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.
The Black-Scholes semigroup is studied on spaces of continuous functions on (0, ∞) which may grow at both 0 and at ∞, which is important since the standard initial value is an unbounded function. We prove that in the Banach spaces Y s,τ := {u ∈ C((0, ∞)) : lim x→∞ u(x) 1 + x s = 0, lim x→0 u(x) 1 + x −τ = 0} with norm u Y s,τ = sup x>0 u(x) (1+x s)(1+x −τ) < ∞, the Black-Scholes semigroup is strongly continuous and chaotic for s > 1, τ ≥ 0 with sν > 1, where √ 2ν is the volatility. The proof relies on the Godefroy-Shapiro hypercyclicity criterion.
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