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- Aleman, Alexandru, et al.
(author)
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Uniform spectral radius and compact Gelfand transform
- 2006
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In: Studia Mathematica. - 0039-3223. ; 172:1, s. 25-46
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Journal article (peer-reviewed)abstract
- We consider the quantization of inversion in commutative p-normed quasi-Banach algebras with unit. The standard questions considered for such an algebra A with unit e and Gelfand transform x bar right arrow (x) over cap are: (i) Is K-nu = sup{parallel to(e - x)(-1)parallel to(p) : x is an element of A, parallel to x parallel to(p) <= 1, max (x) over cap <= nu} bounded, where nu is an element of (0, 1)? (ii) For which delta is an element of (0, 1) is C-delta = sup{parallel to x(-1)parallel to(p) : x is an element of A, parallel to x parallel to(p) <= 1, min (x) over cap >= delta} bounded? Both questions are related to a "uniform spectral radius" of the algebra, r(infinity)(A), introduced by Bjork. Question (i) has an affirmative answer if and only if r(infinity)(A) < 1, and this result is extended to more general nonlinear extremal problems of this type. Question (ii) is more difficult, but it can also be related to the uniform spectral radius. For algebras with compact Gelfand transform we prove that the answer is "yes" for all delta is an element of (0, 1) if and only if r(infinity)(A) = 0. Finally, we specialize to semisimple Beurling type algebras l(w)(p)(D), where 0 < p < 1 and D = N or D = Z. We show that the number r(infinity)(l(w)(p)(D)) can be effectively computed in terms of the underlying weight. In particular, this solves questions (i) and (ii) for many of these algebras. We also construct weights such that the corresponding Beurling algebra has a compact Gelfand transform, but the uniform spectral radius equals an arbitrary given number in (0, 1].
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- Bénéteau, Catherine, et al.
(author)
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Remarks on the Bohr Phenomenon
- 2013
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In: Computational Methods and Function Theory. - 1617-9447. ; 4, s. 1-19
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Journal article (peer-reviewed)abstract
- Bohr’s Theorem [10] states that analytic functions bounded by 1 in the unit disk have power series ∑anzn such that ∑|an||z|n
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- Dahlner, Anders
(author)
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A Wiener Tauberian Theorem for weighted convolution algebras of zonal functions on the automorphism group of the unit disc
- 2006
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In: Bergman Spaces and Related Topics in Complex Analysis, Proceedings. - 0271-4132 .- 1098-3627. ; 404, s. 67-102
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Conference paper (peer-reviewed)abstract
- Our main result gives necessary and sufficient conditions, in terms of Fourier transforms, for an ideal in the algebra L-1(G parallel to K, omega), the convolution algebra of zonal functions on the automorphism group on the unit disc which are integrable with respect to the weight; function omega, to be dense in the algebra, or to have as closure an ideal of functions whose set of common zeros of the Fourier transforms is a finite set on the boundary of the maximal ideal space of the algebra. The weights considered behave like Legendre functions of the first kind.
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- Dahlner, Anders
(author)
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Some Resolvent Estimates in Harmonic Analysis
- 2003
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Doctoral thesis (other academic/artistic)abstract
- This thesis contains three papers about three different estimates of resolvents in harmonic analysis. These papers are: Paper 1. ``A Wiener tauberian theorem for weighted convolution algebras of zonal functions on the automorphism group of the unit disc'' Paper 2. ``Uniform spectral radius and compact Gelfand transform'' Paper 3. ``Decomposable extension of the Cesáro operator on the weighted Bergman space and Bishop's property (b )'' The first paper concerns the classical resolvent transform for a commutative convolution algebra and its applications to tauberian theorems. The second paper concerns uniform estimates of resolvents and of inverses in commutative Banach (and quasi-Banach) algebras, in particular when the Gelfand transform is compact. In the last paper we consider the Cesáro operator and its action on weighted Bergman spaces. Using classical analysis we calculate the spectrum, produce estimates the resolvent and of its left inverse. The results are then used to retrieve operator theoretic information of the Cesáro operator on the weighted Bergman space.
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