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Oxymoron
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Let [itex]\mathcal{H}[/itex] be a Hilbert space over [itex]\mathbb{C}[/itex] and let [itex]T \in \mathcal{B(H)}[/itex].
I want to prove that [itex]\|Tx\| = \|x\| \, \Leftrightarrow \, T^{\ast}T = I[/itex] for all [itex]x \in \mathbb{H}[/itex] and where [itex]I[/itex] is the identity operator in the Hilbert space.
Since this is an if and only if statement I began with the reverse inclusion:
[tex](\Leftarrow)[/tex]
Suppose [itex]T^{\ast}T = I[/itex], then for every [itex]x \in \mathcal{H}[/itex] we have
[tex]\|Tx\|^2 = \langle Tx\,|\,Tx \rangle = \langle x\,|\,T^{\ast}Tx \rangle = \langle x\,|\, x \rangle = \|x\|^2 [/tex]
And therefore [itex]T[/itex] is isometric (after taking square roots of both sides).
But now with the forward inclusion I am having some difficulties.
[tex](\Rightarrow)[/tex]
Suppose that [itex]T[/itex] is isometric. Then for every [itex]x \in \mathcal{H}[/itex] we have
[tex]\|Tx\|^2 = \|x\|^2[/tex]
[tex]\Rightarrow \, \langle Tx\,|\,Tx \rangle = \langle x\,|\,x \rangle[/tex]
[tex]\Rightarrow \, \langle T^{\ast}Tx \,|\, x \rangle = \langle x\,|\,x \rangle[/tex]
[tex]\Rightarrow \, \langle (I-T^{\ast}T)x\,|\,x \rangle = 0[/tex]
Now since [itex]I-T^{\ast}T[/itex] is self-adjoint and we know that for self-adjoint operators
[tex]\|T\|_{op} = \sup \{\langle Tx\,|\,Tx \rangle \,:\, \|x\| = 1\}[/tex]
then [itex]\|I - T^{\ast}T\|_{op} = 0[/itex] and therefore [itex]T^{\ast}T = I[/itex].
This last bit in red I don't understand. Does anyone know how to explain it?
I want to prove that [itex]\|Tx\| = \|x\| \, \Leftrightarrow \, T^{\ast}T = I[/itex] for all [itex]x \in \mathbb{H}[/itex] and where [itex]I[/itex] is the identity operator in the Hilbert space.
Since this is an if and only if statement I began with the reverse inclusion:
[tex](\Leftarrow)[/tex]
Suppose [itex]T^{\ast}T = I[/itex], then for every [itex]x \in \mathcal{H}[/itex] we have
[tex]\|Tx\|^2 = \langle Tx\,|\,Tx \rangle = \langle x\,|\,T^{\ast}Tx \rangle = \langle x\,|\, x \rangle = \|x\|^2 [/tex]
And therefore [itex]T[/itex] is isometric (after taking square roots of both sides).
But now with the forward inclusion I am having some difficulties.
[tex](\Rightarrow)[/tex]
Suppose that [itex]T[/itex] is isometric. Then for every [itex]x \in \mathcal{H}[/itex] we have
[tex]\|Tx\|^2 = \|x\|^2[/tex]
[tex]\Rightarrow \, \langle Tx\,|\,Tx \rangle = \langle x\,|\,x \rangle[/tex]
[tex]\Rightarrow \, \langle T^{\ast}Tx \,|\, x \rangle = \langle x\,|\,x \rangle[/tex]
[tex]\Rightarrow \, \langle (I-T^{\ast}T)x\,|\,x \rangle = 0[/tex]
Now since [itex]I-T^{\ast}T[/itex] is self-adjoint and we know that for self-adjoint operators
[tex]\|T\|_{op} = \sup \{\langle Tx\,|\,Tx \rangle \,:\, \|x\| = 1\}[/tex]
then [itex]\|I - T^{\ast}T\|_{op} = 0[/itex] and therefore [itex]T^{\ast}T = I[/itex].
This last bit in red I don't understand. Does anyone know how to explain it?
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