Stochastic Processes and Spectral Analysis for Hilbert $$C^*$$ C ∗ -Module-Valued Maps

Abstract

In this paper, we consider Hilbert \(C^*\)-module-valued random variables and discuss about their expectations, covariance operators and correlation operators by introducing some adjointable operators on Hilbert \(C^*\)-modules. We also study Hilbert \(C^*\)-module-valued stochastic processes instead of Hilbert space-valued processes as a generalization. Using the characterization of \(C^*\)-algebra-valued bimeasures, we prove the equivalence between V-boundedness and harmonizability of Hilbert \(C^*\)-module-valued stochastic processes. Finally, we consider Hilbert \(C^*\)-module operator-valued stochastic processes and construct spectral distributions associated with Hilbert \(C^*\)-module operator-valued stationary stochastic processes.