Right inverse for a convolution operator in space of germs of analytic functions on connected subsets of $\mathbb C$

Yu. F. Korobeinik

Sbornik. Mathematics, Tome 187 (1996) no. 1, pp. 53-80

Résumé

Several general results concerning the existence of a continuous linear right inverse (CLRI) of a continuous linear operator are established. Using these results it is possible to obtain first (in a more general situation) necessary and then sufficient conditions (and in several cases, a test) for the existence of a CLRI in spaces of analytic germs on certain classes of connected sets for the convolution operator $L_b$ whose symbol $b(z)$ is an entire function of order 1 and minimal type.

Bibliographie

[1] Martineau A., “Sur la topologie des espaces de fonctions holomorphes”, Mat. Ann., 163:1 (1966), 68–88 | MR

[2] Kantorovich L. V., Akilov G. P., Funktsionalnyi analiz v normirovannykh prostranstvakh, GIFML, M., 1959 | MR

[3] Epifanov O. V., “Kriterii epimorfnosti svertki v proizvolnykh oblastyakh kompleksnoi ploskosti”, Matem. zametki, 31:5 (1982), 695–705 | MR | Zbl

[4] Korobeinik Yu. F., “O resheniyakh nekotorykh funktsionalnykh uravnenii v klassakh funktsii, analiticheskikh v vypuklykh oblastyakh”, Matem. sb., 75:2 (1968), 225–234 | MR | Zbl

[5] Miggly H., “Differenzialgleichungen unendlich hoher Ordung mit constanten Koeffizienten”, Comment. Math. Helv., 1938, no. 11, 151–179 | DOI | MR

[6] Korobeinik Yu. F., “O resheniyakh differentsialnogo uravneniya beskonechnogo poryadka, analiticheskikh v nekrugovykh oblastyakh”, Matem. sb., 71 (113):4 (1966), 535–544 | MR | Zbl

[7] Martineau A., “Equations differentielles d'ordre infini”, Bull. Soc. Math. France, 95 (1967), 109–154 | MR | Zbl

[8] Korobeinik Yu. F., “Suschestvovanie analiticheskogo resheniya differentsialnogo uravneniya beskonechnogo poryadka i kharakter ego oblasti analitichnosti”, Matem. sb., 80:1 (1969), 52–75 | MR

[9] Napalkov V. V., “Ob odnom klasse neodnorodnykh uravnenii beskonechnogo poryadka”, UMN, 29:3 (1974), 217–218 | MR | Zbl

[10] Epifanov O. V., “O probleme epimorfizma operatora svertki v kompleksnykh oblastyakh”, Matem. zametki, 16:3 (1974), 415–422 | MR | Zbl

[11] Znamenskii S. V., Ob oblastyakh suschestvovaniya analiticheskikh reshenii differentsialnykh uravnenii beskonechnogo poryadka s postoyannymi koeffitsientami, Preprint IFSO6M, Institut fiziki im. L. V. Kirenskogo Sib. otd. AN SSSR, Krasnoyarsk, 1976

[12] Znamenskii S. V., Netraditsionnaya vypuklost v napravlenii ploskikh oblastei i kompaktov i svoistva golomorfnykh reshenii differentsialnykh uravnenii beskonechnogo poryadka, Depon. v VINITI No 3063-D80 ot 15.07.80, SO AN SSSR Institut fiziki im. L. V. Kirenskogo, Krasnoyarsk, 1980

[13] Epifanov O. V., “Uravneniya svertki v kompleksnoi oblasti”, Issledovaniya po teorii operatorov, Ufa, 1988, 48–58

[14] Korobeinik Yu. F., “O razreshimosti operatora svertki v klassakh veschestvenno-analiticheskikh funktsii”, Matem. zametki, 49:2 (1991), 74–83 | MR | Zbl

[15] Napalkov V. V., Rudakov I. A., “Operator svertki v prostranstvakh veschestvenno analiticheskikh funktsii”, Matem. zametki, 49:3 (1991), 57–65 | MR | Zbl

[16] Maltsev I. M., “Ob epimorfnosti operatora svertki v prostranstvakh funktsii, analiticheskikh na svyaznykh mnozhestvakh”, DAN, 336:3 (1994), 297–300 | MR | Zbl

