37 research outputs found

### On powers of operators with spectrum in cantor sets and spectral synthesis

For $\xi \in \big( 0, \frac{1}{2} \big)$, let $E_{\xi}$ be the perfect
symmetric set associated with $\xi$, that is $E_{\xi} = \Big\{ \exp \Big( 2i
\pi (1-\xi) \sum_{n = 1}^{+\infty} \epsilon_{n} \xi^{n-1} \Big) : \,
\epsilon_{n} = 0 \textrm{ or } 1 \quad (n \geq 1) \Big\}$ and $b(\xi) =
\frac{\log{\frac{1}{\xi}} - \log{2}}{2\log{\frac{1}{\xi}} - \log{2}}.$ Let
$q\geq 3$ be an integer and $s$ be a nonnegative real number. We show that any
invertible operator $T$ on a Banach space with spectrum contained in $E_{1/q}$
that satisfies \begin{eqnarray*} & & \big\| T^{n} \big\| = O \big( n^{s} \big),
\,n \rightarrow +\infty \\ & \textrm{and} & \big\| T^{-n} \big\| = O \big(
e^{n^{\beta}} \big), \, n \rightarrow +\infty \textrm{ for some } \beta <
b(1/q),\end{eqnarray*} also satisfies the stronger property $\big\| T^{-n}
\big\| = O \big( n^{s} \big), \, n \rightarrow +\infty.$ We also show that this
result is false for $E_\xi$ when $1/\xi$ is not a Pisot number and that the
constant $b(1/q)$ is sharp. As a consequence we prove that, if $\omega$ is a
submulticative weight such that $\omega(n)=(1+n)^s, \, (n \geq 0)$ and $C^{-1}
(1+|n|)^s \leq \omega(-n) \leq C e^{n^{\beta}},\, (n\geq 0)$, for some
constants $C>0$ and $\beta < b( 1/q),$ then $E_{1/q}$ satisfies spectral
synthesis in the Beurling algebra of all continuous functions $f$ on the unit
circle $\mathbb{T}$ such that $\sum_{n = -\infty}^{+\infty} | \widehat{f}(n) |
\omega (n) < +\infty$

### Weighted Big Lipschitz algebras of analytic functions and closed ideals

We give the smallest closed ideal with given hull and inner factor for some
weighted big Lipschitz algebras of analytic functions

### The commutant of an operator with bounded conjugation orbits and C₀‒contractions

Let A be an invertible bounded linear operator on a complex Banach space, {A}′ the commutant of A and Bᴀ the set of all operators T such that 〖sup〗_(ₙ≥₀)∥AⁿTA⁻ⁿ ∥ < +∞. Equality {A}′ = Bᴀ was studied by many authors for differents classes of operators. In this paper we investigate a local version of this equality and the case where A is a C₀–contraction.peerReviewe

### Unitary equivalence to truncated Toeplitz operators

In this paper we investigate operators unitarily equivalent to truncated
Toeplitz operators. We show that this class contains certain sums of tensor
products of truncated Toeplitz operators. In particular, it contains arbitrary
inflations of truncated Toeplitz operators; this answers a question posed by
Cima, Garcia, Ross, and Wogen

### Closed ideals with countable hull in algebras of analytic functions smooth up to the boundary

We denote by T the unit circle and by D the unit disc. Let B be a semi-simple unital commutative Banach algebra of functions holomorphic in D and continuous on D, endowed with the pointwise product. We assume that B is continously imbedded in the disc algebra and satisfies the following conditions: (H1) The space of polynomials is a dense subset of B.(H2) limn→+∞ kz nk1/nB = 1.(H3) There exist k ≥ 0 and C > 0 such that˛˛1 − ˛k‚‚f‚‚B ≤ C‚‚(z − λ)f‚‚B, (f ∈ B, < 2). When B satisfies in addition the analytic Ditkin condition, we give a complete characterisation of closed ideals I of B with countable hull h(I), where h(I) = ˘z ∈ D : f(z) = 0, (f ∈ I)¯.Then, we apply this result to many algebras for which the structure of all closed ideals is unknown. We consider, in particular, the weighted algebras ℓ1(ω) and L1(R+, ω)

### Explicit bounds for separation between Oseledets subspaces

We consider a two-sided sequence of bounded operators in a Banach space which
are not necessarily injective and satisfy two properties (SVG) and (FI). The
singular value gap (SVG) property says that two successive singular values of
the cocycle at some index $d$ admit a uniform exponential gap; the fast
invertibility (FI) property says that the cocycle is uniformly invertible on
the fastest $d$-dimensional direction. We prove the existence of a uniform
equivariant splitting of the Banach space into a fast space of dimension $d$
and a slow space of co-dimension $d$. We compute an explicit constant lower
bound on the angle between these two spaces using solely the constants defining
the properties (SVG) and (FI). We extend the results obtained in the
finite-dimensional case for bijective operators and the results obtained by
Blumenthal and Morris in the infinite-dimensional case for injective
norm-continuous cocycles, in the direction that the operators are not required
to be globally injective, that no dynamical system is involved, and no
compactness of the underlying system or smoothness of the cocycle is required.
Moreover, we give quantitative estimates of the angle between the fast and slow
spaces that are new even in the case of finite-dimensional bijective operators
in Hilbert spaces

### Problemes de synthese spectrale dans certaines algebres a poids sur le cercle et la droite et applications

SIGLEINIST T 74308 / INIST-CNRS - Institut de l'Information Scientifique et TechniqueFRFranc

### On polynomially bounded operators acting on a Banach space

By the Von Neumann inequality every contraction on a Hilbert space is polynomially bounded. A simple example shows that this result does not extend to Banach space contractions. In this paper we give general conditions under which an arbitrary Banach space contraction is polynomially bounded. These conditions concern the thinness of the spectrum and the behaviour of the resolvent or the sequence of negative powers. To do this we use techniques from harmonic analysis, in particular, results concerning thin sets such as Helson sets, Kronecker sets and sets that satisfy spectral synthesis