Dirichlet Spaces Associated With Locally Finite Rooted Directed Trees
aa r X i v : . [ m a t h . C V ] F e b DIRICHLET SPACES ASSOCIATED WITHLOCALLY FINITE ROOTED DIRECTED TREES
SAMEER CHAVAN, DEEPAK KUMAR PRADHAN, AND SHAILESH TRIVEDI
Abstract.
Let T = ( V, E ) be a leafless, locally finite rooted directed tree. Weassociate with T a one parameter family of Dirichlet spaces H q ( q > D in the complex plane. These spaces can be realized asreproducing kernel Hilbert spaces associated with the positive definite kernel κ H q ( z, w ) = ∞ X n =0 (1) n ( q ) n z n w n P h e root i + X v ∈ V ≺ ∞ X n =0 ( n v + 2) n ( n v + q + 1) n z n w n P v ( z, w ∈ D ) , where V ≺ denotes the set of branching vertices of T , n v denotes the depthof v ∈ V in T , and P h e root i , P v ( v ∈ V ≺ ) are certain orthogonal projections.Further, we discuss the question of unitary equivalence of operators M (1) z and M (2) z of multiplication by z on Dirichlet spaces H q associated with directedtrees T and T respectively. Introduction
The present work is a sequel to [17]. In that paper, a rich interplay betweenthe directed trees and analytic kernels of finite bandwidth has been exploited tostudy the weighted shifts on directed trees. The analysis therein was based onShimorin’s analytic model as introduced in [28] (for an alternate approach to thefunction theory of weighted shifts on directed trees, the reader is referred to [14],[13]). Our work is also motivated partly by the classification theorem [6, Theorem9.9] obtained for 2-isometric weighted shifts on certain directed trees.The objective of the present paper is to introduce Dirichlet spaces associatedwith certain rooted directed trees. This is carried out by introducing the so-calledDirichlet shifts on directed trees with weights being certain functions of depth ofvertices, and thereafter applying Shimorin’s construction [28] to these shifts. Thesespaces can be thought of as vector-valued weighted Dirichlet spaces (cf. [25]). Wealso discuss the spaces Cauchy dual to Dirichlet spaces. These turn out to be vector-valued Bergman spaces, which play a key role in answering the question of unitaryequivalence of Dirichlet shifts associated with two directed trees. We collect belowsome preliminaries required to define the Dirichlet shifts. For a detailed expositionon weighted shifts on directed trees, the reader is referred to [23] and [17].A directed graph is a pair T = ( V, E ), where V is a nonempty set and E is anonempty subset of V × V \ { ( v, v ) : v ∈ V } . An element of V (resp. E ) is called a vertex (resp. an edge ) of T . A finite sequence { v i } ni =1 of distinct vertices is said tobe a circuit in T if n >
2, ( v i , v i +1 ) ∈ E for all 1 i n − v n , v ) ∈ E . Wesay that two distinct vertices u and v of T are connected by a path if there exists Mathematics Subject Classification.
Primary 46E22, 31C25; Secondary 47B20, 05C20.
Key words and phrases.
Dirichlet space, directed tree, q -isometry.The work of the second author is supported through the NBHM Research Fellowship. a finite sequence { v i } ni =1 of distinct vertices of T ( n >
2) such that v = u , v n = v and ( v i , v i +1 ) or ( v i +1 , v i ) ∈ E for all 1 i n −
1. A directed graph T is said tobe connected if any two distinct vertices of T can be connected by a path in T . For a subset W of V , define Chi ( W ) := [ u ∈ W { v ∈ V : ( u, v ) ∈ E} . We define inductively
Chi h n i ( W ) for n ∈ N as follows: Chi h n i ( W ) := ( W if n = 0 , Chi ( Chi h n − i ( W )) if n > . Given v ∈ V , we write Chi ( v ) := Chi ( { v } ). An element of Chi ( v ) is called a child of v. For a given vertex v ∈ V, consider the set Par ( v ) := { u ∈ V : ( u, v ) ∈ E} . If Par ( v ) is singleton, then the unique vertex in Par ( v ) is called the parent of v , whichwe denote by par ( v ) . Let the subset
Root ( T ) of V be defined as Root ( T ) := { v ∈ V : Par ( v ) = ∅} . Then an element of
