Let S I TT H rightarrow H where T is linear and bounded Sh

Let S = I + T*T: H rightarrow H, where T is linear and bounded. Show that S^-1: S(H) rightarrow H exists.

Solution

T is a bounded linear operator. Hence, T*T is also a bounded linear operator as composition of two bounded linear operator is a bounded linear operator.

I is the identity operator. Hence, it is bounded linear operator. So, S = I + T*T is a bounded linear operator as sum of two bounded linear operator is a bounded linear operator.

Now, S : H --> H is a bounded linear operator.

The identity operator I : H --> H can be written as S^-1S : H --> H as I = S^-1S.

Let, h H. Then I(h) = h

This can be written as S^-1S(h) = h

Or, S^-1(S(h)) = h

As h H and S : H --> H is a bounded linear operator, S(h) S(H). Let S(h) = h₁ S(H)

So, S^-1(h₁) = h where h H and h₁ S(H)

As h is arbitrary, we can say that S^-1 : S(H) --> H exists.   Proved