Download Hwk 8, Due April 16th [pdf]

MATH 611 (Spring 2009) Homework #8 Due April 16th
(1) (a) Let T ∈ B(H1 , H2 ) and S ∈ B(H2 , H3 ) where the Hj are Hilbert spaces. Prove that
(ST )∗ = T ∗ S ∗ .
(b) Let T ∈ B(H, K) be invertible with a bounded inverse. Prove that T ∗ is invertible with a
bounded inverse and that (T ∗ )−1 = (T −1 )∗ .
(2) Let T ∈ B(H, K). Prove that T is an isometric isomorphism if and only if T is a unitary
equivalence.
(3) Let {un }∞
n=1 be an orthonormal basis for a Hilbert space H. Prove that for all x, y ∈ H
(x, y) =
∞
X
(x, un )(un , y).
n=1
(4) Let f ∈ L∞ and let Tf ∈ B(L2 ) be the multiplication operator defined by Tf (g) = f · g. Prove
that (Tf )∗ = Tf .
(5) An operator is said to have rank 1 if its range is one-dimensional. Let T ∈ B(H, K) be a
bounded rank 1 operator between Hilbert spaces and let ψ be a non-zero vector in the range of T .
Show that there exists a φ ∈ H so that T (x) = (x, φ)ψ for all x ∈ H and that kT k = kφk kψk. Find
a formula for T ∗ in terms of φ and ψ.
Additional Questions (Not to be handed in)
(6) Prove that T ∗∗ = T and kT ∗ k = kT k.
(7) Prove that en (x) = exp(inx) for n ∈ Z is an orthonormal set in L2C ([−π, π]).
(8) Prove that any separable infinite dimensional Hilbert spaces is unitarily equivalent to `2 .