Root ( T ) is called a root of T . If Root ( T ) is singleton, then itsunique element is denoted by root . We set V ◦ := V \ Root ( T ). A directed graph T = ( V, E ) is called a directed tree if T has no circuits, T is connected and eachvertex v ∈ V ◦ has a unique parent. A directed tree T is said to be(i) rooted if it has a unique root.(ii) locally finite if card( Chi ( u )) is finite for all u ∈ V, where card( X ) standsfor the cardinality of the set X. (iii) leafless if every vertex has at least one child.Let T = ( V, E ) be a rooted directed tree with root root . Let V ≺ be the set { u ∈ V : card( Chi ( u )) > } of branching vertices of T . For each u ∈ V , the depth of u is the unique non-negative integer n u such that u ∈ Chi h n u i ( root ). Define the branching index of T as k T := ( { n w : w ∈ V ≺ } if V ≺ is non-empty , V ≺ is empty . We say that T is of finite branching index if k T < ∞ . We further say that twodirected trees are isomorphic if there exists a bijection between their sets of verticeswhich preserves directed edges.Let T = ( V, E ) be a rooted directed tree with root root . We always assumethat card( V ) = ℵ . In what follows, l ( V ) stands for the Hilbert space of squaresummable complex functions on V equipped with the standard inner product. Notethat the set { e u } u ∈ V is an orthonormal basis of l ( V ), where e u ∈ l ( V ) is theindicator function of { u } . Given a system λ = { λ v } v ∈ V ◦ of non-negative realnumbers, we define the weighted shift operator S λ on T with weights λ by D ( S λ ) := { f ∈ l ( V ) : Λ T f ∈ l ( V ) } ,S λ f := Λ T f, f ∈ D ( S λ ) , where Λ T is the mapping defined on complex functions f on V by( Λ T f )( v ) := ( λ v · f (cid:0) par ( v ) (cid:1) if v ∈ V ◦ , v is a root of T . Remark . If S λ is bounded then S ∗ λ e u = λ u e par ( u ) if u = root , and S ∗ λ e u = 0otherwise. IRICHLET SPACES ASSOCIATED WITH DIRECTED TREES 3
Throughout these paper, we will be interested in weighted shifts which arebounded linear. The reader is referred to [23] for the basic theory of weightedshifts on directed trees. In particular, it may be concluded from [23, Proposition3.5.1(ii)] that for a rooted directed tree T with root root , the kernel of S ∗ λ is givenby E = h e root i ⊕ M v ∈ V ≺ (cid:0) l ( Chi ( v )) ⊖ h λ v i (cid:1) , (1.1)where λ v : Chi ( v ) → C is defined by λ v ( u ) = λ u and h f i denotes the span of { f } .Recall that a bounded linear operator T on a Hilbert space H is finitely multi-cyclic if there are a finite number of vectors h , · · · , h m in H such that H = _ { T k h , · · · , T k h m : k ∈ N } . Remark . Let T be a leafless, locally finite rooted directed tree with finitebranching index and let S λ be a bounded weighted shift on T . It may be concludedfrom [17, Proposition 2.1 and Corollary 2.3] that S λ is finitely cyclic.2. Dirichlet Shifts on Directed Trees
We now introduce the notion of Dirichlet shift on certain directed trees.
Definition 2.1.
Let T = ( V, E ) be a leafless, locally finite rooted directed tree.For a real number q > , consider the weighted shift S λ,q on T with weights givenby λ u,q = 1 p card( Chi ( v )) r n v + qn v + 1 for u ∈ Chi ( v ) , v ∈ V, (2.1)where n v is the depth of v ∈ V in T . We refer to S λ,q as the Dirichlet shift on T .Let S λ,q be a Dirichlet shift on T . Note that the weights of S λ,q can be rewrittenas λ v,q = r n v + q − n v p card( sib ( v )) for v ∈ V ◦ , where, for u ∈ V , the siblings of u is given by sib ( u ) := ( Chi ( par ( u )) if u = root , ∅ otherwise . Note further thatsup v ∈ V X u ∈ Chi ( v ) λ u,q = sup v ∈ V n v + qn v + 1 = q, inf v ∈ V X u ∈ Chi ( v ) λ u,q = inf v ∈ V n v + qn v + 1 = 1 . It now follows from [23, Propositions 3.1.8 and 3.6.1] that S λ,q is bounded linear,and left-invertible (that is, S λ,q is one-one with closed range). Thus the Cauchydual S ′ λ,q of S λ,q given by S λ,q ( S ∗ λ,q S λ,q ) − is well-defined [28]. It turns out that S ′ λ,q is a weighted shift S λ ′ ,q on T with weights given by λ ′ v,q = r n v n v + q − p card( sib ( v )) for all v ∈ V ◦ . (2.2) Remark . In case T is a rooted directed tree without any branching vertex (thatis, T is isomorphic to N ), S λ,q is unitarily equivalent to the classical weighted shift S w,q with weights w := nq n + qn +1 o n ∈ N ( classical Dirichlet shift ). Note that theweights of the Cauchy dual S ′ w,q of S w,q are nq n +1 n + q o n ∈ N ( classical Bergman shift ). IRICHLET SPACES ASSOCIATED WITH DIRECTED TREES 4
The following classification problem is motivated by [6, Theorem 9.9]:
Problem 2.3.
