Using the operator functional relation Sh(t+s)+Sh(t−s) = 2[I +2Sh2(\( \frac{t}{2} \))] Sh(s), Sh(0) = 0, we introduce and study the strongly continuous sine function Sh(t), t ∈ (−∞, ∞), of linear bounded transformations acting in a complex Banach space E. Also, we study the cosine function Ch(t) given by the equation Ch(t) = I + 2Sh2(\( \frac{t}{2} \)), where I is the identity operator in E.
The pair Ch(t) and Sh(t) is called an exponential trigonometric pair (ETP, in brief). For such pairs, we determine the generating operator (generator) by the equation Sh″ (0)φ = Ch″ (0)φ = Aφ and we give a criterion for A to be the generator of ETP. We find a connection between Sh(t) and the uniform well-posedness of the Cauchy problem with Krein’s condition for the equation \( \frac{d^2u(t)}{dt^2} \) = Au(t). This problem is uniformly well-posed if and only if A is the exponent generator of the sine function Sh(t). We introduce the concept of a bundle of several ETPs, which also forms an ETP, and we give a representation for bundle’s generator. The facts obtained significantly expand the applicability of operator methods to study the well-posedness of initial-boundary value problems.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
V. V. Vasil’ev, S. G. Krein, S. I. Piskarev, “Operator semigroups, cosine operator functions, and linear differential equations,” Itogi Nauki Tekhniki, Math. Anal., 28 (1990), 87–202.
J. Goldstein, Semigroups of Linear Operators and Applications, Vyshcha Shkola, Kiev (1989).
V. A. Kostin, “Abstract strongly continuous pairs of trigonometric transformation groups,” Differ. Uravn., 7, No. 8 (1984), 1419–1425.
V. A. Kostin, “On analytic semigroups and strongly continuous cosine functions,” Dokl. Akad. Nauk SSSR, 307, No. 4 (1989), 796–799.
A.V. Kostin, “Exponential cosine operator functions and their bundles,” in: S. G. Krein VZMS Conference Proceedings, Voronezh (2018), pp. 254–255.
V. A. Kostin, “On the solution of a problem associated with an abstract cosine function,” Dokl. Akad. Nauk, 336, No. 5 (1994), 584–586.
S.G. Krein, Linear differential equations in a Banach space, Nauka, Moscow (1967).
S. G. Krein and M. I. Khazan, “Differential equations in a Banach space,” Itogi Nauki i Tekhniki, Mathematical Analysis, 21 (1993), 130–264.
S. Kurepa, “Semigroups and Cosine Functions,” Lect. Notes Math, 948 (1982).
M. Sova, “Cosine operator functions,” Rozprawy Mat., 49 (1966), 1–46.
H.O. Fattorini, “A note on fractional derivatives of semigroups and cosine functions,” Pasif. J. Math., 109, No. 2 (1983), 335–347.
H.O. Fattorini, Second Order Linear Differential Equations in Banach Spaces, North Holland, Amsterdam (1985).
Author information
Authors and Affiliations
Corresponding authors
Additional information
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 480, 2019, pp. 162–169.
Rights and permissions
About this article
Cite this article
Kostin, V.A., Kostin, A.V. & Kostin, D.V. Operator Sine Functions and Exponential Trigonometric Pairs. J Math Sci 251, 267–272 (2020). https://doi.org/10.1007/s10958-020-05087-3
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-020-05087-3