Solutions of Sylvester equation in $C^*$-modular operators
Анотація
УДК 517.9
Розв’язки рiвняння Сiльвестра для $C^*$-модульних операторiв
Розглянуто розв’язнiсть рiвняння Сiльвестра $AX + Y B = C$ та операторного рiвняння $AXD + FY B = C$ при загальних умовах сумiжностi операторiв мiж гiльбертовими $C^*$- модулями. На основi обернених Мура – Пенроуза для зв’язаних операторiв отримано необхiднi та достатнi умови iснування розв’язкiв цих рiвнянь, а також загальнi вирази для розв’язкiв у випадку, коли вони iснують. Крiм того, запропоновано пiдхiд до вивчення додатних розв’язкiв у спецiальному випадку рiвняння Ляпунова.
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