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Koopman Operator

The Koopman operator of a measure-preserving transformation T on a probability space (X,F,mu) is the linear operator on L^2(X,mu) defined by

(U_Tf)(x)=f(Tx).

It is an isometry. If T is invertible and its inverse is a measurable function, U_T is unitary, with inverse U_(T^(-1)). The constant functions form an invariant subspace, and their orthogonal complement is the mean-zero subspace

L_0^2(X,mu)={f in L^2(X,mu):int_Xfdmu=0}.

The restriction to this subspace describes the spectral behavior apart from the constant functions. The probability-preserving form of Banach's simple Lebesgue spectrum problem asks whether this entire restriction can have simple Lebesgue spectrum.

See also

Banach's Simple Lebesgue Spectrum Problem, Ergodic Theory, Invariant Measure, Linear Operator, Measure-Preserving Transformation, Unitary

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References

Eisner, T.; Farkas, B.; Haase, M.; and Nagel, R. Operator Theoretic Aspects of Ergodic Theory. Cham, Switzerland: Springer, 2015. https://doi.org/10.1007/978-3-319-16898-2.

Cite this as:

Weisstein, Eric W. "Koopman Operator." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/KoopmanOperator.html

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