Fall 2007 Qual , Part I : 1 ,

I.1 Two short problems: 1. Suppose that (X,A, μ) is a measure space, and X = ∪j=1Xj with Xj ⊂ Xj+1 for all j. Let χj be the indicator (characteristic) function of Xj, and let Mj denote the multiplication operator by χj acting on Z = Lp(X, dμ), 1 ≤ p < ∞. Show that Mj → I, the identity operator, in the strong operator topology, i.e. the weakest topology on bounded linear operators L(Z) in which the maps Ez : L(Z) 3 A 7→ Az ∈ Z, z ∈ Z, are continuous, but not necessarily in the norm topology. 2. For f ∈ L1(R), let f̂ (ξ) = ∫ R e−ixξ f (x)dx denote the Fourier transform of f . Show that if f has compact support (i.e. has a representative vanishing outside a compact set), then f̂ extends to an analytic function on all of C. 1. To show that Mj → I in the strong operator topology, it is enough to show that Mj f → f for all f ∈ Z, i.e. that ‖Mj f − f ‖p → 0 for each f . Suppose first that μ(X) < ∞. Let e > 0. Recall from measure theory that for f ∈ Lp(X, dμ), there is a δ > 0 such that if μ(A) < δ, then (∫