In  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 ( and ) and analytic continuation of Dirichlet series () to the spectral synthesis of diagonal operators.
Polynomials construction invariant subspaces diagonal operators spectral synthesis
30B10 30B50 47B36 47B38
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J. M. Anderson, D. Khavinson, and H. S. Shapiro, “Analytic continuation of Dirichlet series”, Revista Matematica Iberoamericana, 11(2), 453–476, 1995.CrossRefMATHMathSciNetGoogle Scholar
I. Deters, “A connection between operator topologies, polynomial interpolation, and synthesis of diagonal operators”, J.Math. Anal. Appl., 350, 354–359, 2009.CrossRefMATHMathSciNetGoogle Scholar
I. Deters and S. M. Seubert, “Cyclic vectors of diagonal operators on the space of functions analytic on a disk”, J.Math. Anal. Appl., 334, 1209–1219, 2007.CrossRefMATHMathSciNetGoogle Scholar
I. Deters and S. M. Seubert, “An application of entire function theory to the synthesis of diagonal operators on the space of entire functions”, Houston Journal Of Mathematics, 38, 201–207, 2012.MATHMathSciNetGoogle Scholar
A. S. B. Holland, Introduction To The Theory Of Entire Functions (Academic Press, 1973).MATHGoogle Scholar
G. Latta and G. Polya, Complex Variables (JohnWiley And Sons, Inc., 1973)Google Scholar
J. Marín Jr. and S. M. Seubert, “Cyclic vectors of diagonal operators on the space of entire functions”, J. Math. Anal. Appl., 320, 599–610, 2006.CrossRefMATHMathSciNetGoogle Scholar
W. Rudin, Real And Complex Analysis 3rd Edition (McGraw-Hill, 1987).Google Scholar
S. M. Seubert, “Spectral synthesis of diagonal operators on the space of entire functions”, Houston Journal of Mathematics, 34(3), 807–816, 2008.MATHMathSciNetGoogle Scholar
S. M. Seubert and J. Gordon Wade, “Spectral synthesis of diagonal operators and representing systems on the space of entire functions”, J.Math. Anal. Appl., 344, 9–16, 2008.CrossRefMATHMathSciNetGoogle Scholar
R. V. Sibilev, “Uniqueness theorem for Wolff-Denjoy series”, St. PetersburgMath. J., 7, 145–68, 1996.MathSciNetGoogle Scholar