Unbounded Symmetric Homogeneous Domains in Spaces of Operators

21 Pages·0.21 MB·English

by  Lawrence A. Harris

Unbounded Symmetric Homogeneous Domains in
5
9 Spaces of Operators
9
1

n
Lawrence A. Harris
a
J

0
3
Abstract

]
V
We define the domain of a linear fractional transformation in a space
C of operators and show that both the affine automorphisms and the compo-
.
h sitions of symmetries act transitively on these domains. Further, we show
t that Liouville’s theorem holds for domains of linear fractional transforma-
a
m tions and, with an additional trace class condition, so does the Riemann
[ removable singularities theorem. We also show that every biholomorphic
mapping of the operator domain I < Z∗Z is a linear isometry when the
1
v space of operators is a complex Jordan subalgebra of (H) with the re-
L
1 movable singularity property and that every biholomorphic mapping of the
0 operator domain I +Z∗Z < Z∗Z is a linear map obtained by multiplica-
2 1 1 2 2
1 tion on the left and right by J-unitary and unitary operators, respectively.
0
5
0. Introduction. This paper introduces a large class of finite and infinite
9
/ dimensional symmetric affinely homogeneous domains which are not holomorphi-
h
t cally equivalent to any bounded domain. These domains are subsets of spaces
a
m of operators and include domains as diverse as a closed complex subspace of the
: bounded linear operators from one Hilbert space to another, the identity compo-
v
∗
nent of the group of invertible operators in a C -algebra and the complement of
i
X
a hyperplane in a Hilbert space. Each of our domains may be characterized as
r
a a component of the domain of definition of some linear fractional transformation
which maps a neighborhood of a point in the component biholomorphically onto
an open set in the same space. Thus we refer to our domains as domains of linear
fractional transformations. We show that at any point of such a domain, there
exists a biholomorphic linear fractional transformation of the domain onto itself
which is a symmetry at the point. This linear fractionaltransformation is a gener-
alization of the Potapov-Ginzburg transformation. Moreover, for any two points
Z and W in the domain, there exists an affine automorphism of the domain of
0 0
theformφ(Z) = W +A(Z Z )B andφisacompositionoftheabovesymmetries.
0 0
−
1
In fact, we exhibit many different formulae for linear fractional transformations
that are automorphisms of domains of linear fractional transformations.
We show that any bounded holomorphic function on the domain of a linear
fractional transformation is constant and, more generally, that the Kobayashi
pseudometric vanishes identically on such domains. We also show that if the
domain of a linear fractional transformation satisfies a trace class condition, then
a holomorphic function on the domain which is locally bounded with respect to
points outside the domain has a holomorphic extension to these points. Further,
we give a sufficient condition for two domains of linear fractional transformations
to be affinely equivalent when the subspace of operators considered is the full
∗
space or a C -algebra. As expected, our unbounded symmetric homogeneous
domains fail to have many of the properties of bounded symmetric domains. For
example, Cartan’s uniqueness theorem may fail, symmetries may not be unique
andautomorphismsmaynotbeacompositionofalinearfractionaltransformation
and a linear map.
Next we turn to a discussion of the characterization of the automorphisms of
non-homogeneous domains which are operator analogues of the domain 1+ z 2 <
1
| |
z 2 in C2 (which is included). The domains we consider are circular domains in a
2
| |
space of operators in the sense discussed previously by the author in [14] and are
holomorphically equivalent to the open unit ball of the space with certain singular
points omitted. We show that with some restrictions, these points are removable
singularities and we deduce that all biholomorphic mappings of the domains are
linear. In the case where the domain and range spaces of the operators considered
have different dimensions, these linear maps are given by operator matrices which
are the coefficient matrices of linear fractional transformations of the unit ball
of the space of operators on the domain space. In the case where the operators
considered have the same domain and range, our domains reduce to analogues
of the exterior of the unit disc in spaces of operators and the linear maps are
invertible isometries of these spaces. A result of Arazy and Solel [2] allows us to
treat spaces of operators which are not necessarily closed under adjoints.