Let S (1) λ,q , S (2) λ,q be Dirichlet shifts on T , T respectively. Find suffi-cient and necessary conditions on T and T which ensure that S (1) λ,q and S (2) λ,q areunitarily equivalent.Note that the above problem has the following solution in case q = 1 . The shifts S (1) λ, and S (2) λ, unitarily equivalent if and only if X v ∈ V (1) ≺ (cid:16) card ( Chi ( v )) − (cid:17) = X v ∈ V (2) ≺ (cid:16) card ( Chi ( v )) − (cid:17) , where V (1) ≺ and V (2) ≺ are set of branching vertices of T and T respectively. Indeed,since S (1) λ, and S (2) λ, are isometries (Proposition 2.7 below), by von Neumann-Wolddecomposition [18], S (1) λ, and S (2) λ, unitarily equivalent if and only if dim ker( S (1) λ, ) ∗ =dim ker( S (2) λ, ) ∗ . The desired conclusion is now immediate from (1.1). This has beenrecorded in more generality in [6, Theorem 9.9(ii)].One of the main results of this note provides a solution to the above problem incase q is a positive integer. Theorem 2.4.
Let q be an integer bigger than . For j = 1 , , let S ( j ) λ,q be theDirichlet shift on rooted directed tree T j = ( V j , E j ) with root j , let G ( j ) n := { v ∈ V j : v ∈ Chi h n i ( root j ) } ( n ∈ N ) , and E j = ker( S ( j ) λ,q ) ∗ . Then S (1) λ,q is unitarily equivalentto S (2) λ,q if and only if for every n ∈ N , X v ∈ V (1) ≺ ∩G (1) n (cid:16) card ( Chi ( v )) − (cid:17) = X v ∈ V (1) ≺ ∩G (2) n (cid:16) card ( Chi ( v )) − (cid:17) . (2.3) Remark . Since T j is locally finite, the V ( j ) ≺ ∩ G ( j ) n is finite for every n ∈ N and j = 1 ,
2, and hence the sums appearing in (2.3) are finite. Further, it may happenthat (2.3) holds for two non-isomorphic directed trees (see [6, Figure 2]).The case q = 2 of Theorem 2.4 is a special case of [6, Theorem 9.9(i)]. In whatfollows, we provide an alternative verification of this fact based on modeling S λ, asa multiplication by z on a vector-valued Dirichlet space. With this identification,the problem essentially reduces to classification problem of multiplication operatorson vector-valued Dirichlet spaces (refer to Section 3). This part of the proof relieson the theory of vector-valued Dirichlet spaces as expounded in [25]. The ratherinvolved proof of the general case, as presented in the last section, relies on treeanalogs of weighted Bergman spaces. These spaces can be seen as Cauchy dual ofDirichlet spaces in the sense of S. Shimorin [28].In the remaining part of this section, we derive some structural properties of theweighted shifts S λ,q on T . Before we state formulae for moments of S λ,q and S λ ′ ,q , recall that the Pochhammer symbol is defined by( x ) y = Γ( x + y )Γ( x ) , where Γ is the gamma function defined for all complex numbers except the non-positive integers. Lemma 2.6.
Let S λ,q be a Dirichlet shift on T = ( V, E ) and let S λ ′ ,q be the Cauchydual of S λ,q . Then for k ∈ N and v ∈ V, k S kλ,q e v k = ( n v + q ) k ( n v + 1) k , k S kλ ′ ,q e v k = ( n v + 1) k ( n v + q ) k . IRICHLET SPACES ASSOCIATED WITH DIRECTED TREES 5
In particular, the spectral radii of S λ,q and S λ ′ ,q are . Proof.
