A characterization of pick bodies

A CHARACTERIZATION OF PICK BODIES
B.
COLE, K. LEWIS
AND
J. WERMER
ABSTRACT
Let 2 = (z
v
, z
n
) be an /i-tuple of distinct points in the open unit disk. We define the Pick
body
@(z)
as the totality of points w = (w
v
,w
n
) in C
n
such that there exists
feH
00
with H/l^ <
1
and J(z
f
) = w
jt
for
1
<_/ <

n.
We discuss the properties of Pick bodies and characterize them among compact subsets of
C. We also study related questions concerning certain algebras of operators on Hilbert space.
0. Introduction
We consider C
n
as an algebra with coordinatewise multiplication and let || • ||
denote an algebra norm on C
n
. The closed unit ball in this norm {zeC
n
\
\\z\\
^ 1} is
a compact convex circled subset of C
n
with non-void interior.
In [4] we studied a particular class of such norms which arise as follows: let A be
a uniform algebra on a compact Hausdorff space X, with maximal ideal space Jt, and
fix an H-tuple of points M
l5
, M
n
in M. Let / = {g e A \ g(Mj) = 0,1 ^ j ^
n}.
Then
/is a closed ideal in A. The quotient algebra A/Us then a normed algebra isomorphic
as an algebra to C
n
under the map [/]

H>-
(J[M
X
),
,J(M
n
)) forfeA, where [f\ denotes
the coset of/ in A/1. The image of the closed unit ball of A/1 under this map is
n
| 3feA with/(M,) = w
p
1 <y ^ n, and
||[/]||
^ 1}.
In addition to the above mentioned properties valid for any algebra norm on C
n
, Si
now has the following additional property. Let k be any positive integer and let
A* denote the closed unit polydisk in C
n
. If P is a polynomial in k variables with
||P||
A
*
=
max
(eA
*
1^(01
^ 1, then for each

fc-tuple
of elements x',x", ,x
w
in 2) we
have that P(x',x", ,x
m
) again is in 9.
This statement follows easily from the definition of @) as the closed unit ball of
A/1
and is also a special case of a theorem due to I. Craw and (independently) B. Cole
concerning ^-algebras, which is given in [2, §50, Proposition 5]. As defined in [4], a
(2-algebra is a Banach algebra which is a quotient of a uniform algebra. This suggests
the following definition.
DEFINITION.
A subset K of C" is called hyperconvex if K is compact with
non-empty interior so that, for all positive integers k, whenever P is a polynomial
in k variables with
||P||
A
*
^ 1 and x',x", ,x
w
are in K, then P{x',x", ,x
(k)
)eK.
We note that, expressed in terms of coordinates, if x' = (x[, ,x'
n
),
x"
= (xl ,x"

n
) etc., then P(x',x", ,*<*>) = {P(x'
x
,xj, ,*<*>), ,P{x'
n
,<, ,xf)).
Received 21 October 1991.
1991
Mathematics
Subject
Classification
30E05.
/.
London
Math. Soc. (2) 48 (1993) 316-328
A
CHARACTERIZATION OF PICK BODIES 317
We may obtain an example of a hyperconvex set as follows: we select an n-tuple
of distinct points z
l5
,z
n
in the open unit disk in C and put
®(z) = {vveC"13/e//
00
with/(*,) = "„
1
^j ^ n, and
\\f\\
n

^ 1},
where z represents the point (z
15
,z
n
) in C
n
.
We call the set
3>{z)
a Pick body. It follows at once from the definition that 2)(z)
is hyperconvex.
Let A(A) denote the disk algebra and let / be the ideal in A(A) consisting of
CLAIM.
For z e C
n
, put
9 = {weC
n
\lfeA(A) withf[
Zj
) = w
p
1
^j^n, and MIL < 1},
where
\\[F]\\
denotes
the norm of[F] in
A(A)/I.

Then ®(z) = ®.
Proof.
Fix
w
e ®(z). Choose
<p
e
H™,
\\ <f> || ^ 1, such that w
j
=
(f>(zj)
for each/. Put
A(0 = 0((1-1A)O- Then/
fc
e^(A) and \\[f
k
]\\ ^ 1. Hence
It follows from the definition that 9) is closed. Therefore w = lim^^w
Conversely, let
WE
Si. For k any positive integer, there exists geA(A),
||g||^l + l/fc, with g(Zj) =
Wj
for all / Passing to a pointwise convergent
subsequence, we obtain gsH™ with HgH^ ^ 1 and g(zj) = w
}
for ally. Hence we3)(z).
LEMMA
1. Fix £ = ((

19
,£„) in C
n
wi7/i |C
y
| <
1
for each] and (, # Cy'/' #/• Let
d® denote the boundary of 3){Q. Then the points C = (Cf CS) fo/ong to 5^ for
k= 1, ,«-1.
The following argument is due to Herbert Alexander. If
then there exists geA(A) such that g(Q =
C*
fory=
1, ,«,
and ||g|| < 1. Hence
\g(z)\
< \z
k
\ for z on the unit circle. By Rouche's theorem, then, the functions z
k
and z
k
—g have the same number of zeros in \z\ < 1. But z
k
—g vanishes at the
n points £
p
1 </ < n, while z* has exactly k zeros. This contradiction yields that
>,

as desired.
Thus each Pick body ®(Q has the two properties: (i) @(Q is hyperconvex in C
n
and (ii) the kth power of
C
belongs to the boundary of ®(Q for k =
1, ,«
—
1.
We
shall show that these two properties characterize Pick bodies.
THEOREM
1. Let K be a subset ofC
n
. Assume that
(1) K is hyperconvex,
(2) dK contains a point z = (z
15
, z
n
) with |z^| <
1
for each j and
z<
^ z, if i #y,
such that z, , z
n-1
all belong to dK.
Then K is a Pick body.
1.