∗
A main lemma is the result that every J -algebra with identity which is closed
in the weak operator topology has what we call the removable singularity prop-
erty, i.e., the singular operators in the open unit ball are removable singularities
for bounded holomorphic functions. This is rather surprising since these singu-
lar operators may have non-empty interior in the infinite dimensional case. We
also give two types of homogeneous circular domains where the group of linear
automorphisms of the domains acts transitively.
Unbounded homogeneous domains in Cn have been considered previously by
Penney [22]andWinkelmann [28]. In particular, Penney obtainsa classification of
2
therationally homogeneous domains in Cn analogousto the classification theorem
for bounded homogeneous domains given by Vinberg, Gindikin and Piatetskii-
Shapiro. See also Hua [18].
Our discussion emphasizes the explicit construction of domains and mappings
using ideas from operator theory and functional analysis. We have considered
bounded symmetric homogeneous domains from this point of view previously
in [10]. See [20] and [25] for an exposition of the theory of bounded symmetric
domains in infinite dimensions.
Let H and K be complex Hilbert spaces and let (H,K) denote the Banach
L
space of all bounded linear transformations from H to K with the operator norm.
A B
We write (H) for (H,H). Throughout, and denote any closed complex
L L
subspaces of (H,K). (The reader interested only in the finite dimensional case
L
may take H = Cn, K = Cm and identify (H,K) with the vector space of all
L
m n matrices of complex numbers.)
×
1. Domains of Linear Fractional Transformations. Let C (K,H),
D (H) and suppose (CZ +D)−1 exists for some Z A. Put X∈=L(CZ +
0 0 0 0
D)∈−1LC. If ZX Z A for all Z A, we define ∈
0
∈ ∈
= Comp Z A : (CZ +D)−1 exists (1)
D Z0{ ∈ }
A
to be the domain of a linear fractional transformation on . (Here and elsewhere,
Comp denotes the connected component containing Z .) It is easy to show that
Z0 0
any finite or infinite product of domains of linear fractional transformations is
the domain of a linear fractional transformation. To justify this terminology, we
observe that by [14, Prop. 5], if the linear fractional transformation
T(Z) = (AZ +B)(CZ +D)−1 (2)
A B
maps a domain in containing Z onto a subset of some containing an interior
0
B A
point of and if the coefficient matrix of T is invertible, then ZX Z for all
0
A ∈
Z and T is a biholomorphic mapping of the domain above onto a similar
∈ B D
domain in . We consider components because a linear fractional transformation
may be a biholomorphic mapping of one component of the open set where it is
defined while mapping the other components outside the space. (See Example 1
below.)
A
Theorem 1 The domain of a linear fractional transformation on is a sym-
D
metric affinely homogeneous domain. In particular, if is given by (1), then for
D
each Y ,
∈ D
U (Z) = Y (Z Y)(CZ +D)−1(CY +D) (3)
Y
− −
3
is a biholomorphic mapping of onto itself satisfying U2 = I, U (Y) = Y and
D Y Y
DU (Y) = I. Moreover, for any point W , there is an invertible affine
Y 0
− ∈ D
linear fractional transformation φ of onto itself with φ(Z ) = W and φ can be
0 0
D
chosen to be a composition of mappings U given above.
Y
Winkelmann [28, Prop. 1] has proved a related result for the complement
of an algebraic subvariety in Cn. It is easy to show that U is the only linear
Y
fractional transformation (2) with the mapping property described there which
A
satisfies T(Y) = Y and DT(Y) = I on . The transformations U are used
Y
−
in [15] to obtain factorizations of certain operator matrices.
A
Example 0. Any is the domain of a linear fractional transformation on
A
when C = 0 and D = I since then X = 0 for any choice of Z . In this case,
0 0
U (Z) = 2Y Z.