We verify the first formula by induction on integers k > v ∈ V .The formula is trivial for k = 0 . Suppose the formula holds for some integer k > . Since
Chi h n i ( u ) and Chi h n i ( w ) are disjoint for distinct vertices u and w, it followsfrom [23, Lemma 6.1.1(i)] that { S kλ,q e w } w ∈ Chi ( v ) is mutually orthogonal. Also, since n w = n v + 1 for w ∈ Chi ( v ), we obtain k S k +1 λ,q e v k = (cid:13)(cid:13)(cid:13) S kλ,q X w ∈ Chi ( v ) λ w,q e w (cid:13)(cid:13)(cid:13) = X w ∈ Chi ( v ) λ w,q k S kλ,q e w k = X w ∈ Chi ( v ) Chi ( v )) n v + qn v + 1 ( n w + q ) k ( n w + 1) k = n v + qn v + 1 ( n v + q + 1) k ( n v + 2) k = ( n v + q ) k +1 ( n v + 1) k +1 . The second formula can be verified similarly. To see the remaining part, note that k S kλ,q k = sup n ∈ N s ( n + q ) k ( n + 1) k , k S kλ ′ ,q k = sup n ∈ N s ( n + 1) k ( n + q ) k , and apply the spectral radius formula. (cid:3) The second part of the following generalizes [8, Proposition 8], [3, Theorem 8.6].
Proposition 2.7.
Let S λ,q be a Dirichlet shift on T = ( V, E ) and let S λ ′ ,q be theCauchy dual of S λ,q . If q is a positive integer, then we have the following: (i) S λ ′ ,q is subnormal, that is, S λ ′ ,q admits a normal extension. (ii) S λ,q is a q -isometry, that is, q X k =0 ( − k (cid:18) qk (cid:19) S ∗ kλ,q S kλ,q = 0 , but not a ( q − -isometry, (iii) if T is of finite branching index, then the self-commutator [ S ∗ λ,q , S λ,q ] := S ∗ λ,q S λ,q − S λ,q S ∗ λ,q of S λ,q is of trace-class, (iv) S λ, has wandering subspace property, that is, for any S λ, -invariant sub-space M of l ( V ) , M = _ k ∈ N { S kλ, f : f ∈ M ⊖ S λ, M} . Proof.
Suppose that q is a positive integer.(i) By [23, Theorem 6.1.3], S λ ′ ,q is subnormal if and only if for every v ∈ V, {k S kλ ′ ,q e v k } k ∈ N is a Hausdorff moment sequence. However, by Lemma 2.6, for v ∈ V and k ∈ N . k S kλ ′ ,q e v k = ( n v + 1) k ( n v + q ) k = ( q = 1 , ( n v +1) ··· ( n v + q − n v + k +1) ··· ( n v + k + q − if q > . Since { k + l } k ∈ N is a Hausdorff moment sequence for any integer l > , by generaltheory [10], so is {k S kλ ′ ,q e v k } k ∈ N .(ii) Since the sequence { S kλ,q e v } v ∈ V is orthogonal, it is sufficient to check that q X k =0 ( − k (cid:18) qk (cid:19) k S kλ,q e v k = 0 for every v ∈ V. IRICHLET SPACES ASSOCIATED WITH DIRECTED TREES 6
However, by Lemma 2.6, for v ∈ V and k ∈ N , k S kλ,q e v k = ( n v + q ) k ( n v + 1) k = ( q = 1 , ( n v + k +1) ··· ( n v + k + q − n v +1) ··· ( n v + q − if q > . In any case, the sequence {k S kλ,q e v k } k ∈ N is a polynomial in k of degree q − . By[16, Proof of Lemma 2.5], S λ,q is a q -isometry, but not a ( q − T is of finite branching index. By Remark 1.2, S λ ′ ,q is finitelycyclic. By part (ii) above, S λ ′ ,q is subnormal. Hence, by Berger-Shaw Theorem[18], S λ ′ ,q has trace-class self-commutator. Since A := S λ,q is right Fredholm withright essential inverse ( A ∗ A ) − A ∗ , the desired conclusion may now be derived fromthe following identity: [ A ∗ , A ] A = − A ∗ A ([ A ′∗ , A ′ ] A ) A ∗ A, where A ′ := A ( A ∗ A ) − .(iv) This is immediate from part (ii), [17, Lemma 3.3] and [26, Theorem 1]. (cid:3) Remark . Assume that T is of finite branching index. Since essential spectralpicture of a finitely multicyclic, completely non-unitary q -isometry T coincides withthe unilateral shift of multiplicity dim ker T ∗ [2], it may be concluded from the BDFTheorem [12] that S λ,q is unitarily equivalent to a compact perturbation of theunilateral shift of multiplicity dim ker S ∗ λ,q . The shift operators S λ,q ( q >
1) provide new examples of finitely multicyclic q -isometries in the following sense (cf. [22, Remark 4.5]). Proposition 2.9.
Let S λ,q be a Dirichlet shift on a directed tree T = ( V, E ) offinite branching index and let S w,q be the classical Dirichlet shift (see Remark 2.2).Then S λ,q is unitarily equivalent to any finite orthogonal sum of S w,q if and only ifeither q = 1 or T is isomorphic to N . Proof.
