Download The von Neumann inequality for linear matrix functions of several

Mathematical Notes, Vol. 6,f, No. ~, 1998
The Von Neumann
Inequality
of Several Variables
for Linear
Matrix
Functions
D. S. K a l y u z h n y i
UDC 517.982
ABSTRACT. It is proved that there exist three commuting contractions in Hilbert space a n d a linear homogeneous matrix function of three independent variables for which the generalized yon N e u m a n n inequality is not
satisfied.
KEY WORDS: Hilbert space, generalized yon Neumann inequality, commuting contractions, matrix-valued polynomial.
J. yon Neumann showed in [1] that, if T is a contractive linear operator (i.e.,
and p(z) is a polynomial in one variable, then
ilTll <- 1) in Hilbert space
IIP(T)il < max IP(~)I,
-- zEA
w h ~ e A = {~ ~ C : I~1 --< 1} is the closed unit disk. T. Ando [2] extended this inequality to any
two commuting contractions T1 and T2 in Hilbert space and an arbitrary polynomial t~zl, z2) in two
independent variables:
liP(T1, T2)II < m ~ IP(~)i
-- zEA=
(Here and in what follows, for any positive integer N , A N ---- {z E C N : [Zk[ _~ 1, k = 1 , . . . , N} is the
dosed unit polydisk.) However, N. Varopoulos [3] showed that, in general, a similar inequality does not hold
beginning with three commuting contractions; namely, he constructed a triple of commuting contractive
linear operators T1, T2, Ts in some finite-dimensional Hilbert space and a second-degree homogeneous
polynomial p(zl, z2, z3) in three independent variables such that
IIp(Tx, T2, T~)II > max Ip(z)l.
zEA a
R e m a r k 1. The degree of the polynomial in the Varopoulos example cannot be reduced. Indeed,
suppose that T = { T 1 , . . . , TN} is a set of contractions in Hilbert space 7~ with scalar product (- 9)~
and
N
t(Zl,-.-,gn)=~+~akZk
k----1
is an arbitrary first-degree polynomial in N independent variables, i.e., a linear scalar function in C ~v.
Then
N
h=l
ll=ll=llyll=; i
zEAN
II=II=ll~ll=l i \
,=]
~EA~
.
,
k=l
k=l
~=x
k----;
]
Translated from Mat~naticheskie Zametki, Vol. 64, No. 2, pp. 218-223, August, 1998.
Original article submitted October 16, 1997.
186
0001-4346/98/6412-0186 $20.00
(~1999 Plenum Publishing Corporation
(we applied the Cauchy-Bunyakovsky-Schwarz inequality to estimate the scalar product and used the
maximum modulus principle for analytic functions in the disk). Note that it is unessential here that the
set T is commutative.
However, for matrix-valued polynomials (i.e., polynomials with matrix coefficients) in several independent variables, the notion of polynomial in several commuting contractions can be considered [4]; we show
that the degree of a polynomial for which the corresponding generalized yon Neumann inequality is violated can then be reduced to one. This problem arose in studying multiparameter passive linear scattering
(the definition of such systems is given in [5]), but it is also of intrinsic interest.
Consider a set T = { T 1 , . . . , TN} of commuting contractions in Hilbert space 7"l and a matrix-valued
polynomial P ( z l , . . . , ZN) in N independent variables:
PCz)=
~
A:',
z = C z l , . . . , z N ) e c N,
(1)
tez~ Itl_<-*
where Z N -=- {t 6 Z N : tit _> 0, k = 1, . . . , N } , Itl = E l = l tit for t 6 Z N+, z' = H ~ z i t N" , and
At G Mn(C) = s n); i.e., we identify the algebra of square matrices of order n over the field C with
the C*-algebra of linear operators in the flnlte-dimensional Hilbert space C n . Let us define the operator
P(W)=
P(TI,...,TN)= ~
At@Tt6s
@~) ~ s
(2)
tez~ ItlSm
here T t = l I [ = , T~'. The polynomial P(T) can also be regarded as an element of the C'-algebra
M,,(C) | s
-~ M , , ( s
of all square matrices of order n over s
i.e., a matrix with operator
elements (see [41). We need the following de~nition.
