Jean Esterle

Theorems of Katznelson-Tzafriri type for contractions

For T, a contraction on a Hilbert space, and ƒ, a function in the disc algebra, we show that, if ƒ ≡ 0 on SpT ∩ Γ then limn → ∞∥ƒ(T)Tn∥ = 0. The same method gives a new proof of recent results of Katznelson-Tzafriri. Several applications are given.

Theorems of Gelfand-Mazur type and continuity of epimorphisms from b(K)

We prove that a commutative unital Banach algebra which is a valuation ring must reduce to the field of complex numbers, which implies that every homomorphism from l∞ onto a Banach algebra is continuous. We show also that if b ϵ [b Rad B]− for some nonnilpotent element b of the radical of a commutative Banach algebra B, then the set of all primes of B cannot form a chain, and we deduce from this result that every homomorphism from b(K) onto a Banach algebra is continuous.

Uniqueness, strong forms of uniqueness and negative powers of contractions

for every e 6∈ E. A set of multiplicity is a set which is not a set of uniqueness. Cantor [5] showed in 1870 that finite sets (in particular the empty set) are sets of uniqueness, and more generally that “reducible” countable sets are sets of uniqueness. It was only in 1908 that Young [32] was able to prove that all countable sets are sets of uniqueness. We will be only interested here in closed sets (we refer to Kechris–Louveau [20] for a discussion of the notion of uniqueness for nonclosed sets). In this case E is a set of multiplicity if and only if there exists a nonzero pseudofunction, i.e. a distribution S (see Section 2) such that Ŝ(n)−−−−→ |n|→∞ 0 ,

Distance near the origin between elements of a strongly continuous semigroup

AbstractSet

DISTANCE ENTRE PUISSANCES D'UNE UNIT APPROCHÉE BORNÉE

\theta (s/t): = (s/t - 1)(t/s)^{\frac{{s/t}}{{s/t - 1}}} = (s - t)\frac{{t^{t/(s - t)} }}{{s^{s/(s - t)} }}

Apostol’s bilateral weighted shifts are hyper-reflexive

if 0<t<s. The key result of the paper shows that if (T (t))t>0 is a nontrivial strongly continuous quasinilpotent semigroup of bounded operators on a Banach space then there exists δ>0 such that ║T(t)-T(s)║>θ(s/t) for 0<t<s≤δ. Also if (T(t))t>0 is a strongly continuous semigroup of bounded operators on a Banach space, and if there exists η>0 and a continuous functiont→s(t) on [0, ν], satisfyings(0)=0, and such that 0<t<s(t) and ║T(t)-T(s(t))║<θ(s/t) fort∈(o, η], then the infinitesimal generator of the semigroup is bounded. Various examples show that these results are sharp.

A Banach Algebra Approach to the Weak Spectral Mapping Theorem for Locally Compact Abelian Groups

The conjecture that every algebra norm || • || on C(X) is equivalent to the uniform norm arises naturally from a theorem of Kaplansky in 1949 that necessarily 11/11 > \f\x ( ƒ e C(X)): see [9, 10.1]. The seminal paper on the automatic continuity of homomorphisms from C(X) is the 1960 paper of Bade and Curtis [2] in which, for example, it is proved that there is a discontinuous homomorphism from C(X) if and only if there is a radical homomorphism, a nonzero homomorphism from a maximal ideal of C(X) into a commutative radical Banach algebra. In 1967, it was proved by Johnson that every homomorphism from certain noncommutative C*-algebras is continuous: see [9, 12.4]. Sinclair proved recently that the existence of a discontinuous homomorphism is equivalent to the existence of an algebra norm on C(X)/I for some nonmaximal prime ideal I of C(X) [9, 11.7], and this was proved independently in [4]. It follows from the work of each of the present authors that such a norm exists provided \C(X)/I\ = Kj. Assuming CH, such an ideal exists for each X, and every nonmaximal prime ideal has this property if X is separable, but, if X is not separable, there may exist a prime ideal / such that C(X)/J is not normable [4].

Injection De Semi-Groupes Divisibles Dans Des Algèbres De Convolution Et Construction D'homomorphismes Discontinus De C(K)

Let