Note that l = dim ker S ∗ λ,q is finite by [17, Proposition 2.1]. If T is iso-morphic to N then clearly S λ,q is unitarily equivalent to S w,q . Further, if q = 1then by the von Neumann-Wold decomposition for isometries [18], S λ,q is unitarilyequivalent to orthogonal sum of dim ker S ∗ λ,q copies of S w,q . This gives the suffi-ciency part. To see the necessity part, suppose that S λ,q is unitarily equivalent toorthogonal sum S ( l ) w,q of l copies of S w,q . Note that the Cauchy dual S λ ′ ,q of S λ,q is unitarily equivalent to ( S ′ w,q ) ( l ) . By [16, Example 2.7], the defect operator D S ′ w,q is an orthogonal projection onto ker S ′∗ w,q , where, for a bounded linear operator T , D T is given by D T := q X k =0 ( − k (cid:18) qk (cid:19) T k T ∗ k The essential part of the proof shows that the defect operator D S λ ′ ,q is never anorthogonal projection unless q = 1 or T is isomorphic to N . We may assume that T is not isomorphic to N . Thus V ≺ is nonempty. Further,since T is locally finite with finite branching index, we can choose v ∈ V ≺ such that Chi ( v ) = { u , u , · · · , u m } and card( Chi ( u j )) = 1 ( j = 1 , · · · , m ) for some positiveinteger m >
2. Let
Chi ( u j ) = { w j } ( j = 1 , · · · , m ) and f v = m X j =1 f ( w j ) e w j be suchthat m X j =1 f ( w j ) = 0 , and f ( w j ) = 0 for some j = 1 , · · · , m. IRICHLET SPACES ASSOCIATED WITH DIRECTED TREES 7
Note that S ∗ λ ′ ,q ( f v ) = 0 and S ∗ kλ ′ ,q ( f v ) = 0 for all integers k >
1. It then followsthat q X k =0 ( − k (cid:18) qk (cid:19) S kλ ′ ,q S ∗ kλ ′ ,q ( f v ) = f v − q m X j =1 f ( w j ) λ ′ w j e w j (2.2) = f v − q n X j =1 f ( w j ) n w j n w j + q − e w j = (cid:18) − q (cid:0) n v + 2 n v + q + 1 (cid:1)(cid:19) f v . It is easy to see from (1.1) that f v is orthogonal to ker( S ∗ λ ′ ,q ). Thus if D S λ ′ ,q is theorthogonal projection onto ker( S ∗ λ ′ ,q ) then we must have (cid:18) − q (cid:0) n v + 2 n v + q + 1 (cid:1)(cid:19) f v = 0 , that is, (1 − q )( n v + 1) = 0 . This is possible only if q = 1 . (cid:3) Dirichlet Spaces Associated with Directed Trees
The following result enables us to associate a Dirichlet space with every leafless,locally finite rooted directed tree. It is worth mentioning that certain Hardy-typespaces are associated with some infinite acyclic, undirected, connected graphs in [4](refer also to [7, Section 4.3] for a version of Dirichlet space on the Bergman tree).
Proposition 3.1.
Let S λ,q be a Dirichlet shift on T = ( V, E ) and let E := ker S ∗ λ,q .Then there exist a z -invariant reproducing kernel Hilbert space H q of E -valuedholomorphic functions defined on the disc D and a unitary mapping U : l ( V ) −→ H q such that M z,q U = U S λ,q , where M z,q denotes the operator of multiplication by z on H q . Further, U maps E onto the subspace E of E -valued constant functionsin H q such that U g = g for every g ∈ E. Furthermore, we have the following: (i) the reproducing kernel κ H q : D × D → B ( E ) associated with H q satisfies κ H q ( · , w ) g ∈ H q and h U f , κ H q ( · , w ) g i H q = h ( U f )( w ) , g i E for every f ∈ l ( V ) and g ∈ E, (ii) κ H q is given by κ H q ( z, w ) = ∞ X n =0 (1) n ( q ) n z n w n P h e root i + X v ∈ V ≺ ∞ X n =0 ( n v + 2) n ( n v + q + 1) n z n w n P l ( Chi ( v )) ⊖h λ v i ( z, w ∈ D ) , where P M denotes the orthogonal projection of H onto the subspace M of H , (iii) The E -valued polynomials in z are dense in H q . In fact, H q = _ { z n f : f ∈ E , n ∈ N } . (iv) If B is an orthonormal basis of E then { z n f : f ∈ B , n ∈ N } forms anorthogonal basis of H q . Remark . Note that the reproducing kernel Hilbert space H is nothing but thevector-valued Hardy space associated with the kernel I E − zw ( z, w ∈ D ) , where I E denotes the identity operator on E. In view of the decomposition (1.1) of E, thisis immediate from the result above. IRICHLET SPACES ASSOCIATED WITH DIRECTED TREES 8
Proof.
