The Hardy Operator

The Hardy operator is defined on sequence and function spaces. Its boundedness can be deduced from the so called Hardy inequaltity in both its discrete and continuous forms. We presented different proofs of this inequality that go back to Hardy, Elliot and Ingham among others. In [BHS65] Brown, Halmos and Shields published further aspects on the discrete unweighted Hardy operator h2 on the Hilbert space l2. They proved important statements about the spectra and point spectra of h2 and its dual operator. We investigated how far the results in [BHS65] are applicable to the Hardy operator on the weighted sequence space l2(N, λ). Definition of the Hardy Operator Let p > 1, λ = (λn)n∈N a sequence of positive weights. Let l (N, λ) be the space of all sequences a = (an)n∈N of complex numbers with ‖(an)n∈N‖plp(N,λ) := ∞ ∑