Multiplicity-free homogeneous operators in the Cowen-Douglas class
aa r X i v : . [ m a t h . F A ] J a n MULTIPLICITY-FREE HOMOGENEOUS OPERATORS IN THECOWEN-DOUGLAS CLASS
ADAM KOR ´ANYI AND GADADHAR MISRA
Abstract.
In a recent paper, the authors have constructed a large class of operators in theCowen-Douglas class Cowen-Douglas class of the unit disc D which are homogeneous withrespect to the action of the group M¨ob – the M¨obius group consisting of bi-holomorphicautomorphisms of the unit disc D . The associated representation for each of these operatorsis multiplicity free . Here we give a different independent construction of all homogeneousoperators in the Cowen-Douglas class with multiplicity free associated representation andverify that they are exactly the examples constructed previously. The homogeneous operators form a class of bounded operators T on a Hilbert space H .The operator T is said to be homogeneous if its spectrum is contained in the closed unit discand for every M¨obius transformation g the operator g ( T ), defined via the usual holomorphicfunctional calculus, is unitarily equivalent to T . To every homogeneous irreducible operator T there corresponds an associated unitary representation π of the universal covering group˜ G of the M¨obius group G : π (ˆ g ) ∗ T π (ˆ g ) = ( p ˆ g ) ( T ) , ˆ g ∈ ˜ G, where p : ˜ G → G is the natural homomorphism. In the paper [6] (see also [3]), it wasshown that each homogeneous operator T , not necessarily irreducible, in B m +1 ( D ) admitsan associated representation. The representations of ˜ G are quite well-known, but we arestill far from a complete description of the homogeneous operators. In the recent paper [6],the following theorem was proved. Theorem . For any positive real number λ > m/ , m ∈ N and an ( m + 1) - tuple ofpositive reals µ = ( µ , µ , . . . , µ m ) with µ = 1 , there exists a reproducing kernel K ( λ, µ ) onthe unit disc such that the adjoint of the multiplication operator M ( λ, µ ) on the correspondingHilbert space A ( λ, µ ) ( D ) is homogeneous. The operators ( M ( λ, µ ) ) ∗ are in the Cowen-Douglasclass B m +1 ( D ) , irreducible and mutually inequivalent. In the paper [6], we have presented the operators M ( λ, µ ) in as elementary a way aspossible, but this presentation hides the natural ways in which these operators can be foundto begin with. Here we will describe another independent construction of the operators M ( λ, µ ) . We will also give an exposition of some of the fundamental background material.Finally, we will prove that if T is an irreducible homogeneous operator in B m +1 ( D ) whoseassociated representation is multiplicity free then, up to equivalence, T is the adjoint of ofthe multiplication operator M ( λ, µ ) for some λ > m/ µ ≥ This research was supported in part by a DST - NSF S&T Cooperation Programme. Background material
Although, we intend to discuss homogeneous operators in the Cowen-Douglas class B n ( D ),the material below is presented in somewhat greater generality. Here we discuss commutingtuples of operators in the Cowen-Douglas class B n ( D ) for some bounded open connectedset D ⊆ C m . The unitary equivalence class of a commuting tuple in B n ( D ) is in one to onecorrespondence with a certain class of holomorphic Hermitian vector bundles (hHvb) on D [4]. These are distinguished by the property, among others, that the Hermitian structureon the fibre at w ∈ D is induced by a reproducing kernel K . It is shown in [4] that thecorresponding operator can be realized as the adjoint of the commuting tuple multiplicationoperator M on the Hilbert space H of holomorphic functions with reproducing kernel K .Start with a Hilbert space H of C n - valued holomorphic functions on a bounded openconnected set D ⊆ C m . Assume that the Hilbert space H contains the set of vector valuedpolynomials and that these form a dense subset in H . We also assume that there is areproducing kernel K for H . We use the notation K w ( z ) := K ( z, w ).Recall that a positive definite kernel K : D × D → C n × n on D defines an inner producton the linear span of { K w ( · ) ξ : w ∈ D , ξ ∈ C n } ⊆ Hol( D , C n ) by the rule h K w ( · ) ξ, K u ( · ) η i = h K w ( u ) ξ, η i , ξ, η ∈ C n . (On the right hand side h , i denotes the inner product of C n . We denote by ε , . . . , ε n the natural basis of C n .) The completion of this subspace is then a Hilbert space H ofholomorphic functions on D (cf. [1]) in which the set of vectors { K