The proof is an application of [17, Theorem 2.2]. The first half and parts(i), (iii) follow from [17, Theorem 2.2]. Thus we only need to verify parts (ii) and(iv). Note first that by [17, Theorem 2.2](ii) and Lemma 2.6, κ H q is given by κ H q ( z, w ) = I E + X j,k > C j,k z j w k ( z, w ∈ D ) , (3.1)where I E denotes the identity operator on E , and C j,k are bounded linear operatorson E given by C j,k = P E S ∗ jλ ′ ,q S kλ ′ ,q | E ( j, k = 1 , , · · · )with P E being the orthogonal projection of l ( V ) onto E . To see that C j,k = 0 for j = k , we need the following identity [15, (5.6)]: X u ∈ Chi h k i ( v ) k − Y l =0 s l,u = 1 for v ∈ V and k > , (3.2)where, for a positive integer l and v ∈ V, s l,v := card( sib ( par h l i ( v ))). This identityfor a fixed v ∈ V can be verified by induction on integers k >
1. Now for v ∈ V and j, k > S kλ ′ ,q e v = s ( n v + 1) k ( n v + q ) k X u ∈ Chi h k i ( v ) k − Y l =0 √ s l,u e u ,S ∗ jλ ′ ,q e v = s ( n v + q ) − j ( n v + 1) − j j − Y l =0 √ s l,v e par h j i ( v ) if n v > j, . Let v ∈ V and j > k . Note that if par h j − k i ( v ) is empty, then S ∗ jλ ′ ,q S kλ ′ ,q e v = 0 . Otherwise S ∗ jλ ′ ,q S kλ ′ ,q e v = s ( n v + 1) k ( n v + q ) k X u ∈ Chi h k i ( v ) k − Y l =0 √ s l,u S ∗ jλ ′ ,q e u = s ( n v + 1) k ( n v + q ) k X u ∈ Chi h k i ( v ) (cid:16) k − Y l =0 √ s l,u × s ( n u + q ) − j ( n u + 1) − j j − Y m =0 √ s m,u (cid:17) e par h j i ( u ) = s ( n v + 1) k ( n v + q ) k s ( n v + k + q ) − j ( n v + k + 1) − j j − k − Y l =0 √ s l,v e par h j − k i ( v ) , where the last equality follows from (3.2), n u = n v + k and s l,v = s l + k,u . This alsoshows the following:(a) For k ∈ N , S ∗ kλ ′ ,q S kλ ′ ,q e v = ( n v + 1) k ( n v + q ) k e v . (3.3)Here we used the convention that product over the empty set equals 1 . (b) For non-negative integers j > k , S ∗ jλ ′ ,q S kλ ′ ,q e v = β ( j, k, v ) e par h j − k i ( v ) (3.4)for some positive scalar β ( j, k, v ) such that β ( j, k, ω ) = β ( j, k, v ) for all ω ∈ sib ( v ) . (3.5) IRICHLET SPACES ASSOCIATED WITH DIRECTED TREES 9
Let f ∈ E . Note that by (1.1), f takes the form f = f root + P w ∈ V ≺ f w , where f root = γe root for some γ ∈ C and f w = P v ∈ Chi ( w ) f ( v ) e v such that X v ∈ Chi ( w ) f ( v ) λ v = 0 for w ∈ V ≺ . In view of (2.1), λ v is constant on Chi ( w ), and hence we obtain that X v ∈ Chi ( w ) f ( v ) = 0 for all w ∈ V ≺ . (3.6)It follows that for w ∈ V ≺ and j > k , S ∗ jλ ′ ,q S kλ ′ ,q f w = X v ∈ Chi ( w ) f ( v ) S ∗ jλ ′ ,q S kλ ′ ,q e v = 0 , (3.7)where we used (3.4), (3.5) and (3.6). It may now be concluded from (3.1), (1.1),(3.3) that the reproducing kernel κ H q takes the required form. Note that theconclusion in (3.7) also holds for S λ,q (by the same reasoning), and hence thesequence { z n E : n ∈ N } of subspaces of H q is mutually orthogonal. This combinedwith part (iii) yields (iv). (cid:3) Remark . Note that κ H takes the form κ H ( z, w ) = − zw log(1 − zw ) P h e root i + X v ∈ V ≺ ∞ X n =0 n v + 2 n v + 2 + n z n w n P l ( Chi ( v )) ⊖h λ v i = − zw log(1 − zw ) P h e root i − X v ∈ V ≺ n v + 2 z n v +1 w n v +1 zw log(1 − zw ) + n v X k =0 z k w k k + 1 ! P l ( Chi ( v )) ⊖h λ v i for z, w ∈ D \ { } . A particular case in which T has only branching point at root , κ H simplifies to κ H ( z, w ) = − zw log(1 − zw ) P h e root i − zw (cid:18) zw log(1 − zw ) + 1 (cid:19) P l ( Chi ( root )) ⊖h λ root i for z, w ∈ D \ { } . Note that in case T is a rooted directed tree without any branching vertex (thatis, T is isomorphic to N ), H q is nothing but the classical Dirichlet space D q ( q > ∞ X n =0 (1) n ( q ) n z n w n ( z, w ∈ D ) . (refer to [20] for the basic theory of classical Dirichlet spaces; the reader is alsoreferred to [7] for an interesting exposition on some recent developments related toDirichlet spaces). This motivates the following definition. Definition 3.4.
