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Functional Analysis: Dr. Shiferaw Feyissa

This document appears to be an assignment submitted by five students - Abener Tewodros, Hana Endiris, Habtamu Meressa, Miliyon Tilahun, and Sisai Bekele - to Dr. Shiferaw Feyissa on May 30, 2016. It contains solutions to three functional analysis problems asking for examples of a self-adjoint operator, a unitary operator, and a normal operator. The first solution provides the example of an operator on a finite-dimensional Hilbert space represented by a self-adjoint matrix. The second solution provides the example of a shift operator on the Hilbert space of square-summable sequences. The third solution provides the example of a multiplication operator on a Hilbert space

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Miliyon Tilahun
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206 views2 pages

Functional Analysis: Dr. Shiferaw Feyissa

This document appears to be an assignment submitted by five students - Abener Tewodros, Hana Endiris, Habtamu Meressa, Miliyon Tilahun, and Sisai Bekele - to Dr. Shiferaw Feyissa on May 30, 2016. It contains solutions to three functional analysis problems asking for examples of a self-adjoint operator, a unitary operator, and a normal operator. The first solution provides the example of an operator on a finite-dimensional Hilbert space represented by a self-adjoint matrix. The second solution provides the example of a shift operator on the Hilbert space of square-summable sequences. The third solution provides the example of a multiplication operator on a Hilbert space

Uploaded by

Miliyon Tilahun
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© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Assignment II

FUNCTIONAL ANALYSIS


May 30, 2016

Submitted to

Dr. Shiferaw Feyissa

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Abener Tewodros .
Hana Endiris . . .
Habtamu Meressa
Miliyon Tilahun . .
Sisai Bekele . . . .

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ID No
. GSR/1417/08
. GSR/1419/08
. GSR/1399/08
. GSR/1401/08
. GSR/1426/08

Functional Analysis Problems


1. Give an example of Self-Adjoint operator.
Solution. Let H = Cn and let {e1 , . . . , en } be the standard orthonormal base in H. Let A be an operator represented by matrix (aij ), where
aij = hAej , ei i. Then the adjoint operator A is represented by the matrix bkj = hA ej , ek i. Consequently
bkj = hej , Aek i = hAek , ej i = ajk .
Therefore, the operator A is self-adjoint if and only if aij = aji .
2. Give an example of Unitary operator.
Solution. Let H be the Hilbert space of all sequences
P of complex numbers x = (. . . , x1 , x0 , x1 , . . .) such that kxk = |xn |2 < . The
inner product is defined by
hx, yi =

xn yn .

The operator T defined by T (xn ) = (xn1 ) is a unitary operator. Indeed, T is invertible and
hT x, yi =

xn1 yn =

xn yn+1 = hx, T 1 yi,

which implies T = T 1 .
3. Give an example of Normal operator.
Solution. Let H be a Hilbert space and let T x = ix for all x H.
Since T x = ix = T x, T is not self-adjoint. On the other hand,
kT xk = kT xk for all x H, and thus T is normal.

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