Y
− A A
Example 1. Let beapower algebra[9], i.e., isaclosedcomplex subspace
of (H) containing the identity operator I on H and the squares of each of its
L
elements. Put
G (A) = Comp Z A : Z−1 exists . (4)
I I{ ∈ }
A A
Clearly, G ( ) is the domain of a linear fractional transformation on since
I
X = I for Z = I. Also, U (Z) = YZ−1Y. If A is a W∗-algebra, the argument
0 0 Y
A
in [6, Cor. 5.30] shows that the invertible operators in are connected so it is
A
not necessary to take components in this case. However, if is the closure of the
polynomials in the bilateral shift [6, Example 4.25], then U is a biholomorphic
I
A A
mapping of G ( ) but U does not take any other invertible elements of (such
I I
A
as the bilateral shift) to .
A
Example 2. Let be a closed complex subspace of (H) containing a
A A L
projection E such that ZEZ whenever Z . Then
∈ ∈
= Comp Z A : (EZ +I E)−1 exists
D E{ ∈ − }
A
is the domain of a linear fractional transformation on since X = E for Z = E.
0 0
The previous examples are the cases E = 0 and E = I. Moreover,
U (Z) = [(E I)Z +E](EZ +I E)−1
E
− −
isavariantformofthePotapov-Ginzburgtransformation[1, 3]. (Anothervariant
§
is called the Redheffer transform in [3, p. 269]. An early reference is [19, p. 240].)
A basic property is that U is a biholomorphic mapping of onto ,
E 1 2
D ∩D D ∩D
where
A ∗ A
= Z : Z JZ < J , = Z : Z < 1 ,
1 2
D { ∈ } D { ∈ || || }
4
and J = I 2E. This follows directly from [14, Lemma 7] since the coefficient
matrix M o−f U satisfies J = (M−1)∗J M−1, where
E 2 1
J 0 I 0
J = , J = .
1 0 J 2 0 I
" # " #
− −
Example 3. If ranC is closed or if ranD ranC, then
⊆
= Z (H,K) : (CZ +D)−1 exists
D { ∈ L }
A
is the domain of a linear fractional transformation on = (H,K) when is
A L D
not empty since is obviously closed under the required products and since is
D
connected by Proposition 4 below.
Example 4. Let c H with c = 0 and d C. Then
∈ 6 ∈
= z H : (z,c) = d
D { ∈ 6 − }
is the complement of a hyperplane in H and the domain of a linear fractional
transformation on H with
(y,c)+d
U (z) = y (z y).
y
− (z,c)+d −
∗
(Identify H with (C,H) and apply Example 3 with C = c .)
L A
Example 5. Let contain all the rank one operators in (H,K) and let
L
x H, y K be unit vectors. Then if d = 0,
∈ ∈ 6
A
= Z : (Zx,y) = d
D { ∈ 6 − }
A ∗
is the domain of a linear fractional transformation on . Indeed, let C = xy ,
∗ ∗
D = dI and note that CZ+D = dI+x(y Z) is invertible if and only if d+(y Z)x
∗ ∗
is invertible. Also X = xy for Z = (1 d)yx and
0 0
−
∗
Z Y xy Z
∗
U (Z) = Y − I (dI +xy Y).
Y
− d " − d+(Zx,y)#
Since C is compact, is connected by Proposition 4 below.
D
Example 6. Let H be a Hilbert space with conjugation z z¯ and let c H
→ ∈
with c = 0. Then each of the domains
6
= z H : 1+2(z,c)+(z,z¯)(c¯,c) = 0 ,
1
D { ∈ 6 }
= z H : (z,z¯) = 0
2
D { ∈ 6 }
is linearly equivalent to the domain of a linear fractional transformation. To see
this, observe that by [10, 2] there exists an invertible linear map z A of H
z
§ →
5
A ∗
onto a Cartan factor of type IV. For , take C = A , Z = A and D = I, and
D1 c z ∗
note that CZ +D is invertible if and only if z by [10, (9)]. Also, X = A
A A A∈ D1 ∗ 0 c
for Z = 0 so ZX Z for all Z since is a J -algebra. Moreover,
0 0 1
∈ ∈ D
is connected by Proposition 4 below since X Z satisfies a quadratic polynomial
0
equation. A similar argument with C = I and D = 0 establishes the result for
, and in that case,
2
D
2(z,y¯)y (y,y¯)z
U (z) = − .
