Atzmon, Aharon An operator on a Fréchet space with no common invariant subspace with its inverse. (English) Zbl 0552.47006 J. Funct. Anal. 55, 68-77 (1984). The main result of this paper is the construction of a Fréchet-Montel space F of formal Laurent series \(\sum^{\infty}_{-\infty}a_ nz^ n\) in which \((z_ n)^{\infty}_{-\infty}\) is a Schauder basis, such that the operators of formal multiplication by z and \(z^{-1}\) are continuous on F and have no common invariant subspace. This answers in the negative the problem of existence of hyperinvariant subspaces for operators on general Fréchet spaces [the problem for Banach spaces remains open]. The strong dual of the above Fréchet space F also yields an example of an abelian complete locally convex algebra, with unit, which does not have any nontrivial closed ideal. Reviewer: H.Jarchow Cited in 5 Documents MSC: 47A15 Invariant subspaces of linear operators 46A11 Spaces determined by compactness or summability properties (nuclear spaces, Schwartz spaces, Montel spaces, etc.) 46H10 Ideals and subalgebras 46A35 Summability and bases in topological vector spaces 46A45 Sequence spaces (including Köthe sequence spaces) Keywords:Fréchet-Montel space; formal Laurent series; Schauder basis; operators of formal multiplication; common invariant subspace; existence of hyperinvariant subspaces PDFBibTeX XMLCite \textit{A. Atzmon}, J. Funct. Anal. 55, 68--77 (1984; Zbl 0552.47006) Full Text: DOI References: [1] Atzmon, A., Operators which are annihilated by analytic functions and invariant subspaces, Acta Math., 144, 27-63 (1980) · Zbl 0449.47007 [3] Chevreau, B.; Pearcy, C. M.; Shields, A. L., Finitely connected domains of \(G\) representations of \(H^∞(G)\), and invariant subspaces, J. Operator Theory, 6, 375-405 (1981) · Zbl 0525.47004 [4] Edwards, R. E., Functional Analysis. Theory and Applications (1965), Holt, Reinhart & Winston: Holt, Reinhart & Winston New York · Zbl 0182.16101 [5] Gellar, R.; Herrero, D., Hyperinvariant subspaces of bilateral weighted shifts, Indiana Univ. Math. J., 23, 771-790 (1973/1974) · Zbl 0253.46128 [6] Kas’yanyuk, S. A., On functions of class \(A\) and \(H_δ\), in an annulus, Mat. Sb. (N.S.), 42, 84, 301-326 (1957), [Russian] · Zbl 0090.28703 [7] Körber, K. H., Die invarianten Tailräume der stefigen Endomorphismen von ω, Math. Ann., 182, 95-103 (1969) [8] Naimark, M. A., Normed Rings (1970), Wolters: Wolters Groningen · Zbl 0218.46042 [9] Sarason, D., The \(H^p\) spaces of an annulus, Mem. Amer. Math. Soc., 56 (1965) · Zbl 0127.07002 [10] Shields, A. L., A note on invariant subspaces, Michigan Math. J., 17, 231-233 (1970) · Zbl 0187.37806 [11] Shields, A. L., Weighted Shift Operators and Analytic Function Theory, (Topics in Operator Theory (1974), Amer. Math. Soc: Amer. Math. Soc Providence, R.I), 49a-128a [12] Wilansky, A., Functional Analysis (1964), Ginn (Blaisdell): Ginn (Blaisdell) Boston · Zbl 0136.10603 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.