Math Conjectures

Banach simple-Lebesgue-spectrum problem

OPENMajorOpen problemProposed 1932 · Standard version

Canonical statement

There exist a standard nonatomic probability space (X,B,μ)(X,\mathcal B,\mu) and an invertible ergodic measure-preserving transformation T:X→XT:X\to X such that the Koopman operator UTf=f∘TU_Tf=f\circ T, restricted to
L02(X,μ)={f∈L2(X,μ):∫Xf dμ=0}, L^2_0(X,\mu)=\left\{f\in L^2(X,\mu):\int_X f\,d\mu=0\right\},
is unitarily equivalent to the bilateral shift SS on ℓ2(Z)\ell^2(\mathbb Z), defined by S(en)=en+1S(e_n)=e_{n+1}.
View source LaTeX
There exist a standard nonatomic probability space \((X,\mathcal B,\mu)\) and an invertible ergodic measure-preserving transformation \(T:X\to X\) such that the Koopman operator \(U_Tf=f\circ T\), restricted to \[ L^2_0(X,\mu)=\left\{f\in L^2(X,\mu):\int_X f\,d\mu=0\right\}, \] is unitarily equivalent to the bilateral shift \(S\) on \(\ell^2(\mathbb Z)\), defined by \(S(e_n)=e_{n+1}\).
DYN-011:[Banach1932OperationsLineaires][KatokStepin1967ApproximationsErgodic][Ryzhikov2024JoiningsProblems][Ryzhikov2024TensorLebesgueSpectrum]
Dynamical systemsAnalysis and functional analysis

Notes

The problem, going back to Banach's 1932 monograph on linear operators [Banach1932OperationsLineaires], asks whether the simplest possible Lebesgue spectrum can be realized dynamically: is there an invertible ergodic measure-preserving transformation TT of a standard nonatomic probability space whose Koopman operator UTf=f∘TU_Tf=f\circ T, restricted to the mean-zero subspace L02L^2_0, is unitarily equivalent to the bilateral shift on ℓ2(Z)\ell^2(\mathbb Z)? Equivalently, TT should have Lebesgue spectral type with multiplicity one.

Each half of the requirement can be met separately: abstract unitary operators with simple Lebesgue spectrum certainly exist, and Koopman operators of Lebesgue spectral type are plentiful, but in all known dynamical examples the multiplicity is larger than one or infinite. The approximation theory of Katok and Stepin, which links spectral properties to the speed of periodic approximation, is a basic tool for controlling multiplicity [KatokStepin1967ApproximationsErgodic]. Recent work of Ryzhikov studies tensor powers and tensor factorizations of simple Lebesgue spectrum and places the question among the central open problems on spectra and joinings [Ryzhikov2024TensorLebesgueSpectrum] [Ryzhikov2024JoiningsProblems].

The problem is open: what is missing is any construction forcing both Lebesgue type and multiplicity one for the Koopman operator of a single Z\mathbb Z-action.

References (4)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.