I nternat. J. Math.
Math. Sci.
133
(1978) 133-136
Vol.
RESEARCH NOTES
ON VON NEUMANN’S INEQUALITY
ARTHUR LUBIN
Department of Mathematics
Illinois Institute of Technology
Chicago, Illinois 60616
(Received October 31, 19 77)
Von Neumann’s inequality states that for a contraction T acting on a Hilbert
space H
IIp(T) ll <_
(v)
sup
{Ip(z) I: Izl
holds for all polynomials p.
{T I,
< }
The analog for a set of comuting contractions
,Tn},
lp(T
Tn) ll
< sup
{Ip(zl
z
n )I:
z
i
< I}
In fact, for any c > o, there exist
is known to be false for n > 2.
where n is sufficiently large, and a polynomial p such that
ll(P(Tl,...,Tn) ll
> c
sup(Ip(z I
,Zn) l:Iz+/-l
<
,
In this note we establish the following weakened version of
PROPOSITION I.
Let
{TI,...,Tn }
Then for any polynomial p,
{TI,...,Tn}
(2)
(Vn):
be co,muting contractions on a Hilbert space H.
A. LUBIN
13
llP(Tl,...,Tn) ll
i.e.,
< sup
Dn {(Zl’ ’Zn) Izi
<
n1/2}
Zn) l:Izll
{Ip(zl,
<
is a spectral set for
n1/2},
,Tn)
(TI,..
Our proof is an easy consequence of the following proposition.
PROPOSITION 2
n
(3 1.9.2). Let {S I
II sill 2 < i.
i_Z_l
Then
{Sl,...,S n}
S
n
be commuting contractions with
has a commuting unitary dilation (in fact
a regular one) and it therefore follwos immediately that
PROOF OF PROPOSITION i.
n
Z
i--I
Then
[I s i
2
{TI,...,Tn}
Given
-i
n
i--i
Given any polyno.al
I[r+/-l
2
p(zl,...,Zn)
< i
let
{Sl,...,S n}
S.1 n-1/2 Ti’
let
satisfies
l,...,n.
i
holds for {S
(v)
I
n
so
q(zl,...,z n)
p
S
n
(n1/2z I ’’’’’n’Zn )"
Then
l]p(n1/2Sl’’’’’n1/2Sn
I]P(TI’’’’’Tn) I]
lq<sx
COROLLARY 3.
sup
{lq(wl,...,Wn) l: lwil
sup
{Ip(nwl,...,n%n) lwil
sup
{[p(zl,...,Zn) l:lzil
(see (i) p. 279).
on H has the polydisc D
PROOF.
,Sn)
-
< l}
1/2}
< n
Any set {T I
(Zl,... z n) :Izil
n
< i}
T
n
of commuting contractions
as a complete spectral set.
< n
UI,...,Un on
q(UI,...,U n) for all
By proposition 2, there exist commuting unitary operators
a Hilbert space K containing H such that
polynomials q, where S.
we have that {N
I,...,Nn}
n
q(Sl,
To and P projects K
S
n)
P
H.
onto
is a normal dilation of {T
l,...,Tn
Setting N
i
n
Da {(zl’’’’’Zn):Izi
spectral set for all commuting contractions
{TI,...,Tn
<
ai}
i
with joint spectrum
sp(N) contained in the boundary of D and the corollary follows as in (i).
n
Similarly, it follows that
U
is a complete
if Z a
-2
i
< i
VON NEUMANN’S INEQUALITY
Since the common intersection of such D
a
in general a complete spectral set since
COROLLARY 4.
{TI,...,Tn}
If
135
is the unit polydisc
(Vn)
can fall if n
D, which is not
>_ 3,
we have
is a set of commuting contractions such that the
intersection of any two complete spectral sets is also a complete spectral set,
then the unit p.olydisc D is also a complete spectral set.
We note that von
Neumann’s
original paper (4) showed that for a single con-
traction the intersection of two spectral sets need not be a spectral set.
n)
Since (v
holds for n
possible result.
PROBLEM.
V(n)
2, we see that proposition i is not the best
This prompts the following
Find
llp(T l,...,Tn)l
inf{r:
< sup
{Ip(z l,...,zn) l:Izll
< r}}
We note that Theorem 1.2(b) of (2) yields information concerning the growth
of V(n) as n increases.
sup
{Ip(z I
z
n
nomials of degree
)l:Iz i
Since
< r}
r
s
sup
s, we have for any
{Ip(z) l:Iz i
> o, V(n)
< i} for homogeneous poly-
n
(1/4-e) for n
sufficiently
large.
ACKNOWLEDGMENT.
This research was partially supported by NSF Grant MCS 76-06516
A01.
REFERENCES
i.
Arveson, W.
2.
Dixon, P. G. The von Neumann Inequality for Polynomials of Degree Greater
Than 2, J. London Math. Soc. 14 (1976) 369-375.
3.
Sz.-Nagy, B. and C. Foias.
North-Holland, 1970.
4.
von
Subalgebras of C*-algebras II, Acta. Math. 128 (1972) 271-308.
Harmonic Analysis of Operators on Hilbert
Space,
Neumann, J. Eine Spektraltheorie fur Allgemeine Operatoren Eines
Unitaren Raumes, Math. Nachr. 4 (1951) 258-281.
KEY WORDS AND PHRASES.
nequality.
Comming coction on a Hb Space,
AMS(MOS) SUBJECT CLASSIFICATIONS(1970).
47A25o
spe s,