## Transcribed Text

1. (i) Let (E,k · k) be a normed vector space. Consider a sequence
(x(k))1k
=1 in E. What does it mean that (x(k))1k
=1 is a Cauchy
sequence? What does it mean that (E,k · k) is a Banach space?
[3 Marks]
(ii) Let (E,k · k) be a Banach space. For n 2 N, n # 2,
consider the vector space En .
(a) Consider the norm k · k1 on En given by
kxk1 =
Xn
l=1
kxlk
for x = (x1, x2, . . . ,xn) 2 En. Prove that (En,k · k1) is a Banach
space. (You do not need to prove that k·k1 is indeed a norm on En .)
[6 Marks]
(b) Now consider the norm k · k1 on En given by
kxk1= max
l2{1,...,n} kxlk
for x = (x1, x2, . . . ,xn) 2 En. Prove that the norms k · k1 and k · k1
Total marks 12 are equivalent. [3 Marks]
2. (i) Let H be a Hilbert space and let G be a closed subspace
of H. State the theorem on the orthogonal projection of a vector
x 2 H onto G. [2 Marks]
(ii) In the Hilbert space `2(R) , consider the vectors
f = (1, 1, 1, 0, 0, 0, . . . ), g= (1, 0, 2, 0, 0, 0, . . . ).
Consider the closed subspace G of `2(R) given by
G = {af + bg | a, b 2 R}.
(a) Find G?. [4 Marks]
(b) Let P1 and P2 denote the orthogonal projections of the
Hilbert space `2(R) onto the closed subspaces G and G? , respectively.
Let
x = (0, 1, 0, x4, x5, x6, . . . ) 2 `2(R),
where
xn = 2−n for n # 4.
Find P1x and P2x.
I
3. (i) Let E be a Banach space with norm k · k. Let A be a
bounded linear operator on E, i.e., A 2 L(E) . Give two equivalent
definitions of the norm of A.
Let L(E) denote the space of all bounded linear operators
on E, equipped with the operator norm k · k. What can you say
about the space (L(E),k · k)? (You do not need to justify/prove
your statement.) [3 Marks]
(ii) Let p 2 [1,1) . Consider the following linear operator
on `p(R) :
Ax = (0, 0, x3,−x2, x5,−x4, x7,−x6, . . . ), x= (x1, x2, x3, . . . ) 2 `p(R).
Prove that A 2 L(`p(R)) and find kAk. [7 Marks]
4. Let H be a Hilbert space and let A 2 L(H) .
(i) Give the definition of the adjoint operator A⇤ .
[2 Marks]
(ii) State the theorem about the kernel and range of A⇤ .
Moreover, prove the part of this theorem that concerns the
kernel of A⇤. [6 Marks]
(iii) Let now H = L2([1,1) ! C) and let
A 2 L(L2([1,1) ! C))
be the bounded linear operator defined by
(Af)(x) = (2+i)f(x3), x2 [1,1).
Find A⇤ .
5. (i) Let A be a bounded linear operator on a Hilbert space
H. What does it mean for the operator A to be unitary? Give both
equivalent definitions! [3 Marks]
(ii) State the theorem about the inverse of a unitary operator.
(iii) Give an example of a unitary operator on the Hilbert
space `2(C) (which is not a multiple of the identity operator), and
explain why this operator is indeed unitary. [8 Marks]
6. (i) Let E be a Banach space over C, and let A 2 L(E) .
(a) State the definition of both the set of regular points of
A, and the spectrum of A. [3 Marks]
(b) Let z be a regular point of the operator A. What is the
resolvent of A at point z? State the Hilbert identity. [3Marks]
(ii) Define A 2 L(L2([0, 2] ! C)) by
(Af)(x) = ↵(x)f(x), x2 [0, 2], f 2 L2([0, 2] ! C).
Here
↵(x) =
1X
n=1
(−1)n 2
n
#( 2
n+1 , 2
n ](x), x2 [0, 2],
where #( 2
n+1 , 2
n ](·) denotes the indicator function of the set
"
2
n+1 , 2
n
⇤
.
Find the spectrum of A. [11 Marks]
1. (a) Let E1 and E2 be normed vector spaces. Let A : E1 ! E2 be a linear operator. Explain what it means that the operator A
is bounded. State two equivalent definitions of the norm of A.
[3 Marks]
(b) Let p 2 [1,1) . Consider the following operator on the
space E = lp(C) : for any x = (x1, x2, x3, . . .) 2 E,
Ax =
⇣
0, 2x3,−
i
4
x4, x5,−
i
6
x6, x7,−
i
8
x8, . . . ,−
i
2k
x2k, x2k+1, . . .
⌘
.
Prove that A 2 L(E) and find kAk.
[8 Marks]
2. (a) Give two equivalent definitions of an invertible operator.
Prove their equivalence.
[4 Marks]
(b) Formulate Banach’s theorem on inverse operators.
[1 Mark]
(c) Let p 2 [1,1) and E = Lp(R) := Lp(R ! R) . Let the
operator A be defined by
(Af)(x) =
4
2 + x4 f(x), x2 R, f 2 E.
Prove that A 2 L(E) and kAk 2.
[8 Marks]
(d) Prove that the operator A defined in (c) is not invertible.
[12 Marks]
Hint: Find g 2 E of the form
g(x) =
4
(2 + x4)↵, x2 R,
where ↵ 2 R, such that g does not belong to Ran(A) .
II
3. (a) Let H be a Hilbert space over C. Let A 2 L(H) . Give
the definition of the adjoint operator A⇤ of A.
[3 Marks]
(b) Let H = L2([1,1) ! C) . Let A 2 L(H) be given by
(Af)(x) = (2+i) f(3x2 − 2), x2 R, f 2 H.
Find A⇤ .
[12 Marks]
4. (a) Give the definition of a Hilbert–Schmidt operator in a
Hilbert space. Prove that this definition is independent on the choice
of a basis. [4 Marks]
(b) Let H = l2(R) and consider A 2 L(H) given by
A(x1, x2, x3, x4, x5, x6, . . .) = (0, 0, x2, x1, x4, x3, x6, x5, . . .), x2 H.
Prove that A is an isometry, but not a unitary operator.
[8 Marks]
5. (a) Let E be a Banach space over C, and let A 2 L(E) .
State the definition of both the set of regular points of A, and the
spectrum of A.
[3 Marks]
(b) Define A 2 L(L2([0, 2] ! C)) by
(Af)(x) = ↵(x)f(x), x2 [0, 2], f 2 L2([0, 2] ! C).
Here
↵(x) =
1X n=1(−1)n 2
n
↵n(x), x2 [0, 2],
where ↵n denotes the indicator function of the set
✓
2
n + 1
,
2
n
%
, i.e.
↵n(x) =
8<
:
1, x2
✓
2
n + 1
,
2
n
%
,
0, otherwise.
Find the spectrum of A.

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