D e f i n i t i o n 1 [01, A set V = { V ~ , . . . , VN} of COmmutingunitary operator, in m l b e n space K: is
called a unitary dilation of a set T = { T 1 , . . . , TN } of COmmuting c o n t r i t i o n opemtom in Hilbert space
7-/C K: if T t = P n U t [7"l for any t E Z ~ , where P n is the orthogonal projector in K: onto the subspace 7"/.
R e m a r k 2. The general Arveson theorem on dilations (see [7, p. 278]) implies that a set T =
{ T 1 , . . . , TN } of commuting contractions in Hilbert space has a unitary dilation if and only if any matrixvalued polynomial P ( z ) of the form (1) satisfies the generalized yon Neumann inequality
IIP(T)II < zEAN
max IIP(z)ll,
-
(3)
-
where P(T) is defined by (2). In particular, for N = 1 and N = 2, this means (see [8] and [21) that (3)
always holds.
W e state the main assertion of this paper for N = 3.
T h e o r e m . There exist a triple T = {T1, 2"2, Ts } of commuting contract/ons in some f~nite-dimensiona/
Hi]bert space TI and a triple A = {At, A2, As} of square matrices of order n > 1 over the s
C such
that the linear homogeneous matrix function
L(z,, z2, z~) = A,z, + A2z2 + Asz3
(4)
satisfies the inequality
IIL(T)II>
max
zE~ s
llL(z)ll.
(5)
To prove this theorem, we need the following de~n~tions.
D e f i n i t i o n 2 (see, e.g., [9]). Let ,4 and B be C*-algebras. A linear map ~0:,4 -~ B is called positive
if it transforms positive elements of .4 into positive elements of B.
Consider a linear subspaee S in a C*-algebra .4 (possibly coinciding with .A), a C*-algebra B, and
a linear map ~ : ,q ---, B. For each positive integer n, M , ( C ) | .A ~- M,(.A) is the C*-algebra of all
square matrices of order n over .4 and M,(C) @ ~ a linear subspace of this C*- algebra. If i d , is the
identity map of Mn(C) onto itself, then ~,, = id, @7~ is the linear map of M , ( C ) | S into M,,(C) | B
that transforms matrices over S into matrices over B by applying the linear map ~ elementwise.
187
D e f i n i t i o n 3 [9]. A linear map ~o: .4 --* B is called completel~.l positive if the m a p s ~o,, are positive for
all n.
D e f i n i t i o n 4 [4]. A linear map ~0: ,S ~ B is called completelg contractive if t h e maps ~0,, are contractive (i.e., II~,,ll < X) for all n.
P r o o f o f t h e t h e o r e m . Remark 2 implies that the required triple T of contractions does not have a
unitary dilation. Consider such a triple from the well-known Parrott Example [10]. Let X be an arbitrary
Iiilbert space of dimension higher than 1 ( X may be finite-dimensional). Put ~ = X @ ,1" Let us specify
a triple T = {T~, 2"2, Ta } of contractive linear operators in the Hilbert space 7t b y block matrices as
T~ =
(o :)
B~
,
T~=
(o :)
B~
'
Ix
o)
"
where BI and B2 are arbitrary not commuting unitary operators in X and Ix is the identity operator
in g. Obviously, the operators T~ commute, because TkTj = 0 for k,j = 1, 2, 3.
