1
$\begingroup$

I have a Banach space $B$ and a continuous linear transformation $F:B \rightarrow B\times B$. One of the induced transformations $F(1):B \rightarrow B$ and $F(2):B \rightarrow B$ into the factors of the product has closed range. Must F have closed range? I have the max norm on the product, i.e., $\|F(x) \| = max\{\|F(1)(x)\|, \|F(2)(x)\|\}$ for $x$ in $B$. I was hoping to use the minimum moduli of the $F(i)$ to provide an affirmative answer.

$\endgroup$
2
$\begingroup$

No. Consider $F(x)=(f(x),0)$ where $f$ does not have closed range.

| cite | improve this answer | |
$\endgroup$
  • $\begingroup$ I felt sure there was a simple counter-example to the question as stated, but was trying with the identity as the other factor. D'oh! $\endgroup$ – Andrew Stacey Jul 29 '10 at 18:40
  • $\begingroup$ Thanks much, Bill. It is embarassing not to have considered mapping into one of the factors (I should have known better). $\endgroup$ – Chris Leary Jul 29 '10 at 19:03
  • $\begingroup$ Does the answer change if the linear maps $F(i)$ must both be nonzero? $\endgroup$ – Chris Leary Jul 29 '10 at 19:41
  • 1
    $\begingroup$ No; make the mapping into into the second factor rank one instead of rank zero. $\endgroup$ – Bill Johnson Jul 29 '10 at 21:19

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.