Page 329 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes Py = P(x – Px) = Px – P x = Px – Px = 0
2
0 = P*y = P*(x – Px) = P*x – P*Px
P*x = P*Px x H

P* = P*P
Now P = (P*)* = (P*P)* = P*P = P*
P is a self adjoint operator.

2
Also P = P.
Hence P is a projection on H.

30.1.2 Invariance

Definition: Let T be an operator on a Hilbert space H and M be a closed subspace of H. Then M
is said to be invariant under T if T M M. If we do not take into account the action of T on
vectors outside M, then T can be regarded as an operator on M itself. The operator T on H induces
on operator T on M such that T (x) = T(x) for every x M. This operator T is called the
M M M
restriction of T on M.

Further, let T be an operator on Hilbert space H. If M is a closed subspace of H and if M and M
are both invariant under T, then T is said to be reduced by M. If T is reduced by M, we also say
that M reduces T.
Theorem 4: A closed linear subspace M of a Hilbert space H is invariant under the operation
T M is invariant under T*.

Proof: Let M is invariant under T, we show M is invariant under T*.

*
*
Let y be any arbitrary vector in M . Then to show that T y is also in M i.e. T y is orthogonal to
every vector in M.
Let x be any vector in M. Then Tx M because M is invariant under T.

Also y M y is orthogonal to every vector in M.
Therefore y is orthogonal to Tx i.e.

(Tx,y) = 0
(x,Ty) = 0

*
T y is orthogonal to every vector x in M.
T * y is in M and so M is invariant under T*.

Conversely, let M is invariant under T*. Thus to show that M is invariant under T. Since M is

a closed linear subspace of H invariant under T*, therefore by first case M is invariant
under T.

But M M M and T * * T * * T.

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