The development of multi-variate operator theory has been similar to
that of function theory in several complex variables except that the non-
commutating co-ordinates pose an entirely new set of challenges. The
introduction of methods of commutative algebra for studying problems
in this area by viewing a pair (H ,T ), where T := (T1,...,Tn) is a commut-
ing n-tuple of operators on the Hilbert space H , as a Hilbert module via
the multiplication (p,h) → p(T )h, p ∈ C[z1,...,zn], h ∈ H , was the begin-
ning of a systematic new development. In this approach, the vast array
of tools from commutative algebra is available for solving problems in
multi-variate operator theory. However, these techniques don’t apply di-
rectly because of the continuity assumption of the module multiplication
either in just the first variable, or in both the variables. A choice is made
depending on the problem at hand.
A second ingredient has been the action of a group G on the module
(H ,T ). This occurs naturally in several instances. For example, if the
joint spectrum X := σ(T ) of the d-tuple T is a G space, that is, G acts on
X by biholomorphic automorphisms, then defining g · T , the classifica-
¡
¢
©
tion of imprimitivities X ,G,(H ,T ) := T :Ug∗p(T )Ug = (g · p)(T )}, Ug is
a unitary representation of the group G and g · p = p ◦ g−1, is an impor-
tant problem. One may study such imprimitivities by choosing a group G
acting on X . This choice might vary from a transitive action to an action,
where the orbits of G in X are quite big. For a typical example, consider
X = Dn, the n-fold product of the unit closed disc, and the group G to
be either the bi-holomorphic automorphism group of Dn or simply the
permutation group Sn acting on Dn.
The study of these problems are greatly facilitated by the analysis of
a class holomorphic curves in the Grassmannian Gr(H ,n) of rank n in
some separable complex Hilbert space H . These holomorphic curves
arise from a class of operators acting on some Hilbert space H intro-
duced in the paper of M. J. Cowen and R. G. Douglas, "Operator Theory
and Complex Geometry", Acta Math., 141 (1978), 187 - 261. The opera-
tors in this class possess an open set Ω ⊆ C of eigenvalues of (constant)
multiplicity k and characterized by the existence of a holomorphic map
γ : Ω → H such that γ(w) := (γ1(w),...,γk(w)), T γi (w) = wγi (w), 1 ≤ i ≤
k, w ∈ Ω. For k = 1, one of the main features of the operator T in this class
2
¯
is that the curvature KT (w) := −∂∂logkγT (w)k of the holomorphic Her-
mitian line bundle ET determined by the holomorphic map γT equipped
with the Hermitian structure kγT (w)k is a complete unitary invariant for
2
the operator T .
It is easy to see that if T is a contraction in the Cowen-Douglas class of
∗
∗
the unit disc D, then KT (w) ≤ KS (w), where S is the backward unilat-
2
∗
∗
eral shift acting on ` . Choosing a holomorphic frame γS , say γS (w) =
2
2
2 −1
∗
∗
(1,w,w ,...), it follows that kγS (w)k = (1 − |w| ) and that KS (w) =
−(1−|w| ) , w ∈ D. Thus the the operator S∗ is an extremal operator in
2 −2
the class of all contractive Cowen-Douglas operator. R. G. Douglas asked
if the curvature KT of a contraction T achieves equality in this inequality
even at just one point, then does it follow that T must be unitarily equiv-
alent to S∗? It is easy to see that the answer is “no”, in general. However, if
T is homogeneous, namely, Uϕ∗TUϕ = ϕ(T ) for each bi-holomorphic au-
tomorphism ϕ of the unit disc and some unitary Uϕ, then the answer is
“yes”. Of course, it is then natural to ask what are all the homogeneous
operators. These are the holomorphic imprimitivities.
The question of classifying holomorphic imprimitivities on a bounded
symmetric domain Ω amounts to classification of commuting tuples of
homogeneous of operators in the Cowen-Douglas class of Ω. There is a
one to one correspondence between these and the holomomorphic ho-
mogeneous vector bundles on the bounded symmetric domain Ω. The
homogeneous bundles can be obtained by holomorphic induction from
representations of a certain parabolic Lie algebra on finite dimensional
inner product spaces. The representations, and the induced bundles,
have composition series with irreducible factors. In joint work with A.
differential operator intertwining the bundle with the direct sum of its
factors. Next, we study Hilbert spaces of sections of these bundles. We
use this to get, in particular, a full description and a similarity theorem
for homogeneous n-tuples of operators in the Cowen-Douglas class of
the Euclidean unit ball in Cm. A different approach is in [54]. The initial
study of these questions restricted to the case of homogeneous holomor-
phic line bundles is in [18].
