# zbMATH — the first resource for mathematics

A criterion for the fundamental principle to hold for invariant subspaces on bounded convex domains in the complex plane. (English. Russian original) Zbl 1274.46063
Funct. Anal. Appl. 46, No. 4, 249-261 (2012); translation from Funkts. Anal. Prilozh. 46, No. 4, 14-30 (2012).
Summary: Let $$D$$ be a bounded convex domain of the complex plane. We study the problem of whether the fundamental principle holds for analytic function spaces on $$D$$ invariant with respect to the differentiation operator and admitting spectral synthesis. Earlier this problem was solved under a restriction on the multiplicities of the eigenvalues of the differentiation operator. In the present paper, we lift this restriction. Thus, we present a complete solution of the fundamental principle problem for arbitrary nontrivial closed invariant subspaces admitting spectral synthesis on arbitrary bounded convex domains.

##### MSC:
 4.6e+16 Banach spaces of continuous, differentiable or analytic functions
Full Text:
##### References:
 [1] V. V. Napalkov, Convolution Equations on Multidimensional Spaces [in Russian], Nauka, Moscow, 1982. · Zbl 0582.47041 [2] A. F. Leont’ev, Exponential Series [in Russian], Nauka, Moscow, 1976. [3] I. F. Krasichkov-Ternovskii, ”A homogeneous convolution type equation on convex domains,” Dokl. Akad. Nauk SSSR, 197:1 (1971), 29–31. [4] A. S. Krivosheev, ”Fundamental principle for invariant subspaces on convex domains,” Izv. Ross. Akad. Nauk Ser. Mat., 68:2 (2004), 71–136; English transl.: Russian Acad. Sci. Izv. Math., 68: 2 (2004), 291–353. · Zbl 1071.30024 · doi:10.4213/im476 [5] G. Valiron, ”Sur les solutions des équations différentielles linéaires d’ordre infini et à coefficients constants,” Ann. Sci. Ecole Norm. Sup., 46:1 (1929), 25–53. · JFM 55.0857.04 [6] L. Schwartz, ”Théorie générale des fonctions moyenne-périodique,” Ann. Math., 48:4 (1947), 857–929. · Zbl 0030.15004 · doi:10.2307/1969386 [7] A. O. Gelfond, ”Linear differential equations of infinite order with constant coefficients and asymptotic periods of entire functions,” Trudy Mat. Inst. Steklov, 38 (1951), 42–67. [8] D. G. Dickson, ”Expansions in series of solutions of linear difference-differential and infinite order differential equations with constant coefficients,” Mem. Amer. Math. Soc., 1957:23 (1957), 1–72. · Zbl 0079.11305 [9] B. Ya. Levin, ”On some applications of the Lagrange interpolation series to the theory of entire functions,” Mat. Sb., 8:3 (1940), 437–454. · Zbl 0024.21801 [10] Yu. F. Korobeinik, ”Interpolation problems, nontrivial expansions of zero, and representing systems,” Izv. Akad. Nauk SSSR Ser. Mat., 44:5 (1980), 1066–1114; English transl.: Math. USSR-Izv., 17:2 (1981), 299–337. [11] Yu. F. Korobeinik, ”Representing systems,” Uspekhi Mat. Nauk, 36:1 (1981), 73–126; English transl.: Russian Math. Surveys, 36:1 (1981), 75–137. · Zbl 0483.30003 [12] A. V. Bratiščev and Yu. F. Korobeinik, ”The multiple interpolation problem in the spaces of entire functions of given proximate order,” Izv. Akad. Nauk SSSR Ser. Mat., 40:5 (1976), 1102–1127; English transl.: Math. USSR-Izv., 10:5 (1976), 1049–1074. [13] A. V. Bratishchev, Köthe bases, entire functions, and their applications [in Russian], Doctoral Dissertation (Physics-Mathematics), Rostov-on-Don, 1995. [14] A. S. Krivosheev, ”A criterion for the fundamental principle for invariant subspaces,” Dokl. Ross. Akad. Nauk, 389:4 (2003), 457–460. · Zbl 1246.30007 [15] A. F. Leont’ev, Entire Functions. Exponential Series [in Russian], Nauka, Moscow, 1983. [16] K. Leichtweiss, Konvexe Mengen, Springer-Verlag, Berlin-New York, 1980. [17] O. A. Krivosheeva, ”Series of exponential monomials on complex domains,” Vestnik UGATU. Matem., 9:3(21) (2007), 96–103. [18] O. A. Krivosheeva, ”Singular points of the sum of an exponential series on the boundary of the domain of convergence,” Ufimsk. Mat. Zh., 1:4 (2009), 78–109. [19] V. V. Napalkov and O. A. Krivosheeva, ”Abel and Cauchy-Hadamard theorems for series of exponential monomials,” Dokl. Ross. Akad. Nauk, 432:5 (2010), 594–596; English transl.: Russian Acad. Sci. Dokl., 432:5 (2010), 449–451. · Zbl 1205.30006 [20] A. Robertson and W. Robertson, Topological Vector Spaces, Cambridge Univ. Press, New York, 1964. · Zbl 0123.30202 [21] P. Lelong and L. Gruman, Entire Functions of Several Complex Variables, Springer-Verlag, New York, 1986. · Zbl 0583.32001 [22] B. Ya. Levin, Distribution of Zeroes of Entire Functions, Transl. Math. Monographs Series, vol. 5, Amer. Math. Soc., Providence, RI, 1964. · Zbl 0152.06703
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.