Random commutation (Q1130354)

The authors study the commutation between a continuous random operator \(\Phi:X \times\Omega\to X\) and a continuous linear operator \(F:X\to X\), where \(X\) is a Banach space and \((\Omega,\Sigma,P)\) a probability space. For two random operators \(\Phi\) and \(\Psi\) they introduce and justify the notation \[ P[\Phi\equiv\Psi]=\min\biggl\{P\bigl[f(\Phi x)=f(\Psi x)\bigr]:x\in X,\;f\in X'\biggr\}. \] The quantity \(P[\Phi F\equiv F\Phi]\) is a measure of the commutation likelihood between \(\Phi\) and \(F\). For a Hilbert space \(X\) and \(F\) normal they give sufficient conditions for \(P[\Phi F\equiv F\Phi]=P[\Phi F^*\equiv F^*\Phi]\), where \(F^*\) is the adjoint operator of \(F\). This is a random analogue of the Fuglede theorem. The commutator of \(\Phi\) and \(F\) is defined as the random operator \([\Phi,F]=\Phi F-F\Phi\). The authors prove the equality \[ P\biggl[\bigl[[\Phi,F],F\bigr]\equiv 0\biggr]=P\bigl[[\Phi,F]\equiv 0\bigr], \] a probabilistic version of the Putnam theorem. Finally they show that if there is a sufficiently large probabilistic commutation between \(\Phi\) and a compact normal operator \(F\), then there exists a random function \(\varphi\) defined on the spectrum of \(F\) such that \(P[\Phi\equiv\varphi(F)]>0\).