[17] Maltsev I. M., “Ob epimorfizme operatora svertki v kompleksnoi oblasti. I: Neobkhodimye usloviya epimorfnosti”, Izv. VUZov. Matem., 1994, no. 7, 49–58 | MR | Zbl

[18] Maltsev I. M., “Ob epimorfizme operatora svertki v kompleksnoi oblasti. II: Dostatochnye usloviya i kriterii epimorfnosti. Primery”, Izv. VUZov. Matem., 1994, no. 11, 43–52 | MR | Zbl

[19] Taylor B. A., “Linear extension operators for entire functions”, Mich. Math. J., 29 (1982), 185–197 | DOI | MR | Zbl

[20] Schwerdtfeger K., Faltungsoperatoren und Räumen holomorpher und beliebig oft differenzierbarer Funktionen, Thesis, Düsseldorf, 1982 | Zbl

[21] Meise R., Taylor B. A., “Sequence space representation for $(FN)$-algebras of entire functions modulo closed ideals”, Studia Math., 85 (1987), 203–227 | MR | Zbl

[22] Momm S., “Partial differential operators of infinite order with constant coefficients on the space of analytic functions on polydisc”, Studia Math., 96 (1990), 51–71 | MR | Zbl

[23] Korobeinik Yu. F., “O pravom obratnom operatore dlya operatora svertki”, Ukr. matem. zhurn., 43:12 (1991), 1167–1176 | MR

[24] Momm S., “Convex univalent functions and continuous right inverse”, J. Function. Analysis, 103 (1992), 85–103 | DOI | MR | Zbl

[25] Momm S., Convolution equation on the analytic functions on convex domains in the plane, Preprint, Düsseldorf University, 1993 | MR

[26] Korobeinik Yu. F., Melikhov S. N., “Lineinyi nepreryvnyi pravyi obratnyi dlya operatora predstavleniya i konformnye otobrazheniya”, DAN, 323:5 (1992), 826–829 | MR | Zbl

[27] Korobeinik Yu. F., Melikhov S. N., “Lineinyi nepreryvnyi pravyi obratnyi dlya operatora predstavleniya i prilozheniya k operatoram svertki”, Sib. matem. zhurn., 34:1 (1993), 70–84 | MR | Zbl

[28] Melikhov S. N., Momm S., Continuous linear right inverse for convolution operator on spaces of functions analytic on convex sets, Preprint, Düsseldorf, 1993

[29] Langenbruch M., “Continuous linear right inverse for convolution operator in spaces of real analytic functions”, Studia Math., CX:1 (1994), 65–82 | MR

[30] Krasichkov-Ternovskii I. F., “Invariantnye podprostranstva analiticheskikh funktsii. III. O rasprostranenii spektralnogo sinteza”, Matem. sb., 88(130):3(7) (1972), 331–352 | MR | Zbl

[31] Polya G., “Eine Verallegemeinerung des Fabryshen Lückensatzes”, Nachr. Gesel. Wiss. Göttingen Mat.-Phys., 1927, 187–195 | Zbl

[32] Robertson A., Robertson V., Topologicheskie vektornye prostranstva, Mir, M., 1967 | MR | Zbl

[33] Raikov D. A., “Dvustoronnyaya teorema o zamknutom grafike dlya lineinykh topologicheskikh prostranstv”, Sib. matem. zhurn., VII:2 (1966), 353–372 | MR

[34] Epifanov O. V., “O suschestvovanii nepreryvnogo pravogo obratnogo dlya operatora v odnom klasse lokalno vypuklykh prostranstv”, Izv. SKNTs VSh. Ser. estestv. nauk, 1991, no. 3, 3–4 | MR | Zbl

[35] Burbaki N., Topologicheskie vektornye prostranstva, IL, M., 1959

[36] Edvards R. E., Funktsionalnyi analiz: Teoriya i prilozheniya, Mir, M., 1969

[37] Levin B. Ya., Raspredelenie kornei tselykh funktsii, GITTL, M., 1956

[38] Leontev A. F., Posledovatelnosti polinomov iz eksponent, Nauka, M., 1980 | MR

[39] Valiron G., “Sur les solutions des équations différentielles linéares d'ordre infini et à coefficients constants”, Ann. Ec. Norm. Sup., 46 (1929), 25–53 | MR | Zbl