Lemmas concerning operators
Let Jf be a finite dimensional Hilbert space and B a linear operator on df.
318 B.
COLE,
K.
LEWIS
AND J.
WERMER
LEMMA
2. Assume that each eigenvalue of B has modulus less than 1 and that
\\B\\
= ll^""
1
!
=
1. Choose a unit vector
<f>
in
tf
with ||5
n
"VII
=
1- Then
tne
n-tuple
<f>,
B^, ,^'
1
^

is linearly independent.
Proof.
Note that Wy/f = WBy/f is equivalent to {{I-B*B)\j/,
y/)
= 0 and since
I—B*B
is positive this is equivalent to (I—B*B)y/ = 0. Note that
so all the inequalities are equalities. Hence we have that
(I—B*B)B
i
<f>
= 0 for
j =
0,\, ,n-2.
Now suppose that the w-tuple $, i?0, , i?""
1
0 is linearly dependent; then there
exists
k
<
n — 1
such that
B
k
<j>
is a linear combination of
<f>,
B(f>, ,B
k
~

x
<j>.
The space
spanned by these vectors is invariant under B and is contained in the kernel of/-
B*B
by the above. Hence B is unitary on this subspace and thus has an eigenvalue of
modulus 1. This contradicts our hypothesis and so the n-tuple
(f>,
B(J>, ,B
n
~
l
(j>
is
linearly independent.
LEMMA
3.
Let 3ft'bean n-dimensional Hilbert
space,
and let B be
a
linear operator on
Jtf with eigenvalues z
x
, ,z
n
. Assume that
(3)
\
Zj

\<\,forl^j^n,
(4)
z
t
*z
t
ifi*j
t
(5) ||*||-II*-
1
11
=
1.
Then there exists a basis of eigenvectors
v
lt
,
v
n
corresponding to z
1?
, z
n
so that
(v
<5
v,)
= for
1
< i,j

^
n.
(6)
\.—z
i
z
i
Proof.
Choose 0eJ^ with ||0||
=
1
and
H*""
1
^
=
1.
As
in
Lemma
2,
it follows that
(I—B*B)B
1
<f>
= 0, for
j
= 0,l, ,n-2, since
for
each such j,

||B(B
1
0)||
=
1
=
||#0||. Also by Lemma 2, 0,
B<j>, ,B
n
~
l
<f>
are linearly independent.
Hence I—B*B has a null space of dimension at least n—
1.
Since B is not unitary, we
must have rank(/-5*£) = 1.
Let v\, ,v'
n
be any basis of eigenvectors for B, and put T= I—B*B. Since
rank
T=
1, the
nxn
matrix ((7X,v])) has rank 1. This matrix
is
also positive
semidefinite. Hence, there exist scalars c
ls
,c

n
so that
Since |cj
2
=
(1 —
|zj
2
) ||v|||
2
#
0, we see that c
t
#
0. So the vectors v
(
= v'Jc
i
define a
basis of eigenvectors with
and Lemma 3 is proved.
Fix z = (z
v
, z
n
) in C
n
satisfying (3) and (4). We define the inner product (,), on
C
n

by
n
(MX
13
L
t
i
s
j0-~
z
t^})~
1
fort,seC
n
.
A
CHARACTERIZATION
OF PICK BODIES 319
An elementary calculation gives
C>4
=
^
V
l
±
~
ie
dO,
and
so

(, \
is
positive definite and makes C
n
into an n-dimensional Hilbert space which
we designate by C".
For each w = (w
1
, ,w
n
), we denote by P
w
the operator on C" defined by
P
w
(t
1
, ,t
n
) = (w
1
t
1
, ,w
n
t
n
). We call P
w
a Pick operator on C?. We also set

S = P,
Let
e
v
,
e
n
denote the standard basis on C
n
. Then, suppressing the subscript z,
we have
(e
it
e
}
)
= for
1
^ ij ^ n. (7)
\—Z
i
2
j
By the definition of S, we also have
Se
i
=
z
i
e

i
for
1
^y ^ n. (8)
LEMMA
4. Let B
be
a
linear operator
on
JiC satisfying
(3), (4), (5),
and
hence
(6).
Then
there
exists a
unitary
map U from Jtf to C?
such
that S =
UBU'
1
.
Proof Choose v
x
, ,v
n
as in Lemma 3. From (6) and (7), it follows that

(e
(
, e
t
)
=
(v
<5
v,)
for
1
^ ij < n.
We define an operator U from Jf to C" by setting
Uv
}
= e
i
fory =
l, ,n
and
extending linearly. Now
UBv
}
= Vz
i
\
i
=
z
i

e
i
= Se
}
=
SUv
}
,
and so UBU'
1
= S. Also,
if 0e^f,
</>
= Yih
v
p
then
= II
E
hUv
t
f
= ||
£tfyf
=
Et,t
k
(e
}
,e

k
)
= £
t
}
f
k
(v
}
,v
k
)
= || E
^v,||
2
=
||0||
2
.
Hence U is unitary, and we are done.
We now drop the assumption that our Hilbert space
3f?
is
finite-dimensional,
and
we choose an n-tuple of bounded operators E
}
,\ ^j < n on #?. We assume that
E
i

E
i
= ^
if i
^j, E} = E
i
# 0
for
1
< i,j ^ n and
E
1
+
E
2
+

+
E
n
= I. (9)
For each w = (w
lt
,
w
n
)
in C
B
, we put

T
w
= E
w,
^ and let si =
{T
w
\ weC
n
}. Then
sf is an n-dimensional operator algebra on Jf, and the map T
w
^
w is
an isomorphism
of sf onto the algebra C".
LEMMA
5. Let 9 = {weC
n
\ \\TJ ^
1}.
Fix z =
(z
15

,z
n
)
in
C

n
such
that
\z
t
\
< 1
for
1
^7 < n and z
i
# z
}
if
i
^j. Put T
= T
z
and
assume
that
\\T\\
=
1
and
there exists </>e^ with WT^fiW
=
\\<f>\\
*
0.

