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4. Let x∗ ∈ X ∗ and λ ∈ C satisfy T ∗ (x∗ ) = λx∗ . Also let x ∈ X, and assume that J(x, T ) has non-empty interior. Prove that the following holds (where we set r := | x∗ , x |): if |λ| < 1 then x∗ , z = 0 for every z in the interior of J(x, T ); if |λ| = 1 then | x∗ , z | = r for every z in the interior of J(x, T ); if |λ| > 1 then x∗ , Q(T )x = 0 for any polynomial Q. 5. Show that if T is cyclic and J(x, T ) has non-empty interior for some T -cyclic vector x then P (T ) has dense range for any non-zero polynomial P .

Moreover, writing f (w) = and using Schwarz’s inequality, one can estimate f as follows: ⎞⎛ ⎞ ⎛ 2 f 2H 2 |an |2 ⎠⎝ n2 |w|2(n−1) ⎠ ≤ |f (w)|2 ≤ ⎝ (1 − |w|2 )3 n≥1 ∞ 0 an wn n≥1 for any w ∈ D. 8) 26 Hypercyclic and supercyclic operators Here and below, C = C(f, φ) is a constant which does not depend on n and may change from line to line. We now have to make some calculations. Since φ is parabolic and not an automorphism, the associated map ψ is a translation, ψ(s) = s + ia, with Re(a) > 0. An easy computation yields (2 − a)z + a · φ(z) = −az + 2 + a Set ψn := ψ ◦ · · · ◦ ψ (n ≥ 1).

Finally, we show that any Hilbert space operator may be written as the sum of two hypercyclic operators. 1 Mixing operators When considering a strategy for constructing hypercyclic operators on an arbitrary “abstract” separable Banach space, one certainly has to keep in mind that some spaces have very few operators. Indeed, it is now well known that there exist (infinitedimensional) Banach spaces on which any operator has the form λI + S, where S is strictly singular (for example, the so-called hereditarily indecomposable spaces; see Chapter 6 for some results concerning hypercyclic operators on such spaces).

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Analyse Mathématique - Fonctions d'une variable - Tomo II by G. Chilov

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