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 Zap. Nauchn. Sem. POMI, 2019, Volume 480, Pages 162–169 (Mi znsl6769)

Operator sine-functions and trigonometric exponential pairs

V. A. Kostin, A. V. Kostin, D. V. Kostin

Voronezh State University

Abstract: With the help of operator functional relations $Sh(t+s)+Sh(t-s) = 2[ I+2 Sh^2(\frac t2)] Sh(s), Sh(0)=0,$ we introduce and study strongly continuous sine-function $Sh(t), t\in(-\infty, \infty),$ of linear bounded transformations acting in a complex Banach space $E$, together with the cosine-function $Ch(t)$ given by the equation $Ch(t)=I+2Sh^2(\frac t2)$, where $I$ is the identity operator in $E$.
The pair $Ch(t)$, $Sh(t)$ is the exponential of a trigonometric pair (ETP). For such pairs a generating operator (generator) is determined by the equation $Sh"(0)\varphi = Ch"(0) \varphi = A \varphi$, and a criterion for $A$ to be the generator of the ETP is provided.
A relationship of $Sh(t)$ with the uniform well-posedness of the Cauchy problem with the Krein condition for the equation $\frac{d^2 u(t)}{dt^2}=Au(t)$ is described. This problem is uniformly well-posed if and only if $A$ is an exponent generator of the sine-function $Sh(t)$.
The concept of bundles of several ETP, which also forms a ETP, is introduced, and a representation for its generator is given.
The obtained facts expand significantly the possibilities of operator methods in the study of well-posed initial boundary value problems.

Key words and phrases: orthogonal polynomials, operator polynomials, Bessel operator functions, strongly-continuous semigroup generator.

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UDC: 517.518.13; 517.983.5

Citation: V. A. Kostin, A. V. Kostin, D. V. Kostin, “Operator sine-functions and trigonometric exponential pairs”, Investigations on linear operators and function theory. Part 47, Zap. Nauchn. Sem. POMI, 480, ПОМИ, СПб., 2019, 162–169

Citation in format AMSBIB
\Bibitem{KosKosKos19} \by V.~A.~Kostin, A.~V.~Kostin, D.~V.~Kostin \paper Operator sine-functions and trigonometric exponential pairs \inbook Investigations on linear operators and function theory. Part~47 \serial Zap. Nauchn. Sem. POMI \yr 2019 \vol 480 \pages 162--169 \publ ПОМИ \publaddr СПб. \mathnet{http://mi.mathnet.ru/znsl6769}