Limit of Inverse Operators: Proving Convergence for Bounded Linear Sequences

[Boromir]

Let $T_{n}$ be a sequence of invertible bounded linear operators with limit $T$ Prove that $(T_{n})^{-1}$ tends to $T^{-1}$

 

[Opalg]

Let $T_{n}$ be a sequence of invertible bounded linear operators with limit $T$ Prove that $(T_{n})^{-1}$ tends to $T^{-1}$

This is not true without the additional assumption that the limit operator $T$ is invertible (in general it need not be).

As a hint, notice that $T_n^{-1}-T^{-1} = T_n^{-1}(T-T_n)T^{-1}$.

 

[Boromir]

This is not true without the additional assumption that the limit operator $T$ is invertible (in general it need not be).

As a hint, notice that $T_n^{-1}-T^{-1} = T_n^{-1}(T-T_n)T^{-1}$.

How do you get that equality?

Once I have got that equality, take the norm, then $T-T_{n}$ tends to zero. Though what happens to the $T_{n}^{-1}$? Its not neccesarily bounded even though individually they are.

 

[Opalg]

... what happens to the $T_{n}^{-1}$? Its not neccesarily bounded even though individually they are.

Good catch, I was being careless there.

I think what you need to do is something like this. For $n$ large enough, $\|T-T_n\| < \frac12\|T^{-1}\|^{-1}$. It follows that $\|I - T^{-1}T_n\| = \|T^{-1}(T-T_n)\| \leqslant \|T^{-1}\|\|T-T_n\| <\frac12$. It follows from the Neumann series that $T^{-1}T_n$ is invertible, with $\|(T^{-1}T_n)^{-1}\| = \|T_n^{-1}T\| <2.$ Thus $\|T_n^{-1}\| = \|T_n^{-1}TT^{-1}\| \leqslant \|T_n^{-1}T\|\|T^{-1}\| <2\|T^{-1}\|.$

 

[Boromir]

Good catch, I was being careless there.

I think what you need to do is something like this. For $n$ large enough, $\|T-T_n\| < \frac12\|T^{-1}\|^{-1}$. It follows that $\|I - T^{-1}T_n\| = \|T^{-1}(T-T_n)\| \leqslant \|T^{-1}\|\|T-T_n\| <\frac12$. It follows from the Neumann series that $T^{-1}T_n$ is invertible, with $\|(T^{-1}T_n)^{-1}\| = \|T_n^{-1}T\| <2.$ Thus $\|T_n^{-1}\| = \|T_n^{-1}TT^{-1}\| \leqslant \|T_n^{-1}T\|\|T^{-1}\| <2\|T^{-1}\|.$

I don't understand how this implies $||T_{n}^{-1}-T^{-1}||$->0.

 

[Opalg]

Good catch, I was being careless there.

I think what you need to do is something like this. For $n$ large enough, $\|T-T_n\| < \frac12\|T^{-1}\|^{-1}$. It follows that $\|I - T^{-1}T_n\| = \|T^{-1}(T-T_n)\| \leqslant \|T^{-1}\|\|T-T_n\| <\frac12$. It follows from the Neumann series that $T^{-1}T_n$ is invertible, with $\|(T^{-1}T_n)^{-1}\| = \|T_n^{-1}T\| <2.$ Thus $\|T_n^{-1}\| = \|T_n^{-1}TT^{-1}\| \leqslant \|T_n^{-1}T\|\|T^{-1}\| <2\|T^{-1}\|.$

I don't understand how this implies $||T_{n}^{-1}-T^{-1}||$->0.

It answers your criticism of my earlier comment by showing that (for $n$ large enough) $\|T_n^{-1}\|$ has a uniform bound $2\|T^{-1}\|$. That earlier comment then gives you the hint for proving that $T_{n}^{-1}-T^{-1} \to0.$

 

[Boromir]

It answers your criticism of my earlier comment by showing that (for $n$ large enough) $\|T_n^{-1}\|$ has a uniform bound $2\|T^{-1}\|$. That earlier comment then gives you the hint for proving that $T_{n}^{-1}-T^{-1} \to0.$

that makes sense now haha