w : w ∈ D} is dense.The kernel K has the reproducing property, that is, h f, K w ξ i = h f ( w ) , ξ i , f ∈ H , w ∈ D , ξ ∈ C m . Now, for 1 ≤ i ≤ m , we have M ∗ i K w ξ = ¯ w i K w ξ, w ∈ D , where (cid:0) M i f (cid:1) ( z ) = z i f ( z ) , f ∈ H and { K w ε i } ni =1 is a basis for ∩ mi =1 ker( M i − w i ) ∗ , w ∈ D .The joint kernel of the commuting m - tuple M ∗ = ( M ∗ , . . . , M ∗ m ), which we assumeto be bounded, then has dimension n . The map σ i : w K ¯ w ε i , w ∈ ¯ D , 1 ≤ i ≤ n ,provides a trivialization of the corresponding bundle E of Cowen - Douglas (cf. [4]). Here¯ D := { z ∈ C m | ¯ z ∈ D} ).On the other hand, suppose we start with an abstract Hilbert space H and a m -tuple ofcommuting operators T = ( T , . . . , T m ) in the Cowen - Douglas class B n ( D ). Then we havea holomorphic Hermitian vector bundle E over D with the fibre E w = ∩ ni =1 ker( T i − w i ) at w ∈ D . Following [4], one associates to this a reproducing kernel Hilbert space ˆ H consistingof holomorphic functions on ¯ D as follows. Take a holomorphic trivialization σ i : D → H with σ i ( w ) , ≤ i ≤ n , spanning E w . For f ∈ H , define ˆ f j ( w ) := h f, σ j ( ¯ w ) i H , w ∈ ¯ D . Set h ˆ f , ˆ g i ˆ H := h f, g i H . The function K w ε j := \ σ j ( ¯ w ) then serves as the reproducing kernel forthe Hilbert space ˆ H . Note that h K w ( z ) ε j , ε i i C n = h K w ε j , K z ε i i ˆ H = h \ σ j ( ¯ w ) , \ σ i (¯ z ) i ˆ H = h σ j ( ¯ w ) , σ i (¯ z ) i H , z, w ∈ ¯ D . ULTIPLICITY-FREE HOMOGENEOUS OPERATORS IN THE COWEN-DOUGLAS CLASS 3
If one applies this construction to the case where H is a Hilbert space of holomorphicfunctions on D , possesses a reproducing kernel, say K , and the operator M ∗ is in B n ( ¯ D )then using the trivialization σ i ( w ) = K ¯ w ε i , w ∈ ¯ D for the bundle E defined on ¯ D , thereproducing kernel for ˆ H is h K w ( z ) ε j , ε i i C n = h K w ε j , K z ε j i H = h σ j ( ¯ w ) , σ i (¯ z ) i H = h K w ε j , K z ε i i ˆ H , z, w ∈ D . Thus H = ˆ H .Let G be a Lie group acting transitively on the domain D ⊆ C d . Let G L ( n, C ) denote theset of non-singular n × n matrices over the complex field C . We start with a multiplier J ,that is, a smooth family of holomorphic maps J g : D → C n × n satisfying the cocycle relation(1.1) J gh ( z ) = J h ( z ) J g ( h · z ) , for all g, h ∈ G, z ∈ D , Let Hol( D , C n ) be the linear space consisting of all holomorphic functions on D takingvalues in C n . We then obtain a natural (left) action U of the group G on Hol( D , C n ):(1.2) ( U g f )( z ) = J g − ( z ) f ( g − · z ) , f ∈ Hol( D , C n ) , z ∈ D . Let K ⊆ G be the compact subgroup which is the stabilizer of 0. For h, k in K , we have J kh (0) = J h (0) J k (0) so that k J k (0) − is a representation of K on C n .As in [6], we say that if a reproducing kernel K transforms according to the rule(1.3) J ( g, z ) K ( g ( z ) , g ( ω )) J ( g, ω ) ∗ = K ( z, ω )for all g ∈ ˜ G ; z, ω ∈ D , then K is quasi-invariant . Proposition . Suppose H has a reproducing kernel K . Then U defined by (1.2) is a unitary representation if and only if K is quasi-invariant. Let g z be an element of G which maps 0 to z , that is g z · z .For quasi-invariant K we have(1.4) K ( g z · , g z ·
0) = ( J g z (0)) − K (0 , J g z (0) ∗ ) − , which shows that K ( z, z ) is uniquely determined by K (0 , z in D , the positivedefinite matrix K ( z, z ) gives the Hermitian structure of our vector bundle.Given any positive definite matrix K (0 ,
0) such that(1.5) J k (0) − K (0 ,
0) = K (0 , J k (0) ∗ for all k ∈ K , that is, the inner product h K (0 , · | ·i is invariant under J k (0), (1.4) defines a Hermitianstructure on the homogeneous vector bundle determined by J g ( z ). In fact, K ( z, z ), for any z ∈ D is well defined, because if g ′ z is another element of G such that g ′ z · z then g ′ z = g z k ADAM KOR ´ANYI AND GADADHAR MISRA for some k ∈ K . Hence K ( g ′ z · , g ′ z ·
0) = K ( g z k · , g z k · J g z k (0)) − K (0 , J g z k (0) ∗ ) − = (cid:0) J k (0) J g z ( k · (cid:1) − K (0 , (cid:0) J g z ( k · ∗ J k (0) ∗ (cid:1) − = ( J g z (0)) − ( J k (0)) − K (0 , J k (0) ∗ ) − ( J g z (0) ∗ ) − = ( J g z (0)) − K (0 , J g z (0) ∗ ) − = K ( g z · , g z · K ( z, z ), but there is no guarantee (and isfalse in general) that K ( z, z ) extends to a positive definite kernel on D × D . It is, however,true that if there is such an extension then it is uniquely determined by K ( z, z ) (because K ( z, w ) is holomorphic in z and antiholomorphic in w ).This leaves us with the following possible strategy for finding the homogeneous operatorsin the Cowen - Douglas class. Find all multipliers, (i.e., holomorphic homogeneous vectorbundles (hhvb)) such that there exists K (0 ,
