Open AccessOn the compactness of solutions to certain operator inequalities arising from the Likhtarnikov — Yakubovich frequency theorem
Author(s) -
Mikhail Anikushin
Publication year - 2020
Publication title -
vestnik sankt-peterburgskogo universiteta. matematika. mehanika. astronomiâ/vestnik sankt-peterburgskogo universiteta. seriâ 1, matematika, mehanika, astronomiâ
Language(s) - English
Resource type - Journals
eISSN - 2587-5884
pISSN - 1025-3106
DOI - 10.21638/spbu01.2020.405
Subject(s) - compact space , mathematics , compact operator , hilbert space , operator (biology) , pure mathematics , finite rank operator , compact operator on hilbert space , pseudo monotone operator , semigroup , quasinormal operator , scalar (mathematics) , mathematical analysis , banach space , operator space , computer science , biochemistry , chemistry , geometry , repressor , transcription factor , extension (predicate logic) , gene , programming language
We study the compactness property of operator solutions to certain operator inequalities arising from the frequency theorem of Likhtarnikov — Yakubovich for C0-semigroups. We show that the operator solution can be described through solutions of an adjoint problem as it was previously known under some regularity condition. Thus we connect some regularity properties of the semigroup with the compactness of the operator in the general case. We also prove several results useful for checking the non-compactness of operator solutions to Lyapunov inequalities and equations, into which the operator Riccati equation degenerates in certain cases arising in applications. As an example, we apply these theorems for a scalar delay equation posed in a proper Hilbert space and show that the operator solution cannot be compact. This results are related to the author recent work on a non-local reduction principle of cocycles (non-autonomous dynamical systems) in Hilbert spaces.