Abstract
In the paper, the invariance property of characteristics (the order and type) of an operator and of a sequence of operators with respect to a topological isomorphism is proved. These characteristics give precise upper and lower bounds for the expressions ‖A n (x)‖ p and enable one to state and solve problems of operator theory in locally convex spaces in a general setting. Examples of such problems are given by the completeness problem for the set of values of a vector function in a locally convex space, the structure problem for a subspace invariant with respect to an operator A, the problem of applicability of an operator series to a locally convex space, the theory of holomorphic operator-valued functions, the theory of operator and differential-operator equations in nonnormed spaces, and so on. However, the immediate evaluation of characteristics of operators (and of sequences of operators) directly by definition is practically unrealizable in spaces with more complicated structure than that of countably normed spaces, due to the absence of an explicit form of seminorms or to their complicated structure. The approach that we use enables us to find characteristics of operators and sequences of operators using the passage to the dual space, by-passing the definition, and makes it possible to obtain bounds for the expressions ‖A n (x)‖ p even if an explicit form of seminorms is unknown.
Similar content being viewed by others
References
V. P. Gromov, “The order and type of a linear operator, and expansion in a series of eigenfunctions,” Dokl. Akad. Nauk SSSR 288 (1), 27–31 (1986) [Soviet Math. Dokl. 33, 588–591 (1986)].
V. P. Gromov, “The order and type of an operator and entire vector-valued functions,” Uchenye Zapiski OGU (lab. TFFA), No. 1, 6–23 (1999).
A. F. Leont’ev, Exponential Series (Nauka, Moscow, 1976) [in Russian].
A. F. Leont’ev, “Entire Functions. Series of Exponentials,” (Nauka, Moscow, 1983) [in Russian].
S. N. Mishin, “On the order and type of an operator,” Dokl. Ross. Akad. Nauk 381 (3), 309–312 (2001) [Dokl. Math. 64 (3), 355–358 (2001)].
S. N. Mishin, “The order and type of an operator and of a sequence of operators acting on locally convex spaces,” Uchenye Zapiski OGU (lab. TFFA), No. 3, 28–75 (2001).
L. V. Kantorovich and G. P. Akilov, Functional Analysis in Normed Spaces (Fizmatgiz, Moscow, 1959; The Macmillan Co., New York, 1964).
V. P. Gromov, “On the completeness of a system of values of a holomorphic vector function in a Fréchet space,” Mat. Zametki 73 (6), 827–840 (2003) [Math. Notes 73 (6), 783–795 (2003)].
O. D. Solomatin, “Generalization of the exponential function and the completeness of a systemof generalized exponentials,” Uchenye Zapiski OGU (lab. TFFA), No. 2, 90–95 (2001).
O. D. Solomatin, “On the completeness of the system of generalized exponentials in a Fréchet space,” Uchenye Zapiski OGU (lab. TFFA), No. 3, 37–46 (2002).
A. Gelfond [Gel’fond], “Sur les systèmes complets de fonctions analytiques,” Mat. Sb. 4 (46) (1), 149–156 (1938).
A. F. Leont’ev, Generalization of Series of Exponentials (Nauka, Moscow, 1981) [in Russian].
Yu. A. Kaz’min, “On completeness of systems of functions of the form {f(z + α n)} and {f (n)(z)},” Uspekhi Mat. Nauk 12 (2 (74)), 151–154 (1957) [in Russian].
Yu. A. Kaz’min, “On the completeness of a system of analytic functions. I,” Vestnik Moskov. Univ. Ser. IMat. Mekh., No. 5, 3–13 (1960) [in Russian].
O. D. Solomatin, “On the invariant subspaces of locally convex spaces,” Fundam. Prikl. Mat. 3 (3), 937–946 (1997) [in Russian].
V. P. Gromov, “On invariant subspaces of spaces of entire functions,” in Selected Problems of Mathematical Analysis, Collection of Papers. Vol. 1 (MOPI, Moscow, 1980), pp. 36–46 [in Russian].
S. N. Mishin, “Relation of characteristics of a sequence of operators with bornological convergence,” Vestn. RUDN. Ser. Matem., Inform., Fiz., No. 4, 26–34 (2010) [in Russian].
S. N. Mishin, “On the applicability of an operator series to a locally convex space,” Uchenye Zapiski OGU, No. 6 (62), 22–26 (2014).
P. C. Sikkema, Differential Operators and Differential Equations of Infinite Order with Constant Coefficients: Researches in Connection with Integral Functions of Finite Order (Noordhoff, Groningen–Djakarta, 1953).
P. van der Steen, On Differential Operators of Infinite Order, Doctoral dissertation (Technical University of Delft, Delft, 1968).
S. N. Mishin, “On characteristics of growth of operator-valued functions,” Ufimsk. Matem. Zhurn. 5 (1), 112–124 (2013).
S. N. Mishin, “The order and type of a sequence of operators and analytic operator-valued functions,” Uchenye Zapiski OGU, No. 6 (62), 27–30 (2014).
H. A. Aksenov, Application of the Theory of Order and Type of an Operator in Locally Convex Spaces to Differential-Operator Equations, Cand. Sci. (Phys.–Math.) Dissertation (Orel State University, Orel, 2011) [in Russian].
S. N. Mishin, “On a form of differential-operator equations with variable coefficients,” Vestn. RUDN. Ser. Matem., Inform., Fiz., No. 1, 3–14 (2015).
M. Reed and B. Simon, Methods of Modern Mathematical Physics. Vol. 1. Functional Analysis (Mir, Moscow, 1977; Academic Press, New York–London, 1972).
A. N. Kolmogorov and S. V. Fomin, Elements of the Theory of Functions and Functional Analysis (Nauka, Moscow, 1972; Introductory Real Analysis, Dover Publications, Inc., New York, 1975).
A. Pietsch, Nuclear Locally Convex Spaces (Nukleäre lokalkonvexe Räume Akademie-Verlag, Berlin, 1965; Mir, Moscow, 1967; Springer-Verlag, New York-Heidelberg, 1972).
S. V. Panyushkin, “Generalized Fourier transform and its applications,” Mat. Zametki 79 (4), 581–596 (2006) [Math. Notes 79 (3–4), 537–550 (2006)].
I. F. Krasichkov, “Completeness in a space of complex valued functions determined by the behavior of the modulus,” Mat. Sb. 68 (110) (1), 26–57 (1965) [in Russian].
V. P. Gromov, S. N. Mishin, and S. V. Panyushkin, Operators of Finite Order and Differential-Operator Equations (OGU, Orel, 2009) [in Russian].
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © S. N. Mishin, 2016, published in Matematicheskie Zametki, 2016, Vol. 100, No. 3, pp. 399–409.
Rights and permissions
About this article
Cite this article
Mishin, S.N. Invariance of the order and type of a sequence of operators. Math Notes 100, 429–437 (2016). https://doi.org/10.1134/S0001434616090091
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0001434616090091