0) satisfying (1.5) and consider all such K (0 , K ( z, z ) obtained form (1.4) extends to a positive definite kernelon D × D . Then check if the multiplication operator is well-defined and bounded on thecorresponding Hilbert space.Let H be a Hilbert space consisting of C n - valued holomorphic functions on some do-main D possessing a reproducing kernel K . The sections of the corresponding holomorphicHermitian vector bundle defined on D have many different realizations. The connectionbetween two of these is given by a n × n invertible matrix valued holomorphic function ϕ on D . For f ∈ H , consider the map Γ ϕ : f ˜ f , where ˜ f ( z ) = ϕ ( z ) f ( z ). Let ˜ H = { ˜ f : f ∈ H} .The requirement that the map Γ ϕ is unitary, prescribes a Hilbert space structure for thefunction space ˜ H . The reproducing kernel for ˜ H is easily calculated(1.6) ˜ K ( z, w ) = ϕ ( z ) K ( z, w ) ϕ ( w ) ∗ . It is also easy to verify that Γ ϕ M Γ ∗ ϕ is the multiplication operator M : ˜ f z ˜ f on theHilbert space ˜ H . Suppose we have a unitary representation U given by a multiplier J acting on H according to (1.2). Transplanting this action to ˜ H under the isometry Γ ϕ , itbecomes (cid:0) ˜ U g − ˜ f (cid:1) ( z ) = ˜ J g ( z ) ˜ f ( g · z ) , where the new multiplier ˜ J is given in terms of the original multiplier J by˜ J g ( z ) = ϕ ( z ) J g ( z ) ϕ ( g · z ) − . Of course, now ˜ K transforms according to (1.3), with the aid of ˜ J . If we want, we can nowensure that, by passing from H to an appropriate ˜ H , ˜ K ( z, ≡
1. We merely have to set ϕ ( z ) = K (0 , / K ( z, − . Thus the reproducing kernel ˜ K is almost unique. The onlyfreedom left is to multiply ϕ ( z ) by a constant unitary n × n matrix. Once the kernel isnormalized, we have J k ( z ) = J k (0) , z ∈ D , k ∈ K . ULTIPLICITY-FREE HOMOGENEOUS OPERATORS IN THE COWEN-DOUGLAS CLASS 5
In fact, I = K ( z,
0) = J k ( z ) K ( k · z, J k (0) ∗ = J k ( z ) J k (0) − and the statement follows. Therefore, once the kernel K is normalized, we have (cid:0) U k − f (cid:1) ( z ) = J k (0) f ( k · z ) , k ∈ K . Given a multiplier J , there is always the following method for constructing a Hilbertspace with a quasi-invariant Kernel K transforming according to (1.4). We look for a func-tional Hilbert space possessing this property among the weighted L spaces of holomorphicfunctions on D . The norm on such a space is(1.7) k f k = Z D f ( z ) ∗ Q ( z ) f ( z ) dV ( z )with some positive matrix valued function Q ( z ). Clearly, this Hilbert space possesses areproducing kernel K . The condition that U g − in (1.2) is unitary is Z D f ( g · z ) ∗ J ∗ g ( z ) Q ( z ) J g ( z ) f ( g · z ) dV ( z ) = Z D f ( w ) ∗ Q ( w ) f ( w ) dV ( w )= Z D f ( g · z ) ∗ Q ( g · z ) f ( g · z ) (cid:12)(cid:12)(cid:12)(cid:12) ∂ ( g · z ) ∂ ( z ) (cid:12)(cid:12)(cid:12)(cid:12) dV ( z ) , that is,(1.8) Q ( g · z ) = J g ( z ) ∗ Q ( z ) J g ( z ) (cid:12)(cid:12)(cid:12)(cid:12) ∂ ( g · z ) ∂ ( z ) (cid:12)(cid:12)(cid:12)(cid:12) − , which is equation (1.3) with J g ( z ) replaced by ∂ ( g · z ) ∂ ( z ) J g ( z ) ∗− .Given the multiplier J g ( z ), Q ( z ) is again determined by Q = Q (0), and (just as in thecase of K (0 ,
0) = A ) it must be a positive matrix commuting with all J k (0), k ∈ K . (It isassumed that each J k (0) is unitary).In this way, we can construct many examples of homogeneous operators in B n ( D ) butnot all.Even, not all the the homogeneous operators in B ( D ) come from this construction. Thereis a homogeneous operator in the class B ( D ) corresponding to the multiplier J ( g, z ) =( g ′ ( z )) λ , λ ∈ R exactly when λ >
0. The reproducing kernel is K ( z, w ) = (1 − z ¯ w ) − λ . Butsuch an operator arises from the construction outlined above only if λ ≥ / Computation of the multipliers for the unit disc
In the case of B n ( D ), it is shown in [6] that the bundle corresponding to a homogeneousCowen-Douglas operator admits an action of the covering group ˜ G of the group G = M¨obvia unitary bundle maps. This suggests the strategy of first finding all the homogeneousholomorphic Hermitian vector bundles (a problem easily solved by known methods) andthen determining which of these correspond to an operator in the Cowen-Douglas class. ADAM KOR ´ANYI AND GADADHAR MISRA