y
(z,z¯)
Proof of Theorem 1. Given Y , put X = (CY +D)−1C and note the
∈ D
identities
CU (Z)+D = (CY +D)(CZ +D)−1(CY +D), (5)
Y
(CY +D)−1(CZ +D) = I +X(Z Y). (6)
−
The coefficient matrix of U is
Y
(I YX) 2Y YXY
M = − − −
X I XY
" #
−
by (6). Clearly M2 = I so M is invertible and U2 = I. Also, U (Y) = Y. Hence
Y Y
by hypothesis and [14, Prop. 5], U is a biholomorphic mapping of onto itself
Y
A A D
when Y = Z and therefore ZXZ whenever Z . Thus [14, Prop. 5]
0
∈ ∈
applies again to show that U is a biholomorphic mapping of onto itself for
Y
D
d
arbitrary Y . Further, DU (Y) = I since DU (Y)Z = U (Y +tZ)
∈ D Y − Y dt Y |t=0
and U (Y +tZ) = Y tZ(I +tXZ)−1.
Y
Next we show that−if Z,W satisfy (CZ + D)−1C(W Z) < 1, then
there is a Y with U (Z) =∈WD. Put X||= (CZ + D)−1C a−nd R||= W Z.
Y
∈ D −
By the holomorphic functional calculus [17, 5.2], the binomial series defines a
Q (H) such that Q2 = I +XR, σ(Q) doe§s not contain 1, and (I +Q)−1 is
a l∈imLit of polynomials in XR. Take Y = Z +R(I +Q)−1. −Then Y A by [14,
∈
Lemma 6a]. Also, since XR = Q2 I, we have that
−
U (Z) Z = Y Z +(Y Z)[I +X(Y Z)]
Y
− − − −
= R(I +Q)−1[2(I +Q)+XR](I +Q)−1
= R
so U (Z) = W. Moreover, (CY + D)−1 exists by this and (5). To show that
Y
Y , let 0 t 1. The above argument applies to the points Z and W =
t
∈ D ≤ ≤
(1 t)Z + tW and the corresponding Q is a continuous function of t in [0,1].
t
−
6
A
Therefore, there is a curve γ in connecting Z to Y such that Cγ(t) + D is
invertible for all 0 t 1.
≤ ≤
Now given W , there is a curve γ in connecting Z to W . By compact-
0 0 0
ness, there is a nu∈mbDer M such that (Cγ(Dt) + D)−1C M for all 0 t 1
|| || ≤ ≤ ≤
and there is a δ > 0 such that γ(s) γ(t) < 1/M whenever s t δ and
|| − || | − | ≤
0 s,t 1. Choose n > 1/δ and put Z = γ(k/n) for 0 k n. Then for
k
≤ ≤ ≤ ≤
1 k n,
≤ ≤
(CZk−1+D)−1C(Zk Zk−1) < 1, (7)
|| − ||
so by what we have shown there exists a Uk with Uk(Zk−1) = Zk. Thus U =
U U satisfies U(Z ) = W .
1 n 0 0
···
To complete the proof, it suffices to show that U U is an affine linear
W Y
◦
fractional transformation for any W,Y since then we can choose n to be even
in the above argument and take φ = U∈. PDut X = (CY +D)−1C and set
Q = [W U (Z)][I +X(Z Y)].
Y
− −
Then Q = W Y +[I +(W Y)X](Z Y) by (6). Hence by (5) and (6),
− − −
U U (Z) = W +Q(CY +D)−1(CW +D)
W Y
◦
= U (Y)+[I +(W Y)X](Z Y)[I +X(W Y)],
W
− − −
2
as required.
