Estimation optimale du gradient du semi-groupe de la chaleur sur le ...

Therefore,
A. Wi´snicki / Journal of Functional Analysis 236 (2006) 447–456 453
D1 = (un)U : (0,un)U ∈ D
is a subset of Y which is isometric to D.
Define T1 : D1 → D1 as
T1 (un)U = PrY
T
(0,un)U ,
where PrY denotes the standard projection onto Y . Notice that T1 is nonexpansive. By assumption,
X has SFPP, and it follows from Theorem 2.4 that (Y )U = ((X)V)U has FPP. Thus there
exists (vn)U ∈ D1 such that
T1 (vn)U = (vn)U
and consequently
T
(0,vn)U = (0,vn)U .
But this contradicts Lemma 2.6 because (0,vn)U ∈ D ⊂ B(K,r). ✷
Let us consider another, more symmetric, class of norms. A norm ·Z on R 2 is said to be
strictly monotone if
(x1,y1) Z < (x2,y2) Z
whenever |x1| |x2|, |y1| < |y2| or |x1| < |x2|, |y1| |y2|. We say that a norm on X ⊕ Y is
strictly monotone if
(x, y) = x, y Z
and the norm ·Z is strictly monotone.
Strictly monotone norms do not necessarily satisfy the condition (i):
(x, 0) =x and (0,y) =y for every x ∈ X, y ∈ Y.
However, it is not difficult to see that the proof of Theorem 3.2 is also valid in this case.
Theorem 3.3. Let X be a Banach space with the super fixed point property and F be a finitedimensional
space. Then F ⊕ X, endowed with a strictly monotone norm, has the super fixed
point property.
In particular, the result holds for l p -products F ⊕p X, p ∈[1, ∞).