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Functional Analysis Problems

  • Thread starter Oxymoron
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Im having some difficulties proving some basic properties of the adjoint operator. I want to prove the following things:

1) There exists a unique map [itex]T^*:K\rightarrow H[/itex]
2) That [itex]T^*[/itex] is bounded and linear.
3) That [itex]T:H\rightarrow K[/itex] is isometric if and only if [itex]T^*T = I[/itex].
4) Deduce that if [itex]T[/itex] is an isometry, then [itex]T[/itex] has closed range.
5) If [itex]S \in B(K,H)[/itex], then [itex](TS)^* = S^*T^*[/itex], and that [itex]T^*^* = T[/itex].
6) Deduce that if [itex]T[/itex] is an isometry, then [itex]TT^*[/itex] is the projection onto the range of [itex]T[/itex].

Note that [itex]H,K[/itex] are Hilbert Spaces.

There are quite a few questions, and I am hoping that by proving each one I will get a much better understanding of these adjoint operators. Now I think I have made a fairly good start with these proofs, so I'd like someone to check them please.

We'll begin with the first one.
 

Answers and Replies

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I want to prove that there exists a unique mapping [itex]T^*:K\rightarrow H[/itex] such that

[tex]\langle Th,k \rangle = \langle h, T^*k \rangle \quad \forall \, h \in H, \, k\in K[/tex]

For each [itex]k \in K[/itex], the mapping [itex]h \rightarrow \langle Th, k\rangle_K[/itex] is in [itex]H^*[/itex]. Hence by Riesz's theorem, there exists a unique [itex]z \in H[/itex] such that

[tex]\langle Th,k \rangle_K = \langle h,z \rangle_H \quad \forall \, h \in H[/tex].

Therefore there exists a unique map [itex]T^*: K \rightarrow H[/itex] such that

[tex]\langle Th, k\rangle_K = \langle h,T^*k\rangle_H \quad \forall \, h \in H, \, k \in K [/tex].

Therefore there exists a unique [itex]T^*[/itex]. [itex]\square[/itex]
 
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2a) To see that [itex]T^*[/itex] is linear, take [itex]k_1, k_2 \in K[/itex] and [itex]\lambda \in \mathbb{F}[/itex], then for any [itex]h \in H[/itex] we have

[tex]\langle Th,k_1+\lambda k_2 \rangle_K &=& \langle Th,k_1\rangle_K + \overline{\lambda}\langle Th, k_2 \rangle_K \\
&=& \langle h,T^*k_1\rangle_K + \overline{\lambda}\langle h,T^*k_2\rangle_K \\
&=& \langle h, T^*k_1 + \lambda T^*k_2\rangle_K
[/tex]

Hence

[tex]T^*(k_1+\lambda k_2) = T^*(k_1) + \lambda T^*(k_2)[/tex]

[itex]T^*[/itex] is linear. [itex]\square[/itex]
 
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2b) To prove that [itex]T^*[/itex] is bounded note first that

[tex]\|T^*k\|^2 = \langle T^* k, T^*k \rangle_K = \langle T(T^*k), k \rangle \leq \|T(T^*k)\|\|k\| \leq \|T\|\|T^*k\|\|k\| \quad \forall \, k \in K [/tex]

Now suppose that [itex]\|T^*k\| > 0[/itex]. Then dividing the above by [itex]\|T^*k\|[/itex] we have

[tex]\|T^*k\| \leq \|T\|\|k\| \quad \forall \, k \in K[/tex]

Note that this is trivial if [itex]\|T^*k\| = 0[/itex].

Therefore [itex]T^*[/itex] is bounded. [itex]\square[/itex]
 
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Im not sure how to begin part 3 and 4 so I'll skip it for now.

5) Now Im not sure if what I have done here proves anything?

[tex]\langle T^*^* h,k \rangle = \langle x, T^* y\rangle = \overline{\langle T^*y, x\rangle} = \overline{\langle y, Tx \rangle} = \langle Tx,y \rangle [/tex]

Does this prove that [itex]T^*^* = T[/itex]?

[tex]\langle (TS)^*x,y \rangle = \langle y, (TS)x \rangle = \langle T^*x, Sy \rangle = \langle S^*T^*x,y \rangle [/tex]

And does this prove that [itex](TS)^* = S^*T^*[/itex]?
 
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3) We have to prove the following if and only if statement:

[tex]\|Th\| = \|h\| \Leftrightarrow T^*T = I[/tex]

[tex](\Leftarrow)[/tex]

Suppose [itex]T^*T = I[/itex] is true, then

[tex]\|Th\|^2 = \langle Th,Th \rangle = \langle h,T^*Th \rangle = \langle h,h \rangle = \|h\|^2[/tex]

Hence [itex]\|Th\| = \|h\|[/tex] after taking square roots of both sides.

[tex](\Rightarrow)[/tex]

Suppose [itex]\|Th\| = \|h\|[/itex] is true, then we have

[tex]\|Th\|^2 = \|h\|[/tex]

That is

[tex]\langle Th,Th \rangle = \langle h,h \rangle[/tex]

This implies that

[tex]\langle h,T^*Th \rangle = \langle h,h \rangle[/tex]

Which implies that

[tex]T^*T = I[/tex]

Therefore [itex]\|Th\| = \|h\| \Leftrightarrow T^*T = I[/itex]. [itex]\square[/tex]
 
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