Let T = ( V, E ) be a leafless, locally finite rooted directed tree.We refer to the space H q , as constructed in Proposition 3.1, as the Dirichlet spaceassociated with T . IRICHLET SPACES ASSOCIATED WITH DIRECTED TREES 10
Corollary 3.5.
Let H q be a Dirichlet space associated with T = ( V, E ) and let M z,q be the operator of multiplication by z on H q . Then, for any f ( z ) = P ∞ n =0 f n z n in H q , we have k f k H q = ∞ X n =0 (cid:16) | a n | ( q ) n (1) n + X v ∈ V ≺ k b n,v k l ( V ) ( n v + q + 1) n ( n v + 2) n (cid:17) , where f n = a n e root + P v ∈ V ≺ b n,v ∈ ker M ∗ z,q is an orthogonal decomposition with a n ∈ C and b n,v ∈ l ( Chi ( v )) ⊖ h λ v i for every n ∈ N . Thus H q is contained in H . Proof.
To see the first part, in view of Proposition 3.1(iv), it suffices to check thatfor f n = a n e root + P v ∈ V ≺ b n,v ∈ ker M ∗ z,q , k z n f n k = | a n | ( q ) n (1) n + X v ∈ V ≺ k b n,v k l ( V ) ( n v + q + 1) n ( n v + 2) n . (3.8)However, by Proposition 3.1, k z n f n k = k S nλ,q f n k = | a n | k S nλ,q e root k + X v ∈ V ≺ k S nλ,q b n,v k , and hence (3.8) is immediate from Lemma 2.6. The remaining part follows fromthe inequality P ∞ n =0 (cid:16) | a n | + P v ∈ V ≺ k b n,v k l ( V ) (cid:17) k f k H q for every f ∈ H q . (cid:3) Let κ : D × D → B ( E ) be a positive definite kernel such that κ ( z, w ) is invertiblefor every z, w ∈ D and let H be the reproducing kernel Hilbert space associatedwith κ. Following [1], we say that H has complete Pick property if there exists apositive definite function F : D × D → B ( E ) such that I E − κ ( z, w ) − = F ( z, w ) ( z, w ∈ D ) . Corollary 3.6.
Let H q be a Dirichlet space associated with T = ( V, E ) . Then H q has complete Pick property.Proof. By Proposition 3.1(ii), κ H q ( z, w ) is orthogonal direct sum of finitely manypositive definite kernels of the form κ k,l ( z, w ) = ∞ X n =0 ( k ) n ( l ) n z n w n P k,l , where P k,l is a non-zero orthogonal projection and k, l are positive integers suchthat l > k . Thus it suffices to check that 1 − κ k,l ( z,w ) is a positive definite kernel.In view of [1, Theorem 7.33 and Lemma 7.38], it is enough to verify that (cid:16) ( k ) n ( l ) n (cid:17) ( k ) n − ( l ) n − ( k ) n +1 ( l ) n +1 for every n ∈ N . However, this is equivalent to ( l + n )( k + n − ( l + n − k + n ) ( n ∈ N ) , which is true whenever l > k . (cid:3) The vector-valued Dirichlet space H . It turns out that H can be iden-tified with a vector-valued Dirichlet space. Let us first reproduce from [25] thedefinition of vector-valued Dirichlet spaces.Let E be a Hilbert space and µ be a positive B ( E )-valued measure on the unitcircle T . The
Poisson integral P [ µ ] : D → B ( E ) of µ is defined by P [ µ ]( z ) := Z T − | z | | e iθ − z | dµ ( e iθ ) ( z ∈ D ) . IRICHLET SPACES ASSOCIATED WITH DIRECTED TREES 11
For an analytic function f : D → E of the form f ( z ) = P ∞ n =0 f n z n with { f n } n ∈ N ⊆ E , set k f k µ := ∞ X n =0 k f n k E + Z D h P [ µ ]( z ) f ′ ( z ) , f ′ ( z ) i E dA ( z ) , (3.9)where dA denotes normalized area measure on D . The E -valued Dirichlet space isdefined as D ( µ ) := { f : D → E | f is an analytic function such that k f k µ < ∞} . Proposition 3.7.