(10)
Then
2 is the Pick body 2{z).
Proof Assume without loss of generality that ||0|| = 1. We have
- = ||r»-
1
0ll = i. (ii)
320 B.
COLE,
K.
LEWIS
AND J.
WERMER
Let
"W
be the subspace of M spanned by the vectors
<j>,
T<f>, ,
T
n
~
x
(f>.
Since st is n-
dimensional, T
n
is a linear combination of /,
T, ,
T
n

~
l
and so if is invariant under
T. We shall show that iV is ^-dimensional and that the restriction of T to if is
unitarily equivalent to the operator 5 on C
2
.
Define f to be the restriction of T to
HT.
By (11),
so \\f1| = IIT
71
"
1
!! = 1. Let x be an eigenvector of f with corresponding eigenvalue X.
Choose
j
Q
such that E
}
x ^0. Then
Hence
X
= z
}
and so
|A|
< 1. Hence each eigenvalue of f has modulus less than 1. So
Lemma 2 applies to f and gives that the w-tuple of vectors
<j>,

f
<j>, ,
f
n
"
1
0 is linearly
independent and hence is a basis of #". Since for each k, T
k
= £^z*£,, the vectors
E
1
(j>,
E
2
0, , E
n
$ span itr. Also, for each ;, TE
i
<f>
- (£
z
fc
£
fc
) Erf =
z,
E
i
<f>.

So the
set
JE*!
0,
£"
2
(j>, ,
E
n
(f>
is a basis of eigenvectors for T.
Thus Lemma 4 applies to T and so there exists a unitary operator U from T^" to
C
z
n
with S =
UTU*.
Fix a in ^. Then ||2J < 1. Let Q be a polynomial with Q(z
}
) = a
}
for
y =
1, ,«.
If
P
a
denotes the Pick operator introduced above, then
P
a

= Q(S) =
UQ(f)U*
=
UQ(T
Z
)U*
= UT
a
U*,
and so HPJ < ||£|| < 1. Hence for all /eC
z
n
,
||P
a
f||
2
< ||r||
2
, the norms being taken in
C". Thus we have
t
afa^IJLl-ZjZj-
1
^
t
hUl-z^y.
(12)
By Pick's theorem, (12) implies that there exists feH™ with H/H^ ^
1

and/^) = a,
for ally, and so ae^(z). Thus
2>
c
Q)(z).
On the other hand, fix a in
Q)(z).
Then, for each e > 0, there exists a function
/e,4(A), which we may take to be a polynomial, with
\\f\\
m
^
1
+e and^z^) = a
}
for
all j by the Claim before Lemma 1. Then
./TO
=
and so by von Neumann's inequality,
||./(^)||
^ 1+e. Hence
||7^||
< 1+e for all
e > 0, implying that
||
7^||
^ 1, and so a e Si. Therefore
2>{z)
a

S>,
and we have shown
that
Q)(z)
= 2.
2.
Proof of Theorem 1
In the proof of Theorem 1, we shall use Cole's representation theorem, [2,
Theorem 7, p. 272] which states that the quotient of a uniform algebra by a closed
ideal is isometrically isomorphic to an algebra of operators on a Hilbert space.
Furthermore, every operator in the algebra achieves its norm on some unit vector.
Proof of Theorem 1. By hypothesis K is a hyperconvex subset of C
n
, and hence
is the closed unit ball of a Banach algebra
3&
which is algebraically isomorphic to C
n
and which satisfies the following condition: whenever x',x", ,x
{k)
are in the closed
unit ball of ^, then P(x',x", ,x
lk)
) is also in the closed unit ball of
38
provided that
P is a polynomial in k variables with
||P||
A
*

< 1.
A
CHARACTERIZATION OF PICK BODIES 321
Proposition 5 in [2, §50] implies that J
1
is isometrically isomorphic to
the quotient A/I of a uniform algebra A by a closed ideal /. Also there exist
M
x
, ,M
n
zM
A
with /=
{feA\j{M^
= 0, j = 1, ,«}. By Cole's representation
theorem stated above, there exists a further isometric isomorphism of A/1 on an
algebra of operators, d, on a Hilbert space, Jf. Combining these two maps,
we have an isometric isomorphism x'M -> s/ and
for all Test, there exists
</>e3?
with ||0|| =
1
and
\\T</>\\
= ||r||. (13)
Let |/J e
A/1
for 7 =
1, ,«

be such that /,(M
t
) =
<5,
t
,
and put £, = T((/J). Then
E
x
, ,E
n
satisfy condition
(9).
For w e
C
n
,
we put 7; = £,
w,
£
r
Thus, r(z
fc
) = 7J for
A:
=
1,2,
In particular, with z as in Theorem
1
(2), we have that

\\T
Z
\\
=
||
T^\\ = 1
in st since ||z|| = Hz""
1
!! =
1
in @.
By (13) there exists
<pe J4?
with ||0|| =
1
and WT^^W = 1, and so (10) in Lemma
5 is satisfied. Hence, by that lemma,
S>
= {weC
n
\\\TJ < \} = 2{z). Since T is
isometric,
w
is in the closed unit ball of
$1
if and only if
||
£,
w
j

