Page 310 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 28: Self Adjoint Operators

Notes
Now Lim I T x I T lim x  I T is a continuous mapping
n n
I T x M range of I T

Thus the CAUCHY sequence I T x in M converges to a vector I T x in M.
n
every CAUCHY sequence in M is a convergent sequence in M.
M is complete subspace of a complete space is closed.
M is closed.

Now we show that M = H. Let if possible M H.
Then M is a proper closed subspace of H.
Therefore, a non-zero vector x in H s.t. x is orthogonal in M.
0
0
Since I T x 0 M, therefore

x M I T x ,x 0
0 0
x Tx ,x 0
0 0 0
x ,x Tx ,x 0
0 0 0 0
x 2 Tx ,x 0 0
0
2
x Tx ,x
0 0
x 2 0  T positive Tx ,x 0 0
0
x 2 0

x 0  x 2 0

x 0
a contradiction to the fact that x 0 0.
Hence we must have M = H and consequently I+T is onto. Thus I+T is non-singular.

This completes the proof of the theorem.
Cor. If T is an arbitrary operator on H, then the operator I+TT* and I+T*T are non-singular.
Proof: We know that for an arbitrary T on H, T*T and TT* are both positive operators.
Hence by Theorem (8) both the operators I+TT* and I+T*T are non-singular.

28.2 Summary

 An operator T on a Hilbert space H is said to be self adjoint if T*=T.
 A self adjoint operator on H is said to be positive if A 0 in the order relation. That is if
Ax,x 0 x H.

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