MAT 578 HW 5 Due Wednesday, 10/2/03
Choose three from among the following problems and any as-yet unworked problems in
previous assignments.
17. Let (X, Ω, µ) be a σ-finite measure space, let k∈L2(X×X), and let T∈K(L2(X))
be the corresponding integral operator:
(T ξ)(x) = ZX
k(x, y)ξ(y)dµ(y).
Suppose that k(y, x) = k(x, y ).
(i) Prove that Tis self-adjoint.
(ii) Let {fi:i∈N}be an orthonormal basis for L2(X). Prove that {fi⊗fj:i, j ∈N}is
an orthonormal basis for L2(X×X). (Recall that for f,gfunctions on X,f⊗gis the
function on X×Xgiven by (f⊗g)(x, y) = f(x)g(y).) Hint: kf⊗gk2=kfk2· kgk2.
(iii) Let λ1,λ2,. . . be the eigenvalues of T, each repeated according to its multiplicity (i.e.
the number of times λiis repeated equals the dimension of the λi-eigenspace). Prove
that
X
i
|λi|2=kkk2
2.
18. Let Tbe a compact self-adjoint operator on the Hilbert space H. Let {ei:i∈N}be
an orthonormal basis of Hconsisting of eigenvectors of T, and let Tei=λiei. Let y∈H.
(i) Prove that there exists x∈Hwith T x =yif and only if the following both hold:
(a) y⊥ker(T),
(b) Pλ−2
i
hy, eii
2<∞(the sum taken only over {i:λi6= 0}).
(ii) Find a formula for the most general vector xfor which Tx =y.
19. Let Tbe a compact self-adjoint operator on the Hilbert space H. Let {ei:i∈N}be
an orthonormal basis of Hconsisting of eigenvectors of T, and let T ei=λiei. Let c∈C
be such that c6= 0 and c6=λifor all i.
(i) Prove that for each y∈Hthere exists a unique x∈Hsuch that (T−cI)x=y.
(ii) Find (with proof) a formula for xin terms of c,y,{λi}and {ei}.
(iii) Prove that the map y7→ xdefined by the result of part (i) is a bounded linear map.