De nition 21Let T be a linear operator on a Hilbert space Then Essay

De

nition 2.1

Let T be a linear operator on a Hilbert space. Then is called an Eigen

values of T if there exist a non zero vector v .

De

nition 2.2

Let T be a linear operator on a Hilbert space. Then the non zero vector

v 2 H is said to be an Eigen vector of T if there exists a scalar such that

vT = v .

De

nition 2.3

Let be an eigen value of T and let M be the set of all eigen vectors of T

corrseponding to the eigen value together with the zero vector.

Then M is called

the Eigen Space of T corresponding to .

It is denoted by M = f v : (T- I ) v =0g

Theorem 2.4

If T is normal,then v is an eigen vector of T with eigen value if and only

if v is an eigen vector of T with eigen value .

Proof:

Given T is normal.

To prove that ( T – I ) is normal.

Let be any scalar.