Constructing polynomials of minimal growth

Real and Complex Analysis
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Abstract

In [2] a cyclic diagonal operator on the space of functions analytic on the unit disk with eigenvalues (λn) is shown to admit spectral synthesis if and only if for each j there is a sequence of polynomials (pn) such that limn→∞pn(λk) = δj,k and lim supn→∞ supk>j |pn(λk)|1/k ≤ 1. The author also shows, through contradiction, that certain classes of cyclic diagonal operators are synthetic. It is the intent of this paper to use the aforementioned equivalence to constructively produce examples of synthetic diagonal operators. In particular, this paper gives two different constructions for sequences of polynomials that satisfy the required properties for certain sequences to be the eigenvalues of a synthetic operator. Along the way we compare this to other results in the literature connecting polynomial behavior ([4] and [9]) and analytic continuation of Dirichlet series ([1]) to the spectral synthesis of diagonal operators.

Keywords

Polynomials construction invariant subspaces diagonal operators spectral synthesis 

MSC2010 numbers

30B10 30B50 47B36 47B38 

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Copyright information

© Allerton Press, Inc. 2014

Authors and Affiliations

  1. 1.Farmers Insurance GroupAkronUSA

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