E
}
\\
^
1
which occurs
if and only if
we&>(z),
and so the closed unit ball of
38
is
Q)(z).
Thus K =
Q){z)
and
Theorem 1 is proved.
3.
Operator algebras
In this section
we
give an alternative approach to some of the questions considered
in the previous sections of this paper. These results are independent of those obtained
earlier and are somewhat more general. Theorem 2 implies Theorem 1, and
other results generalize Theorem 1 in various ways. The price to be paid is that
these calculations are less explicit and elementary than those used in the proof of
Theorem 1.
The main objects of interest here are finite dimensional commutative Banach
algebras, principally operator algebras on Hilbert space (which are not assumed to be
self-adjoint). When $t is any semisimple Banach algebra of dimension
n,

the Gelfand
theory identifies the closed unit ball of
stf
with a subset
Q)^
of
C
n
.
Specifically, we put
%
=
{{x{M
1
), ,x{M
n
))\xesf,\\x\\ ^ 1},
where the maximal ideal space of
$0
consists of
the n
distinct points M
ls
, M
n
. Since
the Gelfand representation is essentially the map
<D:
sf -> C
n

with O(x) =
(x(M
x
), ,
x(M
n
)),
we see that O is an isomorphism when C
n
is viewed as an algebra under
coordinatewise operations. Furthermore, under this map, the closed unit ball of s#
corresponds to ^.
The hyperconvex set Si, associated with the Q-algebra s$ = All in the
Introduction, is precisely the set
S>^
defined above. Moreover, as discussed in the
proof of Theorem 1, every hyperconvex set arises in this way. Thus, from the Banach
algebra point of view, Theorem
1
can be interpreted as a statement about the unit ball
of a finite dimensional semisimple Q-algebra.
We now turn our attention to a broader class of Banach algebras: finite
dimensional commutative algebras of operators on Hilbert
space.
Since
we
know that
every Q-algebra is in fact an operator algebra, results about operator algebras
automatically apply to Q-algebras. And consequently, general results about the unit
ball of an operator algebra provide information about hyperconvex sets.

11 JLM 48
322
B.
COLE,
K.
LEWIS
AND J.
WERMER
For
the
remainder
of
this section,
we let A
denote
the
disk algebra
A(A) and let
n
be a
positive integer.
All
Banach algebras considered
are
assumed
to
contain
an
identity.
DEFINITION. A

Pick
algebra
is a
finite dimensional Banach algebra
J/
which
is
isometrically isomorphic
to A/J
where
/
is
a
closed ideal
in A
with hull(/)
£
int(A).
Furthermore,
an
element xesf
is
said
to be a
Pick
generator
if
{p(x)\p
is a
polynomial,

\\p\\
A
^ 1}
constitutes
a
dense subset
of
the closed unit ball
of
$0.
Note that
J
in
the
above definition
is of
the form bA where
b is a
finite Blaschke
product
and
that
the
Pick algebra
A/J
has
the
Pick generator
[idj
where

id
A
is the
identity
map on A (the
complex coordinate function). Moreover,
s/ = A/J is
semisimple
if and
only
if b has
simple zeros,
and
this
is
exactly
the
situation
considered
in the
Introduction;
so we see
that Pick bodies arise
as
S>^
where
si a
semisimple Pick algebra.
We shall prove
a

slight modification
of
a theorem
due to D.
Sarason which gives
a representation
of
Pick algebras
as
singly generated algebras
of
operators
on a
Hilbert space. Sarason's theorem uses
if
00
in
place
of A and an
arbitrary inner
function
in
place
of
b.
Our
proof will also work
in
this situation; however,
we

need
the disk algebra version
in
what follows.
We review some basic facts concerning Hardy spaces, denoted
by H
p
. Let m
denote Lebesgue measure
on the
unit circle.
Let S be the
shift operator
of
multiplication
by z on H
2
:
Sf{z)
=
zj{z) for
feH
2
. A
function
beH
2
is
called
inner

if
\b\
=
1
a.e.
[m]
and a
function
geH
2
is
called
outer
if
g
is a
cyclic vector
for S,
that
is,
the set of
all
pg,
p a
polynomial,
is
dense
in H
2
. A

theorem
of
Beurling states that
every closed subspace
of H
2
invariant under
S
has
the
form
bH
2
,
where b
in an
inner
function,
and
every function/ei/
2
can be factored
as/=
bg
with
b
inner and g outer.
Also,
for
every

heH
1
,
there exists
keH
2
with
\h\
=
\k\
2
a.e.
[m].
Following Sarason [8],
we
fix
an
inner function b,
let Jf
b
= H
2
Q
bH
2
, denote
by
Pjr
b
the

orthogonal projection from
H
2
to J^, and put
T
b
=
P
jCb
S\
:t
-
b
,
where
S is
multiplication by
z on H
2
.
The orthogonal projection from
L
2
to H
2
is denoted
by P.
ForfeA,
define AS)
on H

2
by AS)g
=fg for geH
2
and
define
<b
b
(f)
=
P^JiS)
\
Xb
.
Note that
O
6
is a
contractive
map
from
A to
$8(JQ,
the
space
of
operators
on
JfT
b

.
Since
bH
2
is
invariant under
5, we
have that
tf
b
is
invariant under
S*; so
7?
=
5*1^ and
p(T
b
)*y =
p(S)*y
for ye
JT
b
.
Hence
for
x,yeJr
b
,
(p(T

b
)x,y)
=
(x,p(T
b
)*y)
=
(x,p(S)*y)
=
(p(S)x,y)
=
(P
Xb
p(S)x,y).
That
is,
p(T
b
)
=
0>
b
(p).
This shows that O
6
is multiplicative
on
polynomials,
and so by
norm continuity of multiplication,

<D
6
is
an
algebra homomorphism. From here on we
shall write <P
6
(/) as
J[T
b
).
Not only
do we
have
\\J{T
b
)\\
^
\\f\\,feA,
but
since
b(T
b
)
= 0, we
have
iiyroii
=
MTj+bmgmw
^

\\f+bg\\
for
all geA, so
||/(7;)||
<
||/+M||,
the
coset norm of/in
A/bA. The
content
of
the
next theorem
is
that this
is
always
an
equality. Sarason [8] used this result
to
prove
Pick's theorem.
A
CHARACTERIZATION
OF
PICK BODIES
323
PROPOSITION
1
(Sarason).