We are going to use the method of holomorphic induction. For this, first we describesome basic facts and fix our notation. We follow the notation of [7] which we will use as areference.The Lie algebra g of ˜ G is spanned by X = ! , X = i − i ! and Y = − ii ! . The subalgebra k corresponding to ˜ K is spanned by X . In the complexifiedLie algebra g C , we mostly use the complex basis h, x, y given by h = − iX = 12 − ! x = X + iY = ! y = X − iY = ! We write G C for the (simply connected group) S L (2 , C ). Let G = S U (1 ,
1) be thesubgroup corresponding to g . The group G C has the closed subgroups K C = n(cid:16) z z (cid:17) : z ∈ C , z = 0 o , P + = n(cid:16) z (cid:17) : z ∈ C o , P − = n(cid:16) z (cid:17) : z ∈ C o ; the corresponding Liealgebras k C = n(cid:16) c − c (cid:17) : c ∈ C o , p + = n(cid:16) c (cid:17) : c ∈ C o , p − = n(cid:16) c (cid:17) : c ∈ C o arespanned by h , x and y , respectively. The product K C P − = n a b a ! : 0 = a ∈ C , b ∈ C o is a closed subgroup to be denoted T ; its Lie algebra is t = C h + C y . The product set P + K C P − = P + T is dense open in G C , contains G , and the product decomposition of eachof its elements is unique. ( G C /T is the Riemann sphere, g ˜ K → gT, ( g ∈ G ) is the naturalembedding of D into it.)According to holomorphic induction [5, Chap 13] the isomorphism classes of homogeneousholomorphic vector bundles are in one to one correspondence with equivalence classes oflinear representations ̺ of the pair ( t , ˜ K ). Since ˜ K is connected, here this means just therepresentations of t . Such a representation is completely determined by the two lineartransformations ̺ ( h ) and ̺ ( y ) which satisfy the bracket relation of h and y , that is,(2.9) [ ̺ ( h ) , ̺ ( y )] = − ̺ ( y ) . The ˜ G -invariant Hermitian structures on the homogeneous holomorphic vector bundle (mak-ing it into a homogeneous holomorphic Hermitian vector bundle), if they exist, are givenby ̺ ( ˜ K )-invariant inner products on the representation space. An inner product is ̺ ( ˜ K )-invariant if and only if ̺ ( h ) is diagonal with real diagonal elements in an appropriate basis.We will be interested only in bundles with a Hermitian structure. So, we will assumewithout restricting generality, that the representation space of ̺ is C d and that ̺ ( h ) is areal diagonal matrix. ULTIPLICITY-FREE HOMOGENEOUS OPERATORS IN THE COWEN-DOUGLAS CLASS 7
Furthermore, we will be interested only in irreducible homogeneous holomorphic Hermit-ian vector bundles, this corresponds to ̺ not being the orthogonal direct sum of non-trivialrepresentations. Suppose we have such a ̺ ; we write V α for the eigenspace of ̺ ( h ) witheigenvalue α . Let − η be the largest eigenvalue of ̺ ( h ) and m be the largest integer suchthat − η, − ( η + 1) , . . . , − ( η + m ) are all eigenvalues. From (2.9) we have ̺ ( y ) V α ⊆ V α − ; thisand orthogonality of the eigenspaces imply that V = ⊕ mj =0 V − ( η + j ) and its orthocomplementare invariant under ̺ . So, V is the whole space, and have proved that the eigenvalues of ̺ ( h ) are − η, . . . , − ( η + m ).¿From this it is clear that ̺ can be written as the tensor product of the one dimensionalrepresentation σ given by σ ( h ) = − η , σ ( y ) = 0, and the representation ̺ given by ̺ ( h ) = ̺ ( h ) + ηI , ̺ ( y ) = ̺ ( y ). Correspondingly, the bundle for ̺ is the tensor product of a linebundle L η and the bundle corresponding to ̺ .The representation ̺ has the great advantage that it lifts to a holomorphic representationof the group T . It follows that the homogeneous holomorphic vector bundle it determinesfor D , ˜ G , can be obtained as the restriction to D of the homogeneous holomorphic vectorbundle over G C /T obtained by ordinary induction in the complex analytic category. So, (asa convenient choice) take the local holomorphic cross section z s ( z ) := (cid:16) z (cid:17) of G C /T over D . In the trivialization given by s ( z ), the multiplier then appears for g = (cid:16) a bc d (cid:17) ∈ G C as J g ( z ) = ̺ (cid:0) s ( z ) − g − s ( g · z ) (cid:1) = ̺ cz + d − c ( cz + d ) − ! = ̺ (cid:16) exp (cid:0) − ccz + d y (cid:1)(cid:17) ̺ (cid:0) exp(2 log( cz + d ) h ) (cid:1) . (2.10)The last two equalities are simple computations.For the line bundle L η , the multiplier is g ′ ( z ) η (we write g ′ ( z ) = ∂g∂z ( z )). Consequently,the multiplier corresponding to the original ̺ is(2.11) J g ( z ) = (cid:0) g ′ ( z ) (cid:1) η J g ( z ) . Conditions imposed by the reproducing kernel
