37 research outputs found

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

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    For ξ(0,12)\xi \in \big( 0, \frac{1}{2} \big), let EξE_{\xi} be the perfect symmetric set associated with ξ\xi, that is Eξ={exp(2iπ(1ξ)n=1+ϵnξn1):ϵn=0 or 1(n1)}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(ξ)=log1ξlog22log1ξlog2.b(\xi) = \frac{\log{\frac{1}{\xi}} - \log{2}}{2\log{\frac{1}{\xi}} - \log{2}}. Let q3q\geq 3 be an integer and ss be a nonnegative real number. We show that any invertible operator TT on a Banach space with spectrum contained in E1/qE_{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 Tn=O(ns),n+.\big\| T^{-n} \big\| = O \big( n^{s} \big), \, n \rightarrow +\infty. We also show that this result is false for EξE_\xi when 1/ξ1/\xi is not a Pisot number and that the constant b(1/q)b(1/q) is sharp. As a consequence we prove that, if ω\omega is a submulticative weight such that ω(n)=(1+n)s,(n0)\omega(n)=(1+n)^s, \, (n \geq 0) and C1(1+n)sω(n)Cenβ,(n0)C^{-1} (1+|n|)^s \leq \omega(-n) \leq C e^{n^{\beta}},\, (n\geq 0), for some constants C>0C>0 and β<b(1/q),\beta < b( 1/q), then E1/qE_{1/q} satisfies spectral synthesis in the Beurling algebra of all continuous functions ff on the unit circle T\mathbb{T} such that n=+f^(n)ω(n)<+\sum_{n = -\infty}^{+\infty} | \widehat{f}(n) | \omega (n) < +\infty

    Weighted Big Lipschitz algebras of analytic functions and closed ideals

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    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

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    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

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    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

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    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

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    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 dd admit a uniform exponential gap; the fast invertibility (FI) property says that the cocycle is uniformly invertible on the fastest dd-dimensional direction. We prove the existence of a uniform equivariant splitting of the Banach space into a fast space of dimension dd and a slow space of co-dimension dd. 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

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

    On polynomially bounded operators acting on a Banach space

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    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