Composition operator

In mathematics, the composition operator ${\displaystyle C_{\phi }}$ with symbol ${\displaystyle \phi }$ is a linear operator defined by the rule

where ${\displaystyle f\circ \phi }$ denotes function composition.

The study of composition operators is covered by AMS category 47B33.

In physics, and especially the area of dynamical systems, the composition operator is usually referred to as the Koopman operator, named after Bernard Koopman. It is the left-adjoint of the transfer operator of Frobenius–Perron.

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