Spectral synthesis of Jordan operators
Let \(X\) be a complex Banach space and \(T\) a bounded linear operator on \(X\). A vector \(x\) in \(X\) is called a root vector for \(T\) if there exists a complex number \(\lambda\) and a positive integer \(n\) such that \((T-\lambda I)^nx= 0\). We say that \(T\) is complete if the root vectors for \(T\) have dense linear span in \(X\). Let \(Y\) be a closed linear subspace of \(X\) with \(TY\subseteq Y\). If for every closed invariant subspace \(Y\) for \(T\), \(T|Y\) (the restriction of \(T\) to \(Y\)) is complete, then \(T\) is said to admit spectral synthesis. \textit{J. Wermer} [Proc. Am. Math. Soc. 3, 270-277 (1952; Zbl 0046.33704)] studied the problem of spectral synthesis for normal operators on a Hilbert space. For a review of progress on the general problem of spectral synthesis up to 1984, see \textit{N. K. Nikol'skij} [``The present state of the analysis-synthesis problem. I'', in: ``Fifteen papers on functional analysis'', Am. Math. Soc. Transl. (2) 124, 97-129 (1984; Zbl 0568.47004)]. In the present paper, the author seeks necessary and sufficient conditions for a Jordan operator to admit spectral synthesis.
scientific article; zbMATH DE number 1533190
scientific article