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Expert answer:Q. #2Compact operators in functional Analysis from the book a Course in Functional Analysis Second Edition
hw_3_4.pdf

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Homework 4
1. Let K : L2 [0, 1] ! L2 [0, 1] be given by
(Kf )(x) =
(a)
(b)
(c)
(d)
Z
1
f (y)(x
y)2 dy.
0
Show that K is self-adjoint.
Show that ran K ✓ span{1, x, x2 }.
Find an orthonormal basis for the range of K consisting of eigenvectors of K.
Find kKk.
2. Let ‘ 2 L1 [0, 1]. Show that the multiplication operator M’ : L2 [0, 1] ! L2 [0, 1] is
compact if and only if ‘(x) = 0 for almost ever x 2 [0, 1]. [Hint: Find an orthonormal
sequence in the range of M’ .]
3. Let T be a compact operator on an infinite dimensional Hilbert space H. Prove the
following:
(a) If {en }1
n=1 is an orthonormal sequence, then lim kT en k = 0
n!1
(b) Show that T is not invertible, that is, there is no operator S 2 B(H) such that
ST h = T Sh = h for all h 2 H.
(c) Show that 0 is not necessarily in the point spectrum, that is, find a compact operator
T so that 0 2
/ p (T ).
4. Let H be a Hilbert space, and let M ⇢ H be a linear manifold. Let f : M ! C be a
linear functional such that |f (x)|  kxk. Without using the Hahn-Banach theorem, show
that there is a linear functional F : H ! C such that F (x) = f (x) for all h 2 M and
|F (h)|  khk for all h 2 H. [Hint: First, define F on M . Show that F has the desired
properties, then use the projection onto M to define F on all of H.]
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