Let b
be
a
finite
Blaschke product
and letfeA. Then
= ||/TO
|| •
In
particular,
the
algebra
st
generated
by T
b
is a
Pick
algebra
and
has
Pick
generator
T
b
.
Proof.
By
the
Hahn-Banach Theorem, there exists

a
complex measure
ji
with
\\f+bA\\
=
jfdfi, ||//||
< 1, and
jbgd/i
= 0 for all geA. We
shall show that there
exist x,yeki,, \\x\\,
\\y\\
^ 1,
with ffdfi
=
ffxydm. This will prove
the
theorem
since
we
shall have
\\f+bA\\
=
jfd
M
=
(fx,y)
=
(P

Kb
fx,y)
=
<J[T
b
)x,y)
^
\\f[T
b
)\\
^
||/+M||.
By
the
Riesz Theorem, there exists
heH
1
with bdfi
= hdm.
There exists
geH
2
with
\h\
=
|g|
2
,
so
dfi

=
u
\g\
2
dm,
where
u is
unimodular.
By
Beurling's theorem,
we
may also assume that g
is
outer. This implies that
ug
_l_
bH
2
since
for
all
keA
we have
0
=
jbkdfi
=
J
bku
\g\

2
dm
=
(bkg,
ug)
and the set of
all
bkg for
keA
is
dense
in
bH
2
.
It follows that
y
=
P(ug)
1
bH
2
and
so
yeX
b
.
Also,
Put
x

=
P
Xb
g; then
||x||
<
1
also.
Now
we
show that jfd/n
=
(fx,y).
We
have
If dfi
=
jfu
\g\
2
dm
=
(fg,
ug)
=
(P(fg),
ug)
since fg e
H
2

.
So
(P(fg),ug)
=
(fg,P(ug))
=
(fg,y)
=
(f{S)g,y)
=
(g,f[S)*y)
= (P^
since
yeJf
b
and
Jf
b
is
invariant under
S*.
Finally,
(P^Asry)
=
(x,AS)*y)
=
(f{S)x,y)
=
{fx,y)
and

we are
done.
Our methods rely
on a
result
in
[3],
and
so for
completeness
we
include
a
proof.
LEMMA 6 (de
Branges
and
Rovnyak).
If T
is
an
operator
on
a
Hilbert
space 3V
with
||31
=
1,

rank(7-
T*T) = 1, and
\\T
m
x\\
-•
0
as
m
-•
oo
for
all
xeJf,
then
T
is
unitarily
equivalent
to the
backward shift
S* on H
2
restricted
to an
S*-invariant
subspace.
Proof.
For
two vectors u,

v
in
2tf,
let
u
® v
be the
operator defined
by
(u <g> v)w
= (w,v)u.
The
hypotheses
on
Timply that there exists teJf with I—T*T=
t®t.
Yoxxetf,
\\T
n
x\\
2
-\\T
n+1
x\\
2
= (T
n
x,T
n
x) -(T*TT

n
x,T
n
x)
and
so
£»\(T
n
x,
t)\
2
=
^o(\\T
n
x\\
2
-\\T
n+1
x\\
2
)
=
\\x\\
2
since \\T
n
x\\^0. Hence,
the
function defined by
f

x
(z)
=
f^{T
n
x,
t)z
n
belongs
to H
2
and
satisfies
||/J =
||x||.
Let
Ube the isometry
Jf^H
2
given by U(x)
=f
x
.
Since/
(Tx)
=
S*f
x
,
we

have UT=S*U
on
Jf,
and
so the subspace
Jf =
XJ{3^)
is invariant under
S*.
Viewing
U as
a
unitary
map onto
Jf, it
follows that
UTU*
=
5*1^.
11-2
324
B.
COLE,
K.
LEWIS
AND J.
WERMER
The next result provides
the key
link between operator algebras

and
quotients
of
the disk algebra.
PROPOSITION
2.
Let
JF be n-dimensional Hilbert space, and let s&bea commutative
subalgebra
of
@{tf).
Let
Test
so
that
\\T\\ = 1,
rank(/-T*r)
= 1, and T has
no eigenvalue
of
modulus
1.
Then,
$0 is an
n-dimensional Pick algebra with Pick
generator
T.
Proof Since rank(/-
T*T) =
rank(7-

TT*) and
since
the
spectral radius
of T,
and hence
T*, is
less than
1, the
hypotheses
of
Lemma
6 are
satisfied
by T*. So T*
is unitarily equivalent
to S*
restricted
to an
invariant subspace which,
by
Beurling's
theorem,
has the
form
Jf
b
for
some inner function
b.

