanih za dobijene rezultate, kao i Ƭihov istorijat, mogu se na i u [11].…”