A little more can be said beyond holomorphic imprimitivities in the
case of a single operator T acting on a Hilbert space H . Assume that
there is a projective unitary representation σ of Möb such that ϕ(T ) =
σ(ϕ)?T σ(ϕ) for all ϕ in Möb. If this is the case, the operator T is ho-
mogeneous and we say that σ(ϕ) is associated with the operator T . A
Möbius equivariant version of the Sz.-Nagy–Foias model theory for com-
pletely non-unitary (cnu) contractions is developed in [67]. As an appli-
cation, we prove that if T is a cnu contraction with associated (projective
1
The numbers in square brackets refer to the list on the Publications page.
unitary) representation σ, then there is a unique projective unitary rep-
ˆ
resentation σ, extending σ, associated with the minimal unitary dilation
ˆ
of T . The representation σ is given in terms of σ by the formula
+
−
ˆ
σ = (π⊗D1 )⊕σ⊕(π? ⊗D1 ),
where D1± are the two Discrete series representations (one holomorphic
and the other anti-holomorphic) living on the Hardy space H2(D), and
π,π? are representations of Möb living on the two defect spaces of T and
defined explicitly in terms of σ and T . Moreover, a cnu contraction T has
an associated representation if and only if its Sz.-Nagy–Foias character-
istic function θT has the product form θT (z) = π?(ϕz)∗θT (0)π(ϕz), z ∈ D,
where ϕz is the involution in Möb mapping z to 0. We obtain a concrete
realization of this product formula for a large subclass of homogeneous
cnu contractions from the Cowen-Douglas class. These are the holomor-
phic imprimitivities among the homogeneous contractions.
Typically, contractive homomorphisms induced by an operator (or, even
a commuting tuple of operators) in the Cowen-Douglas class give rise to
curvature inequalities. However, given the curvature inequality, it is clear
that it can only provide information about the second order jets of the
holomorphic curve. Strengthening the curvature inequality to obtain ad-
ditional information about the holomorphic curve remains an intriguing
problem. Some partial answers to this question are in [51].
A complete set of invariants, originally obtained by Cowen and Dou-
glas, have been refined to provide a tractable set of invariants for large
class of Cowen-Douglas operators [55,56]. In these papers, after impos-
ing a mild condition on the Cowen-Douglas bundles, it is shown that the
curvature together with the second fundamental form serves as a com-
plete set of invariants.
In general, determining the moduli space for the isomorphism classes
of sub-modules of a Hilbert module is a difficult problem. Thanks to
Beurling’s theorem, the moduli space is a singleton for the Hardy module
H2(D) of the unit disc, that is all sub-modules of H2(D) are isomorphic.
However, a rigidity phenomenon occurs in H2(Dn), namely, no two sub-
modules of H2(Dn) are isomorphic barring a very few exceptions. This
is typical of the multi-variable situation. The determination of the iso-
morphism classes of sub-modules of a Hilbert module for anlytic Hilbert
modules seems intractable at the moment. Some partial results have
been obtained by using the monoidal transform to resolve the singulari-
ties of the holomorphic curves corresponding to sub-modules of an an-
alytic Hilbert module. More recently, the submodules of analytic Hilbert
modules defined over certain algebraic varieties in bounded symmetric
domains, the so-called Jordan-Kepler varieties V` of arbitrary rank ` have
been studied. For ` > 1, the singular set of V` is not a complete intersec-
tion. Hence the usual monoidal transformations do not suffice for the
resolution of the singularities. Instead, we describe a new higher rank
version of the blow-up process, defined in terms of Jordan algebraic de-
terminants, and apply this resolution to obtain the rigidity of the sub-
modules vanishing on the singular set [44, 47, 64]. The accompanying
question of finding a model for the quotient modules might be thought
of the model theory of Sz. - Nagy and Foias to the more general con-
text of Hilbert modules over function algebras. Finding a complete set of
invariants for the unitary equivalence classes of such canonical models
is an important problem. Both of these questions have been addressed
partially in [22, 37].
Given a pair of positive real numbers α,β and a sesqui-analytic func-
tion K on a bounded domain Ω ⊆ Cm, we investigate the properties of the
real-analytic function
¡¡
¢¢
2
K (α,β)(z,z) :=
K (α+β) logK (z,z) 1≤i,j≤m, z ∈ Ω,
∂
∂z ∂z
j
i
taking values in m × m matrices. The kernel K (α,β) is non-negative def-
inite whenever K α and K β are non-negative definite. In this case, a re-
alization of the Hilbert module determined by the kernel K (α,β) is ob-
tained. Let Mi , i = 1,2, be two Hilbert modules over the polynomial ring
C[z1,...,zm]. The tensor product M1 ⊗ M2 is clearly a module over the
ring C[z1,...,z2m]. This module multiplication restricts to C[z1,...,zm]
via the diagonal map z → (z,z), z ∈ Ω. Now, a natural decomposition
of the tensor product M1 ⊗ M2 very similar to the Clebsch-Gordon de-
composition of the tensor product of two irreducible unitary representa-
tions occurs relative to the multiplication restricted to C[z1,...,zm]. Two
of the initial pieces in this decomposition have been identified in [65].
The first of these is simply the restriction of M1 ⊗ M2 to the diagonal
set 4 := {(z,z) : z ∈ Ω}, while the second piece in the decomposition is
the module determined by the kernel K (α,β). Moreover, if Ω is a bounded
symmetric domain, then K (α,β) is covariant whenever K is covariant un-
der the action of the bi-holomorphic automorphism group of Ω.