Consequently,
$0 is
unitarily
equivalent
to a
commutative subalgebra
si of
88{X
b
) containing T
b
,
and,
under this
equivalence,
T
b
corresponds
to T.
Also, since
tf has
finite dimension,
b is a
finite
Blaschke product.
For
the
vector
<j>
= P

x
1,
{<j>,
T
b
$, , T
b
~
x
0} is a
basis
for X
b
, and
therefore
R
s
j/with
R<f>
=
0 implies that
R =
0.
It
follows that dim(j/)
= n and
j/is the algebra
generated
by T
b

. So, by
Proposition
1, s/ is a
Pick algebra with Pick generator
T.
A Banach algebra
s&
satisfies
the von
Neumann Inequality
of
order
1
if,
whenever
xestf with
||x|| < 1,
||/?(x)||
<
maxj/jl
for
every polynomial/?. Von Neumann showed
that this condition
is
satisfied
if
$$
is a
subalgebra
of &(&) for a

Hilbert space 3f?\
see [4]
for
more details.
The
next result
is
used
in
this paper only when
sd
x
is
such
an
algebra.
LEMMA
7. Let s^ be a
commutative Banach algebra satisfying
the von
Neumann
Inequality of order
1, and let s^be a
Pick algebra.
Let
Q>
be a
norm
1
homomorphism

from
s%
into «#£,
and let xes^
with \\x\\
^ 1 so
that
O(x) is a
Pick generator
for s^.
Then,
<f>
induces
an
isometric isomorphism between j^/kerCO)
and s^. In
particular,
if
O
is
one-to-one, then
O
establishes
an
isometric isomorphism between
s^ an
Proof. Let
<£>(*)
= y. For
every polynomial

p,
<X>
maps
p(x) to p(y).
Restricting
attention
to
polynomials
in the
unit ball of the disk algebra,
the
hypotheses guarantee
that
O
maps
the
unit ball
of s^,
onto
a
dense subset
of the
unit ball
of
s%.
The
assertion
now
follows.
LEMMA

8. Let n ^ 2. If T
is
an operator on
a
Hilbert space
#?
with
dim(jf) ^ n,
II
T
k
\\
= \ for 0 < k < n, and T
has
no
eigenvalue of modulus
1,
then
I— T*T
has rank
1
and
dim(^f)
= n.
Proof.
Choose
a
unit vector
<p
e

34?
with
||
J
1
""
1
01|
= 1.
From Lemma 2,
we
know
that
</>,
T(f>, ,
T
n
~
x
0 is a
basis
for 2tf.
Repeating
the
argument
of
the first paragraph
of
the
proof

of
Lemma
3, we
conclude that rank(7—
T*T) = 1.
Observe that,
if T
belongs
to a
finite-dimensional subalgebra
of
@}(3ff),
then
the
spectrum
of
T coincides with
the set of
eigenvalues.
For
such
a
T with
||
T\\
= 1,
Thas
no eigenvalue of modulus
1
if and only if its spectral radius

is
less than
1
which occurs
if
and
only
if
||!T*||
< 1 for
some
k.
A
CHARACTERIZATION OF PICK BODIES 325
THEOREM
2. Let n^2,
and let
si be a
commutative Banach algebra
of operators
on a Hilbert space tf with dim(j^) < «. If there exists Tesi with \\P\\ =
1
for
0 <y < n and if T
has
no
eigenvalue
of
modulus
1, then si is an

n-dimensional
Pick
algebra
with Pick
generator
T.
Proof.
First we show that we may assume that there exists a unit vector
<j>
e
2tf
with ir""
1
^!! = 1. By the Gelfand-Naimark-Segal Theorem [1, Corollary to
Theorem 1.7.2], there exists a Hilbert space «#", a unit vector
y/eJtf",
and a
representation
n:@(J{?)->#(JT)
with WniT^y/W = WT^W = 1. Define
by
T(R)
= R © n(R) for Resi. Then
T
is an isometric algebra homomorphism, x{T)
satisfies all of the original hypotheses, and
\\x{T)
n
~\Q
©

^)||
= ||0 ©
y/\\
= 1.
The subspace X =
{R(j) \
R e
jaf} is
invariant under d and has dimension at most n.
The algebra J/ = st\
x
and the operator f
=
T\^ satisfy the hypotheses of Lemma 8
and Proposition 2; so si is an n-dimensional Pick algebra with Pick generator f.
Since the restriction homomorphism
Q>:si
-^ si satisfies the conditions of Lemma 7
with T in the role of x,
<X>
is an isometric isomorphism, and the proof is complete.
When si is semisimple, the above theorem shows that 2^ is a Pick body. So, we
are led to a strengthened form of Theorem
1.
Note that z
(
^
z^
if/ #
j

:
is
a consequence
of
the
theorem, not an assumption. This follows since, by the conclusion of Theorem
2,
T must have n distinct eigenvalues.
THEOREM
V. Let Kbea hyper
convex subset
ofC
n
.
Assume that
K
contains
a point
z = (z
15
,z
n
) with
\z
t
\
<
1
for
each

j and
with
z
n
~
x
edK. Then K is a Pick body.
We now identify all Pick generators for a Pick algebra. Let Aut(A) denote the set
of conformal automorphisms of the unit disk.
LEMMA
9.
Let si be a Pick
algebra
A/Jfor
which
dim(j^) ^ 2.
Then,
x
is
a Pick
generator for si if and
only
if x = [y/]for ^eAut(A). In
particular,
if
x
x
,
x
2

are both
Pick generators for si, then x
2
=
y/(x^)
for some
Proof Let x =
[a]
be a Pick generator for A/J; so
||[fl]||
< 1. Observe that since
2,
the coset x cannot contain a constant function. We write / = bA for a
finite Blaschke product b.
Let y =
[y/
0
]
where
y/
0
= id
A
, and note that ||_y|| ^ 1. Select polynomials g
n
with
||gj|
^
1
so that g

n
(x)-*y in A/J, and select
f
n
e[a]
with ||/J| <
1
+ 1/n. Using a
normal families argument, we obtain /, g, h, keH^ so that ||/|| < 1, ||g|| < 1,
f=a
+
bh,
and gof=
y/
0
+
bk.
Clearly / cannot be constant; so the composition
T
= go/is well defined as a holomorphic map on int(A) with values in
A.
Since
T
leaves
fixed the zeros of
b
and has derivative
1
at each non-simple zero, and since the sum
of the multiplicities of the zeros of b is at least 2, Schwarz's Lemma implies that

