Let $X$ be a reflexive Banach space and $T:X\to X$ is a linear operator. Is it true that $$ X = \mathrm{ker}(I-T) \oplus \overline{\mathrm{range}(I-T^\ast)}, $$ where $\oplus$ is the direct sum?

I know this is true when $X$ is a Hilbert space (due to the fact that $Tx=x$ implies $T^\ast x =x$), and I would suspect something like that is true when $T$ is a compact operator. How about the general case?

  • $\begingroup$ This isn't even true in finite dimensional spaces unless you take some further assumptions (such as $T = T^*$) $\endgroup$ – Omnomnomnom Dec 20 '18 at 0:01
  • $\begingroup$ @Omnomnomnom What if range(I-T) is changed to range(I-T*)? $\endgroup$ – user58955 Dec 20 '18 at 0:05
  • $\begingroup$ Then that's at least correct in the finite dimensional case $\endgroup$ – Omnomnomnom Dec 20 '18 at 1:28
  • $\begingroup$ What does it mean to add a subspace of $X$ and a subspace of $X^{*}$? $\endgroup$ – Kavi Rama Murthy Dec 20 '18 at 5:58

This fails even in finite dimension. Let $X=\mathbb C^2$, and $$ T=\begin{bmatrix} 1&1\\0&1\end{bmatrix}. $$ Then $(I-T)^2=0$, so $$\overline{\operatorname{ran}(I-T)}\subset\ker(I-T).$$

When $X$ is a Hilbert space, it is true that $$\tag1 X=\ker T\oplus \overline{\operatorname{ran} T^*} $$ for any bounded linear operator $T$, since $\overline{\operatorname{ran} T^*}=(\ker T)^\perp$.

For a general Banach space, the equality $(1)$ makes no sense, as $\ker T\subset X$ and $\operatorname{ran} T^*\subset X^*$.

More dramatically, it is known that any Banach space that is not isomorphic to a Hilbert space has a non-complemented subspace $M$. And if $X$ is separable, it is known that there exists a bounded linear $T:X\to X$ with $\ker T=M$. This impedes $(1)$ in general, and there is nothing you can put on the second summand that will make it work.

  • $\begingroup$ For a general Banach space, will it make sense in the sense of isomorphism? $\endgroup$ – user58955 Dec 20 '18 at 2:00
  • $\begingroup$ No. I have edited the answer. $\endgroup$ – Martin Argerami Dec 20 '18 at 2:32

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