Page 330 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 30: Projections

Hence M is invariant under T. Notes
This completes the proof of the theorem.

Theorem 5: A closed linear subspace M of a Hilbert space H reduces on operator M is invariant
under both T and T*.

Proof: Let M reduces T, then by definition both M and M are invariant under T*. But by
theorem 4, if M is invariant under T then M i.e. M is invariant under T*. Thus M is invariant
under T and T*.

Conversely, let M is invariant under both T and T*. Since M is invariant under T*, therefore M
is invariant under T * * = T (by theorem 4). Thus both M and M are invariant under T.
Therefore M reduces T.
Theorem 6: If P is the projection on a closed linear subspace M of a Hilbert space H, then M is
invariant under an operator T TP PTP.
Proof: Let M is invariant under T.

Let x H. Then Px is in the range of T, Px M TPx M.
Now P is projection and M is the range of P. Therefore TPx M TPx will remain unchanged
under P. So, we have
PTPx = TPx
PTP = TP (By equality of mappings)
Conversely, let PTP = TP. Let x M. Since P is a projection with range M and x M , therefore

Px = x
TPx = Tx
PTPx =Tx  PTP TP

PTPx = TPx  TPx Tx
But P is a projection with range M.
P TPx TPx TPx M Tx M

Since TPx = Tx.
Thus x M Tx M
M is invariant under T.

Theorem 7: If P is the projection on a closed linear subspace of M of a Hilbert space H, then M
reduces an operator TP PT.

Proof: M reduces T M is invariant under T and T*.
TP PTP and T * P PT * P

TP PTP and T * P * PT * P *

TP PTP and P * T * * P * T * *P *

TP PTP and PT PTP  P is projection P* P. AlsoTT* T

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