T
= y/
0
.
Hence,
/e
Aut(A)
and x = [f], as desired.
Since, for ^eAut(A), {po y/\p is a polynomial,
\\p\\
<
1}
is a dense subset of the
closed unit ball of A, the converse is clear.
To prove the last statement of the lemma, suppose that
x
lf
x
2
are both Pick
generators. Then, x
x
= [y/^ and x
2
= [^
2
] for
y/
x
,

^
2
eAut(A). So, x
2
= ^(x
x
) for
y/
= y/
2
o
y/^
1
e Aut(A).
326
B.
COLE,
K.
LEWIS AND
J.
WERMER
The next result extends Lemma
8 by
replacing
T
n
~
l
with
a

product
T
x
'~
T
n
_
x
of
commuting operators.
LEMMA
10. Let n^2, and let 2tf be a
Hilbert space with dim(^f)
<
n.
Let
T
x
, ,T
n
_
x
belong
to a
commutative subalgebra
of &(3f) so
that ||7J||
=
1
and

T
t
has no
eigenvalue
of
modulus 1
for i=
1, ,
w—
1, while
\\T
X
-• -
T
n
_
x
\\
= 1.
Then,
=
n
andrank(/= T* 1$
=
1
for i =
1, ,n-1.
Proof Selective xeJif
so
that

||x|| =
||>>||
=
1
for y =
T
x
•••
T
n
_
x
x.
Let
E
r
=
{T
ti
-
T
it
x\0
^k,l^i
x
<
•••
< i
k
^

r)
(interpret
E
o
to
mean {*}), and
let
V
r
=
span(2s
r
)
for 0 ^ r ^
n
—
1.
For
£eE
r
,y
= R£
for some
R of
the form
T
}
•••
7J
; so

1
=
\\y\\
<
||£||
^
1
and
hence ||f ||
=
1
for all
Now
let
l^r^n-l.
For
£eE
r
_
x
,Z'
=
T
r
(Z)eE
r
.
Since
K\\ = Kl = h we
deduce that T?T

r
£
= £ for all
£eE
r
_
lt
and
therefore
/-T*
T
r
= 0on
V
r
_
x
.
It is
clear
that V
r
_
x
e
V
r
and
T
r

(V
r
_
x
)
£
V
r
.
If
V
r
_
x
=
V
r
,
then
V
r
is
^-invariant. Consequently,
T
r
is unitary
on
V
r
_

x
, implying that
T
r
has an
eigenvalue
of
modulus
1,
contrary
to
assumption.
So
1
=
dim(^)
< ••• <
&\m(V
n
_
x
)
^
dim(Jf)
^ n, and in
particular,
=
n and
dim(F
n

_
2
)
= n-1.
Since
/- r*^ 7;^ = 0 on V
n
_
2
,
and equahty holds,
as
claimed, because T
n
_
x
is
not unitary
on Jf.
After re-labeling
the
operators,
the
above argument can
be
applied
to any
T
{
in

place
of
T
n
_
x
.
The preceding lemmas lead
to a
generalization
of
Theorem
2.
THEOREM
3. Let Jf be a
Hilbert space,
and let n^2.
Suppose that
$f
is
a
commutative subalgebra
of
fflffl) with dim(j^)
<
n.
Let
T
x
, ,T

n
_
x
belong
to
s#
so
that
\\Tf\\
=
1
and T
t
has no
eigenvalue
of
modulus
1
for i=
\, ,n—
1, while
\\T
x
-T
n
_
x
\\
= \
(i)

s/ is an
n-dimensional Pick algebra,
(ii) each
T
t
is a
Pick generator for
s/,
(hi)
T
x
=
i//i(T
x
)for
2 ^ i ^ n-1
where y/
t
e Aut(A).
Proof We repeat the proof of Theorem 2, with the following changes. Enlarging
if necessary,
we can
assume that there exists
0
e
3f?
so
that
Then, Lemma 10
is

used
in
place
of
Lemma 8. Assertion (iii)
is a
consequence
of
the
last sentence
of
Lemma
9.
As discussed
at the
beginning
of
this section, operator-theoretic results
can be
applied
to
finite-dimensional, semisimple Q-algebras
to
yield information about
hyperconvex sets,
and
those hyperconvex sets associated with semisimple Pick
algebras are just the Pick bodies. So, we have the following extension
of
Theorem

1.
A
CHARACTERIZATION
OF
PICK
BODIES
327
COROLLARY
1. Let K be a hyperconvex subset of C
n
, and let
x
a)
,x
{2)
, ,
x
{n
-
X)
edKbesuch that
(i)
x
w
x
{2)
• •
•
JC
(B

~
1)
e dK
(multiplying
coordinatewise),
(ii) \xf\ <
1
for
1
^ i ^ n-\ and
1
^j < n.
Then,
K is a Pick body
such
that
(iii) K=
®(x™)
= •••=
9(x
(n
~
X)
),
(iv) for 2 < / <
n —
1,
there exists ^
(
eAut(A) so that x

