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YCor
YCor
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An example of non-invertible operator F$F$ such that $P_nF$ is invertible on $ImP_n$$\operatorname{Im}P_n$ or proving that It is impossible

Given:

$X$ - any Banach space
$F : X \to X$ (linear bounded and non-invertible)
$P_n$, which 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:

$P_nFP_n$ -is invertible on $ImP_n$$\operatorname{Im}P_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.

I've tried a lot of things. I'm already desperate, I really hope for your help, thanks.

An example of non-invertible operator F such that $P_nF$ is invertible on $ImP_n$ or proving that It is impossible

Given:

$X$ - any Banach space
$F : X \to X$ (linear bounded and non-invertible)
$P_n$, which 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.

I've tried a lot of things. I'm already desperate, I really hope for your help, thanks.

An example of non-invertible operator $F$ such that $P_nF$ is invertible on $\operatorname{Im}P_n$ or proving that It is impossible

Given:

$X$ - any Banach space
$F : X \to X$ (linear bounded and non-invertible)
$P_n$, which 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:

$P_nFP_n$ is invertible on $\operatorname{Im}P_n$
$\|(P_nFP_n)^{-1} \| < C$ (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.

I've tried a lot of things. I'm already desperate, I really hope for your help, thanks.

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TorteDeline
TorteDeline
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Given:

$X$ - any Banach space
$F : X \to X$ (linear bounded and non-invertible)
$P_n$, which 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!.

I've tried a lot of things. I'm already desperate, I really hope for your help, thanks.

Given:

$X$ - any Banach space
$F : X \to X$ (linear bounded and non-invertible)
$P_n$, which 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!

I've tried a lot of things. I'm already desperate, I really hope for your help, thanks.

Given:

$X$ - any Banach space
$F : X \to X$ (linear bounded and non-invertible)
$P_n$, which 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.

I've tried a lot of things. I'm already desperate, I really hope for your help, thanks.

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TorteDeline
TorteDeline
  • 11
  • 3

An example of non-invertible operator F such that $P_nF$ is invertible on $ImP_n$ or proving that It is impossible

Given:

$X$ - any Banach space
$F : X \to X$ (linear bounded and non-invertible)
$P_n$, which 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!

I've tried a lot of things. I'm already desperate, I really hope for your help, thanks.