Abstract
An unbounded operator is said to be bisectorial if its spectrum is contained in two sectors lying, respectively, in the left and right half-planes and the resolvent decreases at infinity as 1/λ. It is known that, for any bounded function f, the equation u′ − Au = f with bisectorial operator A has a unique bounded solution u, which is the convolution of f with the Green function. An example of a bisectorial operator generating a Green function unbounded at zero is given.
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References
O. Perron “Die Stabilitätsfrage bei Differentialgleichungen,” Math. Z. 32(5), 703–728 (1930).
Yu. L. Daletskii and M. G. Krein, Stability of Solutions of Differential Equations in a Banach Space, in Nonlinear Analysis and Its Applications (Nauka, Moscow, 1970) [in Russian].
M. A. Krasnoselskii, V. Sh. Burd, and Yu. S. Kolesov, Nonlinear Almost-Periodic Oscillations (Nauka, Moscow, 1970) [in Russian].
H. Bart, I. Gohberg, and M. A. Kaashoek “Wiener-Hopf factorization, inverse Fourier transforms and exponentially dichotomous operators,” J. Funct. Anal. 68(1), 1–42 (1986).
C. V. M. van der Mee, Exponentially Dichotomous Operators and Applications, in Oper. Theory Adv. Appl. (Birkhäuser Verlag, Basel, 2008), Vol. 182.
A. V. Pechkurov, “Bisectorial operator pencils and the problem of bounded solutions,” Spektr. iÉvolutsion. Zadachi 21(2), 76–87 (2011).
A. V. Pechkurov, “On the invertibility in the Schwartz space of the operator generated by a pencil of moderate growth,” Vestn. Voronezhsk. Iniv. Fiz. Mat. 2, 116–122 (2011).
A. V. Pechkurov, “Bisectorial operator pencils and the problem of bounded solutions,” Russ. Math. 56, No. 3, 26–35 (2012); Izv. Vyssh. Uchebn. Zaved. Mat., No. 3, 31–41 (2012) [Russian Math. (Iz. VUZ) 56 (3), 26–35 (2012)].
D. Henry Geometric Theory of Semilinear Parabolic Equations (Springer-Verlag, Heidelberg, 1981; Mir, Moscow, 1985).
A. G. Baskakov and K. I. Chernyshev, “Spectral analysis of linear relations, and degenerate semigroups of operators,” Mat. Sb. 193(11), 3–42 (2002) [Sb. Math. 193 (11), 1573–1610 (2002)].
A. G. Baskakov and K. I. Chernyshev, “On distribution semigroups with a singularity at zero and bounded solutions of differential inclusions,” Mat. Zametki 79(1), 19–33 (2006) [Math. Notes 79 (1–2), 18–30 (2006)].
A. G. Baskakov, “Spectral analysis of differential operators with unbounded operator-valued coefficients, difference relations and semigroups of difference relations,” Izv. Vyssh. Uchebn. Zaved. Mat. 73(2), 3–68 (2009) [Russian Math. (Iz. VUZ) 73 (2), 215–278 (2009)].
M. S. Bichegkuev, “To the theory of infinitely differentiable semigroups of operators,” Algebra Anal. 22(2), 1–13 (2011) [St. Petersburg Math. J. 22 (2), 175–182 (2011)].
L. D. Kudryavtsev, A Short Course in Mathematical Analysis (Nauka, Moscow, 1989) [in Russian].
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Original Russian Text © A. V. Pechkurov, 2015, published in Matematicheskie Zametki, 2015, Vol. 97, No. 2, pp. 249–254.
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Pechkurov, A.V. An example in the theory of bisectorial operators. Math Notes 97, 243–248 (2015). https://doi.org/10.1134/S0001434615010253
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DOI: https://doi.org/10.1134/S0001434615010253