Geodesic
Functional calculus generated by a~square pencil
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 39, Tome 389 (2011), pp. 113-130

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A linear transformation $\Upsilon$ is introduced that assigns an element of a special Banach algebra to each analytic function defined on the spectrum of the pencil $\lambda\mapsto\lambda^2E+\lambda F+H$. The transformation $\Upsilon$ maps the product of two functions into the product of two elements of the algebra. As an application, a formula for a solution of the differential equation $E\ddot x(t)+F\dot x(t)+Hx(t)=f(t)$ is given. 
@article{ZNSL_2011_389_a6,
     author = {I. V. Kurbatova},
     title = {Functional calculus generated by a~square pencil},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {113--130},
     publisher = {mathdoc},
     volume = {389},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2011_389_a6/}
}
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I. V. Kurbatova. Functional calculus generated by a~square pencil. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 39, Tome 389 (2011), pp. 113-130. http://geodesic.mathdoc.fr/item/ZNSL_2011_389_a6/