9
$\begingroup$

The space $( \ell^2 ,\lVert \cdot \rVert _2 )$ is a Hilbert space. The space $X=(\ell^2 \oplus \ell^2 , \lVert \cdot \rVert_\infty )$ is a Banach space. Does X have fixed point property? (For any closed convex bounded subset $C\subseteq X $ and any nonexpansive map $T:C\to C $ there is a $x\in C$ such that $T(x)=x$)

The space $X$ isn't uniformly convex so I can't use theorems about uniformly convex. There is no theorems about F.P.P in products of spaces and I have no another idea. Does someone have any idea?

$\endgroup$
  • $\begingroup$ Maybe use (somehow) the fact that the norm $\|\cdot\|_\infty$ is the limit of uniformly convex norms $\|\cdot\|_p$ for $p\to\infty$? $\endgroup$ – Taras Banakh Nov 12 '18 at 8:43
  • $\begingroup$ No. this is not solution. I tried for ten days but i didn't achieve solution $\endgroup$ – Darman Nov 12 '18 at 8:46
  • $\begingroup$ How is your norm defined? Is $$|| \,\cdot \,||_{\infty} = \max \left( || \,\cdot \,||_2, \; || \,\cdot \,||_2 \right)?$$ $\endgroup$ – Francesco Polizzi Nov 12 '18 at 9:34
  • $\begingroup$ Yes . .the norm is this $\endgroup$ – Darman Nov 12 '18 at 9:36
8
$\begingroup$

I think that the answer is yes, and that it should follow from the following facts:

  1. every Hilbert space is uniformly convex, hence it has normal structure;

  2. the direct sum of two Banach spaces with normal structure, endowed with the infinity norm, has again normal structure (Belluce-Kirk-Steiner, Pacific Journal Math. 1968);

  3. the finite direct sum of separable Banach spaces (with any of the possible equivalent norm on it) is itself reflexive, in particular $X = \ell^2 \oplus \ell^2$ is reflexive;

  4. normal structure on a reflexive Banach space implies FPP (Kirk, Amer. Math. Monthly 1965).

$\endgroup$
  • $\begingroup$ But normal structure implies W.F.P.P And I didn't see number (3) somewhere. $\endgroup$ – Darman Nov 12 '18 at 10:13
  • 2
    $\begingroup$ Wouldn't WFPP and FPP coincide on reflexive spaces? $\endgroup$ – Dirk Werner Nov 12 '18 at 21:09
  • $\begingroup$ Yes. But X maybe isn't reflexive. $\endgroup$ – Darman Nov 13 '18 at 2:12
  • 2
    $\begingroup$ The finite direct sum of reflexive Banach spaces is itself reflexive. Note also that your norm is equivalent to the (finite) $L^p$-norm $$||(x, \, y)||_p = \left( ||x||^p + ||y||^p \right)^{1/p},$$ see en.wikipedia.org/wiki/Banach_space#Reflexivity $\endgroup$ – Francesco Polizzi Nov 13 '18 at 6:48

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.