TR/04/85 February 1985
The Complex Grothendieck for 2x2 matrices
Andrew Tonge by
Summary.
for 2x2 We show that the complex Grothendieck constant matrices is 1
-1-
Grothendieck's inequality asserts that if (ai j) is an n×n matrix and if x 1 . . . ,xn ,y1 , . .. ,yn are elements of the unit ball of a Hilbert space, then there is an absolute constant K such that n aij xi ,yj k|| a || ,1 .
1 j
i,
|
|
∑ < > ≤ ∞=
We have written ||a||∞ , 1 for the norm of the matrix (ai j) considered as a linear operator from ℓ∞n to ℓ1n. Thus
⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧ ∑ ≤ ≤ ≤ ≤
= =
∞ n :|si| 1 |,tj | 1,1 i,j n
1 j
i, aij si tj ,1 sup
||
a
||
When we say that K is an absolute constant, we mean that K is independent of n , of the matrix (ai j) and of the vectors
x1,...,xn ,y1,...,yn . The reader may consult [2] for a proof of Grothendieck's inequality and for supplementary information.
The best value of K is unknown, and there are differences between the real and complex cases. However, in both cases it is known that K > 1 · More precise regults may be found in [1] and [3].
Allan Sinclair (Edinburgh) has suggested that it would be of interest to determine the best universal constants Km ,n such that if (a i j ) is an n×n matrix and if x1 ,...,xn ,y1,...,yn are elements of the unit ball of a Hilbert space, then
n aij xi ,yj km,n || a || ,1
1 j
i,
|
|
∑= < > ≤ ∞For example, this would be a way to obtain information about the best value of K ·
Sinclair has obtained various estimates for Km ,n (private communication) and has conjectured that in the complex case
K2 ,2 = 1. (Results from [l] show that in the real case K2 ,n = √2 for every n > 1.) We prove this conjecture and show, more
generally, that in the complex case K m ,2 = 1 for every m .
THEOREM. Let (ai j) be a complex 2×2 matrix and let x 1 , x2, y1 and y2 be elements of the unit ball of a complex Hilbert space. Then
2 aij xi,yj ||a || ,1
1 j
i,
|
|
∑ < > ≤ ∞=
- 2 -
Proof. Simple manipulations reduce the problem to that of proving
sup
{
|a11 s1 a21 s2 | |a12 s1 a22 s2|:| s1| | s2 | 1}
.} 1 2 ||
x
| 1||
x
||
: 2||
22 x 1 a
12 x a
||
2 ||
21 x 1 a
11 x a
||
{ sup
=
= +
+ +
≤
=
= +
+ +
Writing norms (and absolute values) in terms of inner products, this inequality becomes
{ }
{
u Re vz p Re qz : | z| 1}
.sup
1
| z
| : qz Re p vz
Re u sup
that show can
we if follow therefore
will result The
22 . 12 a 2a q
2 and 22 | a 2 | 12 | a
| p 21 , 11 a 2a v 2 , 21 | a 2 | 11 | a
| u where
, 1 2 | s
| 1| s
| 2 : s 1, s q Re 2 p
s 1, s v Re u sup
1 2 ||
x
||
1||
x
||
2 : x 1, x q Re 2 p
x 1, x v Re u sup
= +
+ +
=
≤ +
+ +
=
+
=
= +
=
=
=
>
+ +
>
<
+
≤
=
=
>
<
+ +
>
<
+
⎭⎬
⎫
⎩⎨
⎧
⎭⎬⎫
⎩⎨⎧
Now write v , q and z in terms of their real and imaginary parts, so that v = α + iβ , q=λ +iμ and z = x + iy . Then if we set
f(x,y) = u + αx −βy + p + λx − μy we have to prove that
sup
{
f(x,y):x2 + y2 ≤1}
= sup{
f(x,y):x2 + y2 = 1}
.[Note that when x2 + y2 ≤ 1, f(x,y) is well-defined as a real number, since u > |v| and p > |q| .]
This identity will certainly follow if we can show that f has no local maximum in the open disc x 2 + y2 < 1 . In this disc f is continuously differentiable, provided that u ≥ |v| > 0 and P ≥ |q| > 0 . The partial derivatives are
2 . 1 ) uy λx (p 2u 2 1 1 ) βy αx (u 2β y 1
/ f
2 and 1 ) uy λx (p 2λ 2 1 1 ) βy αx (u 2 α x 1 / f
− − +
− −
− +
−
=
∂
∂
− − +
− +
− +
=
∂
∂
A local maximum can only occur if ∂f/∂x = ∂f/∂y = 0 , and this is impossible when the following conditions are satisfied.
(i) α , β , λ and μ are all non-zero, and (ii) αu ≠ βλ .
- 3 -
However, if these conditions are not satisfied , then for any given ε > 0 we can select a matrix 'j) such that
(ai , i (1 ε/8
| ' j a i
aij − ≤ ≤ ≤
| j 2)
and such that the numbers α' , β ' , λ ' and μ ' associated with ( 'j ) are all non-zero and do satisfy α' μ' ≠ β' λ' . But then
ai
⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧ ∑ < > ≤ ≤ ≤ ≤
= aij xi,y j : || xi || 1, || y j || 1, 1 i, j 2 2
1 j sup i,
⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧ ∑ < > ≤ ≤ ≤ ≤
+ =
≤ ' xi,yj : || xi || 1, || yj || 1, 1 i, j 2
aij 2
1 j sup i, 2 ε
1
≤ 2
1 ε + || a' || ∞ ,1
≤ ε + ||a ||∞,1
Let ε → 0 to get the result we want.
References.
[1] J.L .Krivine. Constantes de Grothendieck et fonctions de type positif sur les spheres. Adv. in Math, 31 (1979) 16-30.
[2] J. Lindenstrauss and L.Tzafriri. Classical Banach spaces I.
Springer (1977).
[3] G.Pisier. Grothendieck' s theorem for non-cotntnutative C*-algebras with an appendix on Grothendieck's constants. J. Funct. Anal.
29 (1978) 397-415.