We now assume that we have a homogeneous holomorphic vector bundle induced by ̺ asin the preceding sections and that it has a reproducing kernel. Then we derive conditionsabout the action of ˜ G that follow from this hypothesis. In the final section, we will show thatthese conditions are sufficient: they lead directly to the construction of all homogeneousoperators the Cowen-Douglas class with multiplicity free representations.Under our hypothesis there is a Hilbert space structure on our sections in which theaction of ˜ G given by (1.4) is unitary. We will study this representation through its K -types (i.e., its restriction to ˜ K ). We first compute the infinitesimal representation. ADAM KOR ´ANYI AND GADADHAR MISRA
For X ∈ g , and holomorphic f , we have( U X f )( z ) := (cid:0) ddt (cid:1) | t =0 (cid:0) U exp( tX ) f (cid:1) ( z )= (cid:0) ddt (cid:1) | t =0 n(cid:16) ∂ (cid:0) exp( − tX ) · z (cid:1) ∂z (cid:17) η J − tX ) ( z ) f (exp( − tX ) · z ) o . (3.12)There is a local action of G C , so this formula remains meaningful also for X ∈ g C . Thereare three factors to differentiate. For the last one, (cid:0) ddt (cid:1) | t =0 f (exp( − tX ) · z ) = − ( Xz ) f ′ ( z ),and we see that exp( tx ) · z = t ! · z = z + t gives x · z = 1; by similar computations, y · z = − z , h · z = z . For the first factor, we interchange the differentiations and get − η ∂∂z ( X · z ), i.e., 0 , ηz, − η , respectively for x, y and h .To differentiate the factor in the middle, we use its expression (2.10). First for X = y ,we have ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 ̺ (cid:0) exp( − t ( tz + 1) − y ) (cid:1) = ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 (cid:0) exp( − t ( tz + 1) − ̺ ( y ) (cid:1) = − ̺ ( y )(3.13)and ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 ̺ (exp(2 log( tz + 1) h )) = ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 exp(2 log( tz + 1) ̺ ( h ))= 2 z̺ ( h )(3.14)¿From these, following the conventions of [7] in defining H,E,F, it follows that( F f )( z ) := ( U − y f )( z ) = ddt (cid:12)(cid:12)(cid:12)(cid:12) | t =0 J exp( ty ) ( z ) f (exp( ty ) · z )= (cid:0) − ηzI + 2 z̺ ( h ) − ̺ ( y ) (cid:1) f ( z ) − z f ′ ( z ) . (3.15)Similar, simpler computations give, for g = exp( tx ) = t ! (3.16) ( Ef )( z ) := (cid:0) U x f (cid:1) ( z ) = − f ′ ( z ) . Finally, for g = exp( th ) = e t/ e − t/ ! , we have J exp( th ) ( z ) = ̺ e − t/ e t/ ! = exp( − t ) ̺ ( h ) . Hence it is not hard to verify that(3.17) ( Hf )( z ) := (cid:0) U h f (cid:1) ( z ) = (cid:0) − ηI + ̺ ( h ) (cid:1) f ( z ) − zf ′ ( z ) . Under our hypothesis, we have a reproducing kernel and U is unitary. From our computa-tions above, we can determine how U decomposes into irreducibles. The infinitesimal rep-resentation of U acts on the vector valued polynomials; a good basis for this space is { ε j z n : n ≥ } ; ε j is the j th natural basis vector in C d . We have H ( ε j z n ) = − ( η + j + n )( ε j z n ), sothe lowest K - types of the irreducible summands are spanned by the ε j . This space is also ULTIPLICITY-FREE HOMOGENEOUS OPERATORS IN THE COWEN-DOUGLAS CLASS 9 the kernel of E . So, U is direct sum of discrete series representations ( U η + j , in the notationof [7]), each one appearing as many times as − ( η + j ) appears on the diagonal of ̺ ( h ).4. The multiplicity-free case
In order to be able to use the computations of [6] without confusion, we introduce theparameter λ = η + m .From the last remark of the preceding section, it is clear that if U is multiplicity-freethen ̺ ( h ) is an ( m + 1) × ( m + 1) matrix with eigenvalues − λ + m , − λ + m − , . . . , − λ − m .As ̺ ( h ) ε j = − ( λ − m + j ) ε j , (2.9) shows that ̺ ( h ) (cid:0) ̺ ( y ) ε j (cid:1) = − ( λ + m j + 1) ̺ ( y ) ε j , that is , ̺ ( y ) ε j = const ε j +1 . So, ̺ ( y ) is a lower triangular matrix (with non-zero entries, otherwise we have a reduciblebundle). The homogeneous holomorphic vector bundle determines ̺ ( y ) only up to a conju-gacy by a matrix commuting with ̺ ( h ), that is, a diagonal matrix. So, we can choose therealization of our bundle by applying an appropriate conjugation such that ̺ ( y ) = S m , thetriangular matrix whose ( j, j −
1) element is j for 1 ≤ j ≤ m .By standard representation theory of SL(2 , R ), the vectors ( − F ) n ε j are orthogonal andthe irreducible subspaces H ( j ) for U are span { ( − F ) n ε j : n ≥ } for 0 ≤ j ≤ m . There isalso precise information about the norms.Using this, we can construct an orthonormal basis for our representation space.For any n ≥
