Local description of closed ideals and submodules of analytic functions of one variable.~I

I. F. Krasichkov-Ternovskii

Izvestiya. Mathematics , Tome 14 (1980) no. 1, pp. 41-60.

Résumé

Let $P$ be a topological module (over the ring of polynomials) of vector-valued functions $f\colon G\to\mathbf C^q$, holomorphic in a domain $G\subset\mathbf C$. A closed submodule $I\subset P$ is local (that is, uniquely determined by the collection $I_\lambda$, $\lambda\in G$, of its localized submodules) if and only if $I$ is stable and saturated. A submodule is said to be stable if it admits division by binomials: $f\in I$, $\frac f{z-\lambda}\in I_\lambda\Rightarrow\frac f{z-\lambda}\in I$. Being saturated amounts to possessing sufficiently many elements. Bibliography: 26 titles.

Bibliographie

[1] Schwartz L., “Théorie générate des fonctions moyenne-périodique”, Ann. Math., 48:4 (1947), 857–929 | DOI | MR | Zbl

[2] Krasichkov-Ternovskii I. F., “Invariantnye podprostranstva analiticheskikh funktsii. I: Spektralnyi sintez na vypuklykh oblastyakh”, Matem. sb., 87(129):4 (1972), 459–488

[3] Krasichkov-Ternovskii I. F., “Invariantnye podprostranstva analiticheskikh funktsii. II: Spektralnyi sintez na vypuklykh oblastyakh”, Matem. sb., 88(130):1 (1972), 3–30

[4] Krasichkov-Ternovskii I. F., “Invariantnye podprostranstva analiticheskikh funktsii. III: O rasprostranenii spektralnogo sinteza”, Matem. sb., 88(130):3 (1972), 331–352

[5] Nikolskii N. K., Izbrannye zadachi vesovoi approksimatsii i spektralnogo analiza, Tr. Matem. in-ta im. V. A. Steklova AN SSSR, SKhKh, 1974 | MR

[6] Nikolskii N. K., “Invariantnye podprostranstva v teorii operatorov i teorii funktsii”, Matematicheskii analiz. Itogi nauki, 12, 1974, 199–412

[7] Rudin W., “The closed ideals in an algebra of analytic functions”, Canad. J. Math., 9:3 (1957), 426–434 | MR | Zbl

[8] Korenblyum B. I., “Zamknutye idealy koltsa $A^n$”, Funktsionalnyi analiz i ego prilozheniya, 6:3 (1972), 38–52 | MR

[9] Taylor B. A., Williams D. L., “Idelas in rings of analytic functions with smooth boundary values”, Canad. J. Math., 12:6 (1970), 1266–1283 | MR

[10] Beurling A., “A critial topology in harmonic analysis on semigroups”, Acta Math., 112:3,4 (1964), 215–228 | DOI | MR | Zbl

[11] Rashevskii P. K., “O zamknutykh idealakh v odnoi schetno-normirovannoi algebre tselykh analiticheskikh funktsii”, Dokl. AN SSSR, 162:3 (1965), 513–515

[12] Krasichkov I. F., “O zamknutykh idealakh v lokalno vypuklykh algebrakh tselykh funktsii, I”, Izv. AN SSSR. Ser. matem., 31 (1967), 37–60 | Zbl

[13] Krasichkov I. F., “O zamknutykh idealakh v lokalno vypuklykh algebrakh tselykh funktsii, II”, Izv. AN SSSR. Ser. matem., 32 (1968), 1024–1032 | Zbl

[14] Krasichkov I. F., “O zamknutykh idealakh v lokalno vypuklykh algebrakh tselykh funktsii. Algebry minimalnogo tipa”, Sib. matem. zh., 9:1 (1968), 77–96 | Zbl

[15] Nikolskii N. K., “Zamknutye idealy v nekotorykh algebrakh tselykh funktsii”, Sib. matem. zh., IX:1 (1968), 211–215

[16] Taylor B. A., “Locally convex spaces of entire functions”, Entire functions and related parts of analysis, Proc. Symp. Pure Math., 11, Amer. Math. Soc., 1968

[17] Shamoyan F. A., “O zamknutykh idealakh v odnoi algebre bystro rastuschikh analiticheskikh funktsii”, Izv. AN Arm. SSR. Ser. “Matematika”, IV:4 (1969), 267–277

[18] Matsaev V. I., Magulskii E. Z., “Teorema deleniya dlya analiticheskikh funktsii s zadannoi mazhorantoi i nekotorye ee prilozheniya”, Zap. nauchn. seminarov LOMI, 56, 1976, 73–89 | Zbl

[19] Kelleher J. J., Taylor B. A., “Closed ideals in locally convex algebras of analitic functions”, J. für die Reine und Angewandte Mathematik, 225 (1972), 190–209 | MR

[20] Nikolskii N. K., “Kriterii slaboi obratimosti v prostranstvakh analiticheskikh funktsii, vydelyaemykh ogranicheniem na rost”, Zap. nauchn. seminarov LOMI, 30, 1972, 106–129

[21] Cartan H., “Idéaux et modules de fonctions analytiques de variables complexes”, Bulletin de la Société. Mathematique de France, 78:1 (1950), 29–64 | MR | Zbl

[22] Hörmander L., “Generators for some rings of analytic functions”, Bull. Amer. Math. Soc., 73:6 (1967), 943–949 | DOI | MR | Zbl

[23] Ferrier J. P., Spectral Theory and Complex analysis, North-Holland Publ. Company, Amsterdam, London, 1973 | MR | Zbl

[24] Waelbroek L., “Etude spectrale des algèbres complètes”, Acad. Roy. Belg. Cl. Sci. Mém., 31 (1960) | MR

[25] Hörmander L., “$L^2$ estimates and existence theorems for the $\overline\partial$ operator”, Acta Mathematica, 113:1,2 (1965), 89–152 | DOI | MR | Zbl

[26] Shefer X., Topologicheskie vektornye prostranstva, Mir, M., 1971 | MR