Page 303 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes = 2 A – A
n
0 as n

A A * 0 or A A* 0 A A *

A is self-adjoint operator.
This completes the proof of the theorem.
Theorem 2: Let S be the set of all self-adjoint operators in (H). Then S is a closed linear subspace
of (H) and therefore S is a real Banach space containing the identity transformation.
Proof: Clearly S is a non-empty subset of (H), since O is self adjoint operator i.e. O S.

H , since O is self adjoint operator i.e. O S.

Let A ,A 2 S, We prove that A 1 A 2 S.
1
A ,A 2 S A * A and A * 2 A 2 ...(1)
1
1
1
For , R, we have
A 1 A * A * A 2 *
2
1
A * 1 A * 2

A 1 A 2  , are real numbers, ,

A A is also a self adjoint operator on H.
1 2
A ,A S A A S.
1 2 1 2

S is a real linear subspace of H .

Now to show that S is a closed subset of the Banach space H . Let A be any limit point of S.
Then a sequence of operator A is such that A A. We shall show that A S i.e. A A *.
n n
Let us consider

A A * A A n A n A *

A A n A n A *

A A n A n A * n A * n A *

A A n A n A * n A * n A *

A n A A n A n A n A *  A n S A * n A n

A n A 0 A n A

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