0, we let u jn ( z ) = ( − F ) n ε j .To proceed further, we need to find the vectors u jn ( z ) explicitly. This is facilitated by thefollowing Lemma. Lemma . Let u be a vector with u ℓ ( z ) = u ℓ z n − ℓ , ≤ ℓ ≤ m and n ≥ . We then have ( − F u ) ℓ ( z ) = (2 λ − m + ℓ + n ) u ℓ z n +1 − ℓ + ℓu ℓ − z n +1 − ℓ , ≤ ℓ ≤ m. Proof.
We recall (3.15) that − ( F f )( z ) = 2 λzf ( z ) + S m f ( z ) − zD m f ( z ) + z f ′ ( z ) for f ∈ H ( n ), where D m = − ̺ ( h ) is the diagonal operator with diagonal {− m , − m +1 , . . . , m } and S m is the forward weighted shift with weights 1 , , . . . , m . Therefore we have( − F u ) ℓ ( z ) = (cid:0) λu ℓ + ℓu ℓ − − ( m − ℓ ) u ℓ + ( n − ℓ ) u ℓ (cid:1) z n +1 − ℓ completing the proof. (cid:3) Lemma . For ≤ j ≤ m and ≤ ℓ ≤ m , we have u jn,ℓ ( z ) = ( ≤ ℓ ≤ j − (cid:0) nk (cid:1) ( j + 1) k (2 λ − m + 2 j + k ) n − k z n − k if j ≤ ℓ ≤ m, k = ℓ − j, where u jn,ℓ ( z ) is the scalar valued function at the position ℓ of the C m +1 - valued function u jn ( z ) := ( − F ) n ε j .Proof. The proof is by induction on n . The vectors u jn are in H ( n ) for 0 ≤ j ≤ m . For afixed but arbitrary positive integer j , 0 ≤ j ≤ m , we see that u jn,ℓ ( z ) is 0 if n < ℓ − j . Wehave to verify that ( − F u jn )( z ) = u jn +1 ( z ). From the previous Lemma, we have( − F u jn ) ℓ ( z ) = (2 λ − m + ℓ + n + j ) u jn,ℓ z n + j +1 − ℓ + ℓu jn,ℓ − z n + j +1 − ℓ , where ( − F u jn ) ℓ ( z ) is the scalar function at the position ℓ of the C m +1 - valued function( − F u jn )( z ). To complete the proof, we note (using k = ℓ − j ) that( − F u jn ) j + k ( z )= (cid:0)(cid:0) nk (cid:1) ( j + 1) k (2 λ − m + 2 j + k ) n − k (2 λ − m + 2 j + k + n ) + (cid:0) nk − (cid:1) ( j + 1) k (2 λ − m + 2 j + k − n − k (cid:1) z n +1 − k = ( j + 1) k (2 λ − m + 2 j + k ) n − k (cid:0)(cid:0) nk (cid:1) (2 λ − m + 2 j + k + n ) + (cid:0) nk − (cid:1) (2 λ − m + 2 j + k − (cid:1) z n +1 − k = ( j + 1) k (2 λ − m + 2 j + k ) n − k (cid:0) ( (cid:0) nk (cid:1) + (cid:0) nk − (cid:1) (2 λ − m + 2 j + k −
1) + ( n + 1) (cid:0) nk (cid:1)(cid:1) z n +1 − k = ( j + 1) k (2 λ − m + 2 j + k ) n − k (cid:0)(cid:0) n +1 k (cid:1) (2 λ − m + 2 j + k −
1) + (cid:0) n +1 k (cid:1) ( n − k + 1) (cid:1) z n +1 − k = ( j + 1) k (2 λ − m + 2 j + k ) n − k (cid:0)(cid:0) n +1 k (cid:1) (2 λ − m + 2 j + n ) (cid:1) z n +1 − k = ( j + 1) k (cid:0)(cid:0) n +1 k (cid:1) (2 λ − m + 2 j + k ) n +1 − k (cid:1) z n +1 − k = u jn +1 ,j + k ( z )for a fixed but arbitrary j , 0 ≤ j ≤ m and k , 0 ≤ k ≤ m − j . This completes the proof. (cid:3) On H ( j ) , we have the representation U λ j acting (0 ≤ j ≤ m ), where λ j = λ − m + j .Its lowest K - type is spanned by ε j (= u j ) and Hε j = λ j ε j . By [7, Prop 6.14] we have k ( − F ) k ε j k = σ jk k ( − F ) k − ε j k with σ jk = (2 λ j + k − k for all k ≥
1. (Here we used that the constant q in [7, equation (6.33)] equals λ j (1 − λ j ) by[7, Theorem 6.2].) We write σ jn = n Y k =1 σ jk which can be written in a compact form(4.18) σ jn = ((2 λ j ) n (1) n ) , where ( x ) n = ( x + 1) · · · ( x + n − (cid:0) nk (cid:1) as wellas ( x ) n − k are both zero if n < k .The positivity of the normalizing constants (cid:0) σ jn − j (cid:1) ( n ≥ j ) is equivalent to the existenceof an inner product for which the set of vectors e jn − j defined by the formula: e jn − j = ( σ jn ) − u jn − j ( z ) , n ≥ j, ≤ j ≤ m forms an orthonormal set. Of course, the positivity condition is fulfilled if and only if2 λ > m .In this way, for fixed j , each e jn − j has the same norm for all n ≥ j . Hence the only possiblechoice for an orthonormal system is { µ j e jn − j : n ≥ j } for some positive real numbers µ j > ULTIPLICITY-FREE HOMOGENEOUS OPERATORS IN THE COWEN-DOUGLAS CLASS 11 (0 ≤ j ≤ m ). However, we may choose the norm of the first vector, that is, the vector e j ,0 ≤ j ≤ m , arbitrarily. Therefore, all the possible choices for an orthonormal set are µ j e jn − j ( z ) = µ j p (2 λ − m + 2 j ) n − j p (1) n − j u jn − j ( z ) , (4.19) n ≥ j, ≤ j ≤ m, and µ j , ≤ j ≤ m are m + 1 arbitrary positive numbers.Let us fix a positive real number λ and m ∈ N satisfying 2 λ > m . Let H ( λ, µ ) denote theclosed linear span of the vectors { µ j e jn − j : 0 ≤ j ≤ m, n ≥ j } . Then the Hilbert space H ( λ, µ ) is the representation space for U defined in (1.2). Since the vectors u jn ⊥ u kp as longas j = k , it follows that the Hilbert space H ( λ, µ ) is the orthogonal direct sum ⊕ mj =0 1 µ j H ( j ) .We proceed to compute the reproducing kernel by using the orthonormal system { µ j e jn − j : n ≥ j } , 0 ≤ j ≤ m . We point out that for 0 ≤ ℓ ≤ m , the entry e ℓ,jn − j z n − j at the position ℓ of the vector e jn − j ( z ) is 0 for n < ℓ . Consequently, e jn − j is the zero vector unless n ≥ j .The set of vectors { µ j e jn − j : 0 ≤ j ≤ m, n ≥ j } is orthonormal in the Hilbert space H ( λ, µ ) .We note that e jn − j ( z ) = (( e ℓ,jn − j z n − k )) mℓ =0 , (cid:0) e jn − j ( z ) (cid:1) ℓ = , ≤ ℓ ≤ j − q (2 λ +2 j − m + k ) n − j − k (1) n − j − k q ( n − j − k +1) k (2 λ +2 j − m ) k ( j +1) k (1) k z n − k , j ≤ ℓ ≤ m, k = ℓ − j. (4.20)We have under the hypothesis that we have a reproducing kernel Hilbert space on whichthe representation U is unitary, explicitly determined an orthonormal basis for this space.Now we are able to answer the question of whether this space really exists. For this itis enough to show that P e n ( z ) e n ( w ) tr converges pointwise, the sum then represents thereproducing kernel for this Hilbert space. We will sum the series explicitly, and will verifythat it gives exactly the kernels constructed in [6]. This will complete the program ofthis paper by proving that the examples of [6] give all the homogeneous operators in theCowen-Douglas class whose associated representation is multiplicity free.To compute the kernel function, it is convenient to set, for any n ≥ G ( µ , n, z ) = µ e , n z n . . . . . . . . . ... . . . ... µ e j, n z n − j . . . µ j e j,jn − j z n − j . . . . . . ... . . . ... µ e m, n z n − m . . . µ j e m,jn − j z n − m . . . µ m e m,mn − m z n − m = z n . . . . . . z n − m e , n . . . . . . . . . ... . . . ... e j, n . . . e j,jn − j . . . . . . ... . . . ... e m, n . . . e m,jn − j . . . e m,mn − m µ . . . . . . µ m = D n ( z ) G ( n ) D ( µ ) , (4.21) where D n ( z ) , D ( µ ) are the two diagonal matrices and G ( n ) = (( e ℓ,jn − j )) mℓ,j =0 with e ℓ,jn − j = 0 if ℓ < j or if n < ℓ . The nonzero entries of the lower triangular matrix G ( n ), using (4.20), are G j + k,j ( n ) = (cid:0) n − jk (cid:1) ( j + 1) k (2 λ − m + 2 j + k ) n − j − k p (2 λ − m + 2 j ) n − j p (1) n − j = p (2 λ − m + 2 j + k ) n − j − k p (2 λ − m + 2 j ) k ( n − j − k + 1) k p (1) n − j ( j + 1) k (1) k = s (2 λ − m + 2 j + k ) n − j − k (2 λ − m + 2 j ) k s ( n − j − k + 1) k (1) n − j − k ( j + 1) k (1) k (4.22)for 0 ≤ k ≤ m − j .Now, we are ready to compute the reproducing kernel K j for the Hilbert space H ( j ) =span { e jn − j : n ≥ j } , 0 ≤ j ≤ m . Recall that K ( z, w ) = P ∞ n =0 e n ( z ) e n ( w ) ∗ for any orthonor-mal basis e n , n ≥
0. This ensures that K is a positive definite kernel. For our computations,we will use the particular orthonormal basis e jn − j as described in (4.19). Since there are j zeros at the top of each of these basis vectors, it follows that ( ℓ, p ) will be 0 if either ℓ < j or p < j . We will compute (( K j ( z, w ))), at ( ℓ, p ) for j ≤ ℓ, p ≤ m . For ℓ, p as above, we have(( K j ( z, w ))) ℓ,p = ∞ X n ≥ max ( ℓ,p ) e jn − j,ℓ ( z ) e jn − j,p ( w )= ∞ X n ≥ max ( ℓ,p ) G ℓ,j ( n ) G p,j ( n ) z n − ℓ ¯ w n − p . We first simplify the co-efficient G ℓ,j ( n ) G p,j ( n ) of z n − ℓ ¯ w n − p . The values of G ℓ,j ( n ) are givenin (4.22). Therefore, we have G ℓ,j ( n ) G p,j ( n )= (cid:16) (2 λ j + ℓ − j ) n − ℓ (2 λ j ) ℓ − j ( n − ℓ + 1) ℓ − j (1) n − ℓ (2 λ j + p − j ) n − p (2 λ j ) p − j ( n − ℓ + 1) ℓ − j (1) n − p (cid:17) / × ( j + 1) ℓ − j (1) ℓ − j ( j + 1) p − j (1) p − j = (2 λ j + p − j ) n − p ( n − ℓ + 1) ℓ − j (2 λ j ) ℓ − j (1) n − p (cid:16) (2 λ j + ℓ − j ) p − ℓ ( n − p + 1) p − ℓ (2 λ j + ℓ − j ) p − ℓ ( n − p + 1) p − ℓ (cid:17) / × ( j + 1) ℓ − j (1) ℓ − j ( j + 1) p − j (1) p − j = (2 λ j ) p − j (2 λ j + p − j ) n − p ( n − ℓ + 1) ℓ − j ( n − p + 1) p − j (2 λ j ) p − j (2 λ j ) ℓ − j (1) n − p ( n − p + 1) p − j ( j + 1) ℓ − j (1) ℓ − j ( j + 1) p − j (1) p − j = (2 λ j ) n − j ( n − ℓ + 1) ℓ − j ( n − p + 1) p − j (2 λ j ) p − j (2 λ j ) ℓ − j (1) n − j ( j + 1) ℓ − j (1) ℓ − j ( j + 1) p − j (1) p − j . Theorem . Given an arbitrary set µ , . . . , µ m of positive numbers, and λ > m , we have K ( λ, µ ) ( z, w ) = m X j =0 µ j K j ( z, w ) = B ( λ, µ ) ( z, w ) . As a result, the two Hilbert spaces H ( λ, µ ) and A ( λ, µ ) of [6] are equal. ULTIPLICITY-FREE HOMOGENEOUS OPERATORS IN THE COWEN-DOUGLAS CLASS 13
Proof.
