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We construct a functional calculus, g H> g(A), for functions, g, that are the sum of a Stieltjes function and a nonnegative operator monotone function, and unbounded linear operators, A, whose resolvent set contains (-oo, 0), with [\\r(r + A)' 1 1| | r > 0} bounded. For such functions g, we show that -g(A) generates a bounded holomorphic strongly continuous semigroup of angle 6, whenever -A does.We show that, for any Bernstein function f,-f(A) generates a bounded holomorphic strongly continuous semigroup of angle 7r/2, whenever -A does.We also prove some new results about the Bochner-Phillips functional calculus. We discuss the relationship between fractional powers and our construction.1991 Mathematics subject classification (Amer. Math. Soc): 47 A 60, 47 B 44, 47 D 05.

For L equal to the unilateral or bilateral shift on a weighted sequence space, we characterize, in terms of f and the weight function, those f holomorphic on the spectrum of L for which f (L) is a chaotic operator. For B equal to d/dx, the generator of left translation, on weighted L p spaces on [0, ∞) or R, we similarly characterize those polynomials Q for which the differential operator Q(B) generates a chaotic semigroup.

In this paper we generalize the notion of hypercyclic and chaotic semigroups to families of unbounded operators. We study this concept within the frameworks of C-regularized semigroups and of regular distribution semigroups. We then apply our results to unbounded semigroups generated by differential operators with constant coefficients in weighted spaces and to the unbounded semigroup {(−∆) t } t≥0 , where ∆ is the Laplacian operator.

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