Abstract:
Let GG be a convex region in the complex plane and HH be the space of analytic functions on GG with the topology of uniform convergence on compacta of GG. A closed subspace W⊂HW⊂H is said to be invariant if it is invariant with respect to the differentiation operator, i.e. if f∈Wf∈W, then f′∈W. We say that W admits a spectral synthesis if W is the closed linear span of the exponential monomials contained in W. L. Schwartz in 1947 asked the question: Is it true that every invariant subspace admits a spectral synthesis? We find that the answer, generally speaking, is no. In this paper we formulate the precise criteria for the admissibility of spectral synthesis in terms of annihilator submodules of invariant subspaces.
Bibliography: 23 titles.
Citation:
I. F. Krasichkov-Ternovskii, “Invariant subspaces of analytic functions. I. Spectral analysis on convex regions”, Math. USSR-Sb., 16:4 (1972), 471–500
This publication is cited in the following 89 articles:
A. A. Tatarkin, A. B. Shishkin, “Exponential Synthesis in the Kernel of a q-Sided Convolution Operator”, J Math Sci, 282:4 (2024), 581
N. F. Abuzyarova, Z. Yu. Fazullin, “Invariant subspaces in non-quasianalytic spaces of Ω-ultradifferentiable functions on an interval”, Eurasian Math. J., 15:3 (2024), 9–24
A. S. Krivosheev, O. A. Krivosheeva, “Isklyuchitelnye mnozhestva”, SMFN, 69, no. 2, Rossiiskii universitet druzhby narodov, M., 2023, 289–305
N. F. Abuzyarova, “Invariantnye podprostranstva v nekvazianaliticheskikh prostranstvakh Ω-ultradifferentsiruemykh funktsii na intervale”, Izv. vuzov. Matem., 2023, no. 11, 86–91
N. F. Abuzyarova, “Invariant Subspaces in Nonquasianalytic Spaces of Ω-Ultradifferentiable Functions on an Interval”, Russ Math., 67:11 (2023), 75
Yu. S. Saranchuk, A. B. Shishkin, “General elementary solution of a homogeneous q-sided convolution type equation”, St. Petersburg Math. J., 34:4 (2023), 695–713
A. A. Tatarkin, A. B. Shishkin, “Eksponentsialnyi sintez v yadre operatora q-storonnei svertki”, Issledovaniya po lineinym operatoram i teorii funktsii. 50, Zap. nauchn. sem. POMI, 512, POMI, SPb., 2022, 191–222
N. F. Abuzyarova, “Differentiation Operator in the Beurling Space of Ultradifferentiable Functions of Normal Type on an Interval”, Lobachevskii J Math, 43:6 (2022), 1472
A. S. Krivosheev, O. A. Krivosheeva, “Invariant Spaces of Entire Functions”, Math. Notes, 109:3 (2021), 413–426
A. B. Shishkin, “On continuous endomorphisms of entire functions”, Sb. Math., 212:4 (2021), 567–591
A. A. Tatarkin, A. B. Shishkin, “Sintez v yadre operatora trekhstoronnei svertki”, Materialy Voronezhskoi vesennei matematicheskoi shkoly
«Sovremennye metody teorii kraevykh zadach. Pontryaginskie chteniya–XXX». Voronezh, 3–9 maya 2019 g. Chast 4, Itogi nauki i tekhn. Sovrem. mat. i ee pril. Temat. obz., 193, VINITI RAN, M., 2021, 130–141
A. S. Krivosheev, O. A. Krivosheeva, “Invariant subspaces in unbounded domains”, Probl. anal. Issues Anal., 10(28):3 (2021), 91–107
S. N. Melikhov, L. V. Khanina, “Analytic solutions of convolution equations on convex sets in the complex plane with an open obstacle on the boundary”, Sb. Math., 211:7 (2020), 1014–1040
A. S. Krivosheev, O. A. Krivosheeva, “Invariant subspaces in half-plane”, Ufa Math. J., 12:3 (2020), 30–43
O. A. Ivanova, S. N. Melikhov, “Algebry analiticheskikh funktsionalov i obobschennoe proizvedenie Dyuamelya”, Vladikavk. matem. zhurn., 22:3 (2020), 72–84
A. B. Shishkin, “Odnostoronnie skhemy dvoistvennosti”, Vladikavk. matem. zhurn., 22:3 (2020), 124–150
O. A. Krivosheeva, “Basis in invariant subspace of analytical functions”, Ufa Math. J., 10:2 (2018), 58–77
S. G. Merzlyakov, “Systems of convolution equations in complex domains”, Ufa Math. J., 10:2 (2018), 78–92
A. A. Tatarkin, U. S. Saranchuk, “Elementary solutions of a homogeneous q-sided convolution equation”, Probl. anal. Issues Anal., 7(25), spetsvypusk (2018), 137–152
N. F. Abuzyarova, “Spectral Synthesis for the Differentiation Operator in the Schwartz Space”, Math. Notes, 102:2 (2017), 137–148
Citation in format AMSBIB
Contact us: