MAT 578 HOMEWORK 2 Due: 9/6/07
Solve three of the following problems.
5. (a) A topological group is a group that is also a Hausdorff topological space such
that the group operations (composition, inverse) are continuous. Let Gbe a
topological group and let Hbe a subgroup which is an open set. Prove that His
a closed set.
(b) Let Abe a unital Banach algebra in which k1Ak= 1. Let G0be the subgroup
of G(A) generated by elements of the form 1 −xand (1 −x)−1with x∈Aand
kxk<1. Prove that G0equals the connected component of G(A) containing 1A.
(c) With notation as in part (b), prove that G0is a normal subgroup of G(A).
6. Let (an)∞
n=1 be a bounded sequence of complex numbers. Prove that there is a bounded
operator Aon `2such that Aen=anen+1, (where en= (0,0,...,0,1,0,0, . . .) is the
nth standard basis vector in `2). Compute the norm of Ain terms of (an). (The
operator Ais called a unilateral weighted shift.)
7. A unitary operator on a Hilbert space His an operator U∈B(H) such that Uis
an isometry and is invertible (and hence U−1is also a unitary operator). Let Abe a
unilateral weighted shift (as in problem 6).
(a) Prove that for every complex number λwith |λ|= 1, there is a unitary operator
Uλ∈B(`2) such that UλAU−1
λ=λA.
(b) Prove that the spectrum of Aequals the union of (possibly degenerate) circles
with center at 0.
8. Let Abe a unilateral weighted shift corresponding to a sequence (an) as in problem
6. Suppose that limn→∞ an= 0. Prove that Ais quasinilpotent.
9. Prove that the Volterra operator of problem 4 is quasinilpotent.
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