An Interesting Infinite Series and Its Implications to Operator Theory

Abstract

The following is a discussion regarding a specific class of operators acting on the space of entire functions, denoted $H(\mathbb{C})$. A diagonal operator $D$ on $H(\mathbb{C})$ is defined to be a continuous linear map, sending $H(\mathbb{C})$ into $H(\mathbb{C})$, that has the monomials $z^n$ as its eigenvectors and $\{\lambda_n\}$ as the corresponding eigenvalues. A closed subspace $M$ is invariant for $D$ if $Df\in M$ for all $f\in M$. The study of invariant subspaces is a popular topic in modern operator theory. We observe that the closed linear span of the orbit, which we write $\overline{\mbox{span}}\{D^kf:k\geq0\}=\overline{\mbox{span}}\{\sum^{\infty}_{n=0}a_n\lambda_n^kz^n:k\geq0\}$, is the smallest closed invariant subspace for $D$ containing $f$. If every invariant subspace for a diagonal operator $D$ on $H(\mathbb{C})$ can be expressed as a closed linear span of some subset of the eigenvectors of $D$, we say that $D$ admits spectral synthesis on $H(\mathbb{C})$. Until recently, it was not known whether or not every diagonal operator on $H(\mathbb{C})$ admitted spectral synthesis. This article focuses on using techniques from calculus and linear algebra to construct a class of operators which fail spectral synthesis on $H(\mathbb{C})$. If the reader is not familiar with the operator theory definitions provided in the background, he or she can still appreciate the construction of an interesting infinite series relying on properties of logarithms, various convergence tests, and Cramer's Rule.