We have:
$X$ - any Banach space
$F : X \to X$ (linear bounded and non-invertible)
$P_n : X \to X$
Where $P_n$ is projector that strongly converges to the identity operator $I$ as $n \to\infty$
Can you help me come up with a linear bounded and non-invertible operator $F$ such that the conditions are satisfied:
$P_nFP_n$ - invertible on $ImP_n$
$\|(P_nFP_n)^{-1} \| < C$ (some constant)
$P_n \to I$ strongly
If you can help at least with idea it also will be great. Thank you very much! If you think that such example doesn't exist: please can you prove it or help with idea of proving it, thx!
As I understand: F should have some a feature at infinity that does not appear in the finite case. But if I try to come up with this consideration in the operator, I get either an unbounded or nonlinear operator.
I had idea like this:
$X = l2 $ space
$Fx = (x_1f(x),...,x_nf(x),..)$,
where $f(x)=1$ when $x$ has zeros and $f(x)=0$ when $x$ has not zeros. And it partly works: $F$ - non-invertible and $PnFPn$ - invertible (because it always has zeros). But in this case $F$ - is non-linear. Maybe you can help me to improve this example. But I think it is not right way.
This task is related with projection method. There is defenition. Maybe it will be helpfull