We now compare the co-efficients (( K j ( z, w ))) ℓ,p with that of a known Kernel. Let B λ j ( z, w ) = (1 − z ¯ w ) − λ j , where B ( z, w ) = (1 − z ¯ w ) − is the Bergman kernel on the unitdisc. We let ∂ and ¯ ∂ denote differentiation with respect to z and ¯ w respectively. Put˜ B ( λ j ) ( z, w ) = (( ∂ ℓ − j ¯ ∂ p − j (1 − z ¯ w ) − λ j )) j ≤ ℓ,p ≤ m . We expand the entry at the position ( ℓ, p ) of ˜ B ( λ j ) ( z, w ) to see that(( ˜ B ( λ j ) ( z, w ))) ℓ,p = X ν ≥ max ( ℓ − j,p − j ) (2 λ j ) ν (1) ν ( ν − ℓ + j + 1) ℓ − j ( ν + j − p + 1) p − j z ν − ( ℓ − j ) ¯ w ν − ( p − j ) = X n ≥ max ( ℓ,p ) (2 λ j ) n − j (1) n − j ( n − ℓ + 1) ℓ − j ( n − p + 1) p − j z n − ℓ ¯ w n − p , where we have set n = m + j . Comparing these coefficients with that of G ℓ,j ( n ) G p,j ( n ), wefind that(4.23) K j ( z, w ) = D j ˜ B ( λ j ) ( z, w ) D j , where D j is a diagonal matrix with λ j ) ℓ − j ( j +1) ℓ − j (1) ℓ − j at the ( ℓ, ℓ ) position with j ≤ ℓ ≤ m .Hence K j ( z, w ) = B ( λ j ) ( z, w ) which was defined in the equation ([6, equation (4.3)]).Clearly, we can add up the kernels K j to obtain the kernel K ( λ, µ ) for the Hilbert space H ( λ, µ ) = ⊕ mj =0 1 µ j H ( j ) . Hence the proof of the theorem is complete. (cid:3) Corollary . The irreducible homogeneous operators in the Cowen - Douglas class whoseassociated representation is multiplicity free are exactly the adjoints of M ( λ, µ ) constructedin [6] .Proof. In our discussion up to here we proved that the Hilbert space H ( λ, µ ) correspondingto a homogeneous operator in the Cowen - Douglas class has a reproducing kernel given by K ( λ, µ ) = P m µ j K j , 2 λ > , µ , . . . , µ m >
0. It follows from the Theorem that the kernelsobtained this way are the same as (are equivalent to) the kernels constructed in [6]. Theseoperators were shown to be irreducible [6]. (cid:3)
We now consider the action of the multiplication operator M ( λ, µ ) on the Hilbert space H ( λ, µ ) . Let H ( n ) be the linear span of the vectors { e n ( z ) , . . . , e jn − j ( z ) , . . . , e mn − m ( z ) } , where as before, for 0 ≤ ℓ ≤ m , e jn − ℓ ( z ) is zero if n − ℓ <
0. Clearly, H ( λ, µ ) = ⊕ ∞ n =0 H ( n ).We have zG ( n, z ) = D n ( z ) G ( n ) D ( µ )= D n +1 ( z ) G ( n ) D ( µ )= D n +1 ( z ) G ( n + 1) D ( µ ) (cid:0) D ( µ ) − G ( n + 1) − G ( n ) D ( µ ) (cid:1) . If we let W ( n ) = D ( µ ) − G ( n + 1) − G ( n ) D ( µ ), then we see that z e jn − j ( z ) = G ( µ , n +1 , z ) W j ( n ), where W j ( n ) is the j th column of the matrix W ( n ). It follows that the operator M ( λ, µ ) defines a block shift W on the representation space H ( λ, µ ) . The block shift W isdefined by the requirement that W : H ( n ) → H ( n + 1) and W |H ( n ) = W t rn . Here, we have a construction of the representation space H ( λ, µ ) along with the matrixrepresentation of the operator M ( λ, µ ) which is independent of the corresponding resultsfrom [6]. 5. Examples
Recall that G ( µ , n, z ) = D n ( z ) G ( n ) D ( µ ). Once we determine the matrix G ( n ) explicitly,we can calculate both the block weighted shift and the kernel function.We discuss these calculations in the particular case of m = 1. First, it is easily seen that(5.24) G ( n ) = (cid:0) (2 λ − n (1) n (cid:1) / n λ − ) / (cid:0) (2 λ ) n − (1) n − (cid:1) / (cid:0) (2 λ +1) n − (1) n − (cid:1) / ! . The block W n of the weighted shift W is(5.25) W n = ( n +12 λ + n − ) / − µ ( λ λ − ) / ( λ + n − λ + n ) ) / ( n λ + n ) / ! . Finally, the reproducing kernel K ( λ, µ ) with m = 1 is easily calculated:(5.26) K ( λ, µ ) ( z, w ) = − ¯ wz ) λ − z (1 − ¯ wz ) λ ¯ w (1 − ¯ wz ) λ λ − λ −
1) ¯ wz (1 − ¯ wz ) λ +1 ! + µ − ¯ wz ) λ +1 ! . One might continue the explicit calculations, as above, in the particular case of m = 2as well. We begin with the matrix(5.27) G ( n ) = (cid:0) (2 λ − n (1) n (cid:1) / n λ − ) / (cid:0) (2 λ − n − (1) n − (cid:1) / (cid:0) (2 λ ) n − (1) n − (cid:1) / n ( n − λ − λ − ) / (cid:0) (2 λ ) n − (1) n − (cid:1) / n − λ ) / (cid:0) (2 λ +1) n − (1) n − (cid:1) / (cid:0) (2 λ +2) n − (1) n − (cid:1) / . The block W n of the weighted shift W , in this case, is(5.28) (cid:0) n +12 λ + n − (cid:1) / − µ (cid:0) λ − λ − (cid:1) / (cid:0) λ + n − λ + n − (cid:1) / (cid:0) n λ + n − (cid:1) / − µ (cid:0) λ +1(2 λ − (cid:1) / (cid:0) n (2 λ + n − (cid:1) / − µ µ (cid:0) λ +12 λ (cid:1) / (cid:0) λ + n − λ + n ) (cid:1) / (cid:0) n − λ + n (cid:1) / . Finally, the reproducing kernel K ( λ, µ ) with m = 2 has the form: K ( λ, µ ) ( z, w ) = − ¯ wz ) λ − z (1 − ¯ wz ) λ − z (1 − ¯ wz ) λ ¯ w (1 − ¯ wz ) λ − λ −
2) ¯ wz (2 λ − − ¯ wz ) λ z (2+(2 λ −
2) ¯ wz )(2 λ − − ¯ wz ) λ +1 ¯ w (1 − ¯ wz ) λ ¯ w (2+(2 λ −
2) ¯ wz )(2 λ − − ¯ wz ) λ +1 λ −
1) ¯ wz +(2 λ − λ − z ¯ w (2 λ − λ − − ¯ wz ) λ +2 + µ − ¯ wz ) λ z (1 − ¯ wz ) λ +1 ¯ w (1 − ¯ wz ) λ +1 λ λ ¯ wz (1 − ¯ wz ) λ +2 + µ − ¯ wz ) λ +2 . (5.29) ULTIPLICITY-FREE HOMOGENEOUS OPERATORS IN THE COWEN-DOUGLAS CLASS 15
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Lehman College, The City University of New York, Bronx, NY 10468
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