1. Let H be a separable Hilbert space, {eP
n }n an orthonormal basis in H.
(i) Prove that for any A ∈ L(H) the sum n kAen k2 is independent of the choice of the
basis. If it is finite, its square root is denoted by kAk2 and is called the Hilbert-Schmidt
norm of A, and then A is called a Hilbert-Schmidt operator.
(ii) Show that the set of Hilbert-Schmidt operators is a vector subspace of L(H), and k · k2
is a norm on it majorizing the operator norm.
(iii)For Hilbert-Schmidt operators A and B define a scalar product by
X
(A, B) =
(Aen , Ben ).
n
Show that (A, B) is independent of the choice of the basis. Prove that the space of HilbertSchmidt operators is a Hilbert space with respect to this scalar product.
(iv)Let I = [a, b], K ∈ L2 (I 2 ), and A the integral operator on L2 (I) defined by K, so
Z
(Af )(x) = K(x, t)f (t)dt.
I
Prove that A is a Hilbert-Schmidt operator and kAk2 = kKk2 .
2. Let I = [a, b], K ∈ C(I 2 ), A the integral operator on L2 (I) defined by K. Show that
(i) any eigenvector of A with non-zero eigenvalue lies in C[a, b] ⊂ L2 (I);
(ii) A ≥ 0 if and only if for any n ∈ N and any different points t1 , . . . , tn ∈ [a, b] the
operator defined by the matrix (K(ti , tj ))ni,j=1 is positive (in particular, K(s, t) = K(t, s)
and K(t, t) ≥ 0).