{i)
= ^(x
(1)
)
(applying
\f/
t
coordinatewise).
4.
Additional comments
1.
The converse to Theorem 2 is true also. In particular, if s/ is an n-dimensional
Pick algebra with Pick generator T, then
1
= || T\\ =
||
T"**"
1
1|
> ||r
n
|| and st is
isometrically isomorphic to a subalgebra of ^(Jf) where «?f is an w-dimensional
Hilbert space. This can be seen using Proposition
1
and Rouche's theorem as in the
proof of Lemma 1.
2.
Let A be the disk algebra, and let J = bA where b is a finite Blaschke product.
With an argument similar to that in the proof to Proposition 1, it can be shown that

each coset in
A/J
contains an element of minimum norm. Specifically,
ifxeA/J,
then
there exists a unique gex with ||g|| =
||JC||
;
moreover, g is a constant multiple of a
finite Blaschke product of order less than the order of b.
3.
A recurring assumption used in Section 3 is that T
is
a norm
1
operator with
no eigenvalue of modulus
1,
which clearly corresponds to the requirement in Theorem
1 that zedK with
\z
}
\
<
1
for each/ We now analyse this assumption, showing how
our main results take on a simpler form after applying an appropriate reduction.
Proofs are omitted.
In view of the comments before Theorem 2, for each operator T in our
investigation,

we
can deal with its spectrum, a(T), instead of its eigenvalues. We begin
with a result about Banach algebras.
LEMMA.
Let
$4
be a finite
dimensional Banach algebra with identity denoted
by
1.
There
exists x erf so that
\\x\\
= 1, Xea(x) with
\X\
= 1,
and
x #
X
if
and
only
if there
exists yerf so that y
2
= y,
\\y\\
= 1, andy ^ 1.
The proof of this lemma is a straightforward exercise in
finite-dimensional

linear
algebra. For example, y is calculated from x by the formula y = lim
Jfc
_
00
(|(l +Xx))
k
.
Consequently, we call s4
decomposable
if there exists yerf with y
2
—
y,
||
j>||
= 1,
and y # 1; otherwise, s$ is called
non-decomposable.
If s/ is a finite-dimensional
subalgebra of ^(«3f) where Jf is a Hilbert space, then s$ is decomposable exactly
when st contains a non-trivial orthogonal projection. Theorems 2 and 3 can be re-
stated for non-decomposable algebras in a slightly simpler form. For instance, we
have the following theorem.
THEOREM
2''.
Let n ^ 2, and let s/ be a
non-decomposable,
commutative Banach
algebra

of
operators
on
a
Hilbert space
#?
with
dim(j^) ^ n. If
there
exists Test with
||
T
}
\\
= I/or
0
<y <
n and
Tis
not a
multiple
of
the
identity,
then
sf
is
an
n-dimensional
Pick

algebra
with Pick
generator
T.
328
A
CHARACTERIZATION
OF
PICK BODIES
Using minimal orthogonal projections, we obtain the following decomposition for
operator algebras. A similar result holds for Banach algebras satisfying the von
Neumann Inequality of order 1.
PROPOSITION. Let s& be
a
finite dimensional, commutative subalgebra of
where Jff is a Hilbert space. Then, $4 is isometrically isomorphic to $$
x
x ••• xs/
r
,
where each st
{
is a non-decomposable subalgebra of&(3%) and
3%
is a Hilbert space.
The product algebra is normed so that ||(7i, ,
T
r
)\\
= max

1;S(!gr
||7J|| for
T
t
ejtf
t
.
When
s&
is isometrically isomorphic to the ^-algebra AIJ where A is a uniform
algebra and J is a closed ideal in A, this decomposition has the following
interpretation: the minimal norm 1 idempotents of A/J (and hence the algebras s/
t
in the above proposition) are in one-to-one correspondence with those Gleason parts
of A that meet hu\\(J). See [5] for information on Gleason parts.
As in Section 3, it is easy to translate these results into statements about
hyperconvex sets. For a hyperconvex set K £ C
n
, we say that it is decomposable if,
after re-ordering the coordinates, K = K
x
x K
2
where K
x
s C"
1
, K
2
c C"

a
, n = n
x
+
n
2
.
Otherwise, K is said to be non-decomposable.
PROPOSITION.
Let K be a hyperconvex subset of C
n
. Then,
(i) K is non-decomposable if and only if zsK with max,
|z
y
|
= 1 implies that
z = (A, ,X) for some XeC with
\X\
= 1,
(ii) after re-ordering the coordinates, K= K
x
x
• • •
x K
r
, where each K
{
is a non-
decomposable hyperconvex subset of

C
n<
and n = n
x
+
• • •
+ n
r
.
Thus
we
obtain another modified form
of
Theorem
1.
THEOREM 1". Let K be a non-decomposable, hyperconvex subset ofC
n
. Let zeK
so that
(i) z ? (A, ,A) for each AeC with
\X\
= 1,
(ii) z*-
x
edK.
Then K is a Pick body.
References
1. W.
ARVESON,
An

invitation
to
C*-algebras
(Springer-Verlag,
New
York, 1976).
2.
F. F.
BONSALL
and J.
DUNCAN,
Complete normed
algebras
(Springer-Verlag, Berlin, Heidelberg,
New
York, 1973).
3.
L.
DE
BRANGES
and J.
ROVNYAK,
Perturbation theory
and its
applications
in
quantum mechanics
(ed.
Calvin
H.

Wilcox; Wiley,
New
York, 1966).
4.
B.
COLE,
K.
LEWIS
and J.
WERMER,
'Pick conditions
on a
uniform algebra
and von
Neumann
inequalities',
J.
Func.
Anal,
107 (1992) 235-254.
5.
T.
W.
GAMELIN,
Uniform algebras (Prentice Hall, Englewood Cliffs, 1969).
6.
K.
LEWIS
and J.
WERMER,

'On the
theorems
of
Pick
and von
Neumann', Function Spaces
(ed. K.
Jarosz; Marcel Dekker, 1992).
7.
G.
PICK '
Uber
die
Beschrankungen analytischer Funktionen, welche durch vorgegebene
Funtionswerte bewirkt werden', Math.
Ann. 11
(1916)
7-23.
8.
D.
SARASON,
'Generalized interpolation
in
/f
00
',
Trans. Amer. Math.
Soc.
127 (1967) 179-203.
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