Weakly Periodic Sequences of Bounded Linear Transformations: a Spectral Characterization

Let X and Y be two Hilbert spaces, and L(X, Y ) the space of bounded linear transformations from X into Y . Let {An} ⊂ L(X, Y ) be a weakly periodic sequence of period T . Spectral theory of weakly periodic sequences in a Hilbert space is studied by H. L. Hurd and V. Mandrekar (1991). In this work we proceed further to characterize {An} by a positive measure μ and a number T of L(X, X)-valued functions a0, . . . , aT−1; in the spectral form An = ∫ 2π 0 e−iλnΦ(dλ)Vn(λ), where Vn(λ) = T−1 k=0 e −i 2πkn T ak(λ) and Φ is an L(X, Y )-valued Borel set function on [0, 2π) such that (Φ(∆)x, Φ(∆ ′ )x ′ )Y = (x, x)Xμ(∆ ∩∆ ′ ).