Generators with a closure relation

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    Assume that a block operator of the form $\left(\begin{array}{c}A_1\\ A_2\ 0\end{array}\right)$, acting on the Banach space $X_1 \times X_2$, generates a contraction $C_0$-semigroup. We show that the operator AS defined by $A_Sx = A_1 \left(\begin{array}{c}x\\ SA_2x \end{array}\right)$ with the natural domain generates a contraction semigroup on X1. Here, S is a boundedly invertible operator for which $\varepsilon I-S^{-1}$ is dissipative for some ε > 0. With this result the existence and uniqueness of solutions of the heat equation can be derived from the wave equation.
    Original languageUndefined
    Pages (from-to)157-165
    Number of pages9
    JournalOperators and matrices
    Issue number1
    Publication statusPublished - Mar 2014


    • MSC-34G10
    • Block operator
    • EWI-25235
    • Semi-inner-product
    • METIS-305957
    • MSC-47B44
    • MSC-47D06
    • IR-91548
    • Semigroup of operators
    • Dissipative operator

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