Let T N = {A E C N : IAk] = I, k = I,..., N} be the N-dimensional toms. Consider the threedimensional linear subspace 8 = span{A1, A2, I} of the C*-algebra C(T "2) of continuous complex-valued
functions on the two-dimensional toms T 2 generated by the functions A~(A) = A1, A2(A) = A~, and
I(A) = 1. We define a linear map ~0:,9 --* s
by setting
~(aA1 +
,8A2 + 7 I ) =
aB1 + 3B2 + 7I,~
for any a, ~, 7 E C. Let us show that the m a p ~ is contractivebut not completely contractive. According
to Remark 1, we have
II~o(,~A, + ~A2 + 7I)11 = II~n, + ~B= + 7Ixll < max I~=~ + ~== + 71
-- :EA=
=
max
Afi'l~
laA, -k/~A2 + 71 = II'~A, + ~A~ + vXll
(here we have used the maximum modulus principle for analytic functions in t h e bidisk). Thus ~ is a
contractive map. Now, suppose that additionally ~a is completely contractive. By" Theorem 1.2.9 [4], it
can then be extended to a completely positive linear map ~: C('I[a) -* s
a n d ~(I) = I x . Such a
map has the form [9]
~ ( f ) = Pxa-(f) IR',
f E C(T2),
D ,t" and P x is the
where lr is a representation of the C*-algebra C(T u) in some Hilbert space y
orthogonal projector in 32 onto the subspace X . For any z E X , we have
I1=11= IIBk=ll = Iko(Ak)=ll = IIq(Ak)=ll = IIPx=CAk)=ll,
k
=
1,
2.
On the other hand, ~r(Ak) are unitary operators in the space 32, because At, a r e unitary elements of
the C*-algebra C(T2). Therefore, [[z]] = [[Ir(Ak)z]]. Thus []PxTr(Ak)z][ = ]]~r(Ak)z[[; this is only possible
if P x a ' ( A k ) z - a'(Ak)z, i.e., ~r(Ak)z E ~ ' . Since z is a n arbitrary element of , u w'e o b t a i n
7r(Ak) [ ,.t'
=
Px~'CAk)l X = ~(Ak) = ~(A,) = Bk,
k = l, 2-
Therefore,
B~Ba = a'(A,)~r(A~)IX = a'(A,A~)IX = 7r(A2A,)IX = a'(A2)~r(A,)l X = B=B~,
which contradicts the choice of the operators B1 and B2. Hence ~0 is not completely contractive, i.e., there
exists a positive integer rt > 1 for which II~.ll > 1. This means that there exists a t r i p l e A = {A~, Az, As}
of square matrices of order n over C such that
lib.(A, | A, + A2 | A2 + A3 @I)ll > IIA, | A, + A2 @A2 + Aa ~ Ill.
188
(o)
Let us examine the left-hand and right-hand sides of this inequality separately. We have
lib.(At | A~ + A2 @ A2 + Aa | I)ll= IIAI| BI + A~ | B2 § A3 | Ixl[
= I1(~. @ P{o}ex)(A, @T, +A2 @T2 +A3 | T3)[C" |
< I[A1 |
+A2 @T2 +A3 @T3[],
{0})11
where I . is the identity matrix of order n, X $ {0} and {0) (BX are the subspaces of 7"[ = X(BX formed
by all vectors h = =1 (Bm2 with =2 -- 0 and =1 = 0, respectively, and P{0}e,v is the orthoprojector in 7"l
onto the subspace (0} (B X. The right-hand side of (6) satisfies the equality
IIA, | AI § ,42 | A2 § As | Ill
=
max
.XE.r'~
IIA, A~ § A2,X,_ § A:dl,
because the C*-algebra M,(C) | C(T 2) is isomorphic to the C*-algebra C(T 2, Mn(C)) of n x n matrix
functions continuous on T 2 (see, e.g., [11, Proposition 4.7.3]). However,
~
IIARA~§
A2A2 §
Aall =
max
CE'~
IIA~G §
A'2(2 §
A~r
= max
zEA a
IIA~,~§ A2z2 § Aazall
according to the maximum principle for analytic matrix functions of several variables (see, e.g., [12]).
Thus we finallyobtain the inequality
IlA,|
+A2 |
+As |
> m ~ llA~z~+A2z2 + A3z311,
zEA a
which coincides with (5) for the linear homogeneous matrix function (4). This completes the proof of the
theorem. []
This research was supported by INTAS under grant No. 93-0322ext.
References
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(1951).
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operator theory," J. Funct. AnaL, 16, 83--100 (1974).
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E-mQi/address: Dmltriy.KalyuzhnlyOp25.f61.n467.z2.fidonet.org
Translated by O. V. Sipacheva
189