# Generators with a closure relation

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1 Citation (Scopus)

## Abstract

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 language Undefined 157-165 9 Operators and matrices 8 1 https://doi.org/10.7153/oam-08-08 Published - Mar 2014

## Keywords

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