There are many biholomorphic mappings of the domain of a linear fractional
A
transformationon besides (3). Forexample, ifW and X (W Z ) < 1,
0 0 0 0
∈ D || − ||
then
φ(Z) = W +[I +(W Z )X ]1/2(Z Z )[I +X (W Z )]1/2
0 0 0 0 0 0 0 0
− − −
isaninvertible affinelinear fractionaltransformationof onto itself with φ(Z ) =
0
D
W , where the square roots are defined by the binomial series. This follows
0
from [14, Lemma 6c], (6) and the identity
I +X [φ(Z) Z ] = R1/2[I +X (Z Z )]R1/2,
0 0 0 0
− −
where R = I +X (W Z ). Moreover, the above proof shows that finite com-
0 0 0
−
positions of these mappings act transitively on . Note that φ = U U when
W Y
A DA ∗ ◦ A
W = U (Y) andZ = Y. If = (H,K) orif is aC -algebrawithC,D ,
0 W 0
L ∈
asimpler formula fora transitive set ofaffinemappings of isgiven by (12)below
D
with R = I.
7
There is a general class of involutory biholomorphic mappings of which
D
contains the transformations (3). Specifically, if W and X (W Z ) < 1,
0 0 0 0
∈ D || − ||
define
VW0(Z) = Z0 −TW0−Z0(Z −Z0),
where
T (Z) = (I +AX )−1/2(Z A)(I +X Z)−1(I +X A)1/2.
A 0 0 0
−
(Compare [11, p. 146].) Note the identities V = φ U and U = V φ. It
W0 ◦ Z0 Z0 W0 ◦
follows that V is a biholomorphic mapping of onto itself with V2 = I and
W0 D W0
V (Z ) = W and that the composition V V of any two such mappings is
W0 0 0 W1 ◦ W0
affine. In particular, V = U .
Z0 Z0
The domain of a linear fractional transformation may have biholomorphic
mappings which cannot be expressed as the composition of a linear fractional
transformation and an invertible linear map. For example, let A = (C,C2),
L
C = [ 0 1 ] and D = 0. Then the domain of (1) is
D
= z1 A : z = 0
D { " z2 # ∈ 2 6 }
and
z z +g(z )
h 1 = 1 2
z z
" 2 # " 2 #
is a biholomorphic mapping of for any holomorphic function g : C 0 C.
D \{ } →
Note that Cartan’s uniqueness theorem fails for since if g(z) = (z 1)2 and
D −
0
Z = , then h(Z ) = Z and Dh(Z ) = I, but h = I on . Moreover, U
0 " 1 # 0 0 0 6 D Z0
and h−1 U h are distinct symmetries of at Z .
X◦ Z0 ◦ D 0
Let be any Banach space.
A
Theorem 2 If is the domain of a linear fractional transformation on and if
X D
h : is a bounded holomorphic function, then h is constant. In particular,
D →
is not holomorphically equivalent to a bounded domain.
D
A
Proof. Let Z with X (Z Z ) < 1. We will construct an entire
0 0
∈ || − ||
function f : C with f(0) = Z and f(1) = Z. Then g(λ) = h(f(λ)) is
0
→ D
a bounded entire function so h(Z) = h(Z ) since g is constant [17, Th. 3.13.2].
0
Hence h is constant in a neighborhood of Z so h is constant on by the identity
0
D
theorem [17, Th. 3.16.4].
8
1/n
λ
Recall [16, 6.3] that limsup 1 and that the binomial
§ n→∞ n ≤
(cid:12) (cid:18) (cid:19) (cid:12)
series bλ(z) = ∞n=0 λn zn satis(cid:12)(cid:12)(cid:12)fies bλ(z)(cid:12)(cid:12)(cid:12)b−λ(z) = 1 for all λ ∈ C and
(cid:18) (cid:19)
z < 1. Put PW = X (Z Z ). By the holomorphic functional calculus,
0 0
b|λ|(W)b−λ(W) = I and henc−e bλ(W)−1 exists for all λ C. Define f(λ) =
∈
Z +(Z Z ) ∞ λ Wn−1. Then f is entire and f(λ) A for all λ C
0 − 0 n=1 n ∈ ∈
(cid:18) (cid:19)
by [14, LemmaP6a]. Moreover, if λ C, then
∈
(CZ +D)−1[Cf(λ)+D] = I +X [f(λ) Z ] = b (W),
0 0 0 λ
−
so f(λ) . 2
∈ D
Note that it follows from the properties of f(λ) given above and (7) that the
pseudometric ρ assigned to by any Schwarz-Pick System [12] satisfies ρ 0.
D ≡
Thus, in particular, the Caratheodory and Kobayashi pseudometrics for vanish
D
identically. (Compare [28, Cor. 1].)
The next result gives a class of domains where all points outside the domain
are removable singularities for locally bounded holomorphic functions.
Theorem 3 Suppose the set = Z A : (CZ + D)−1 exists is non-empty
D { A∈ }
and C is a trace class operator. Then is an analytic set. Moreover, if is
A X \D E
a domain in and if h : is a holomorphic function which is locally
D ∩E →
bounded in the sense that each operator in has a neighborhood such that
E \D ˆ U X
h is bounded on , then h extends to a holomorphic function h : .
U ∩D E →
It is easy to show that the conclusions of Theorem 2 hold for any satisfying
D
the hypotheses of Theorem 3. See [23] for the definition and properties of analytic
A
sets in Banach spaces. Without some restriction on C or , the complement of
may have non-empty interior and thus is not an analytic set. For example, if
D A
there is a Z such that CZ + D has only a one-sided inverse Q, then an
1 1
∈ A
elementary argument shows that CZ + D is not invertible for any Z with
∈
Z Z < 1/r, where r = C Q .
1
|| − || || |||| ||
Proof. By (6) and a translation, we may suppose that D = I. It suffices to
A
show that there is a holomorphic function f : C satisfying
→
A
= Z : f(Z) = 0 (8)
D { ∈ 6 }
A
since then is an analytic set and the remainder of the theorem follows from
\D
the extension of the Riemann removable singularities theorem given in [23, Th.
II.1.1.5]. It is easy to construct such an f when C has finite rank. Indeed, put
⊥ ⊥ ⊥
H = ran C, K = (kerC) , H = H and K = K . Since these spaces are
1 1 2 1 2 1
9
closed, the operators in (H,K) and (K,H) can be written as associated 2 2
L L ×
operator matrices. In particular, if Z (H,K),
∈ L
Z Z Y 0
Z = 1 2 , C = , (9)
Z Z 0 0
" 3 4 # " #
where Z (H ,K ), Z (H ,K ), Z (H ,K ), Z (H ,K ) and
1 1 1 2 2 1 3 1 2 4 2 2
∈ L ∈ L ∈ L ∈ L
Y (K ,H ). Hence
1 1
∈ L
I +YZ YZ
I +CZ = 1 2 , (10)
0 I
" #
where the successive appearances of I denote the identity operator on H, H and
1
H , respectively. Thus Z if and only if (I +YZ )−1 exists. Since H is finite
2 1 1
∈ D
dimensional, we may take f(Z) = det(I +YZ ) to obtain (8).
1
If C is a trace class operator, there exists a sequence F of finite rank
n
{ }
operators in (H,K) with F C 0, where is the trace class norm [8,
n 1 1
L || − || → || ||
Th. VI.4.1]. Let f be the function defined above where C is replaced by F .
n n
A
Then by [8, p.115-117], the sequence f is locally uniformly bounded on and
n
{A}
converges pointwise toa functionf on which isdenotedbyf(Z) = det(I+CZ).
A
Hence f is holomorphic on by an extension of Vitali’s theorem [17, Th. 3.18.1]
2
and (8) holds by [8, VII.3(13)].
In some cases it is not necessary to take components in the definition of the
domain of a linear fractional transformation.
Proposition 4 The set
= Z A : (CZ +D)−1 exists (11)
D { ∈ }
is connected if at least one of the following conditions holds:
A
a) C contains only compact operators,
A
b) there is some Z such that each operator in X satisfies a non-trivial
0 0
∈ D
polynomial equation,
A
c) = (H,K) and ranC is closed,
L
A
d) = (H,K) and ranD ranC.
L ⊆
10