Let H be a Dirichlet space associated with T = ( V, E ) and let M z, be the operator of multiplication by z on H . Define a positive B ( E ) -valuedmeasure µ T given by dµ T ( e iθ ) (cid:16) ae root + X v ∈ V ≺ b v (cid:17) := (cid:16) ae root + X v ∈ V ≺ b v n v + 2 (cid:17) dσ ( e iθ ) , (3.10) where E := ker M ∗ z, , a ∈ C , b v ∈ l ( Chi ( v )) ⊖ h λ v i , and dσ denotes the normalizedarclength measure on unit circle T . Then H equals the E -valued Dirichlet space D ( µ T ) with equality of norms.Proof. We claim that if f ∈ H then k f k H = k f k µ T . Note that by Corollary 3.5, k f k H = ∞ X n =0 (cid:16) ( n + 1) | a n | + X v ∈ V ≺ (cid:16) nn v + 2 (cid:17) k b n,v k E (cid:17) = k f k H + ∞ X n =0 n (cid:16) | a n | + X v ∈ V ≺ k b n,v k E n v + 2 (cid:17) . (3.11)In view of Proposition 3.1, it suffices to verify k f n k H = k f n k µ T , where f n ( z ) = ae root + X v ∈ V ≺ b v z n ( n > . However, in the light of (3.9) and (3.11), it is enough to verify that n (cid:16) | a | + X v ∈ V ≺ k b v k l ( V ) n v + 2 (cid:17) = Z D h P [ µ T ]( z ) f ′ n ( z ) , f ′ n ( z ) i E dA ( z ) , where f ′ n ( z ) = (cid:16) ae root + P v ∈ V ≺ b v (cid:17) nz n − for integers n > . Note first that for z ∈ D , P [ µ T ]( z ) (cid:16) ae root + X v ∈ V ≺ b v (cid:17) = P [ σ ]( z ) (cid:16) ae root + X v ∈ V ≺ b v n v + 2 (cid:17) = ae root + X v ∈ V ≺ b v n v + 2 . (3.12) IRICHLET SPACES ASSOCIATED WITH DIRECTED TREES 12
Since k g k H = k g k l ( V ) for every g ∈ E (Proposition 3.1), it now follows that Z D h P [ µ T ]( z ) f ′ n ( z ) , f ′ n ( z ) i E dA ( z )= Z D n | z n − | D ae root + X v ∈ V ≺ b v n v + 2 , ae root + X v ∈ V ≺ b v E E dA ( z )= (cid:16) | a | + X v ∈ V ≺ k b v k H n v + 2 (cid:17) Z D n | z n − | dA ( z )= n (cid:16) | a | + X v ∈ V ≺ k b v k l ( V ) n v + 2 (cid:17) . Thus the claim stands verified. The claim implies in particular that H ⊆ D ( µ T ).Since E -valued analytic polynomials are dense in D ( µ T ) ([25, Corollary 3.1]), wemust have H = D ( µ T ). (cid:3) As in the classical case [20, Corollary 1.4.3], any Dirichlet space H associatedwith T admits the conformal invariance property. Corollary 3.8.
Let H be a Dirichlet space associated with T = ( V, E ) and let φ be an automorphism of the unit disc. Then, for every f ∈ H , f ◦ φ ∈ H and Z D h P [ µ T ]( z )( f ◦ φ ) ′ ( z ) , ( f ◦ φ ) ′ ( z ) i E dA ( z ) = Z D h P [ µ T ]( z ) f ′ ( z ) , f ′ ( z ) i E dA ( z ) , where µ T is as given in (3.10) .Proof. Let f ∈ H . Note that by change of variables, Z D h P [ µ T ]( z )( f ◦ φ ) ′ ( z ) , ( f ◦ φ ) ′ ( z ) i E dA ( z )= Z D h P [ µ T ]( z ) f ′ ( φ ( z )) , f ′ ( φ ( z )) i E | φ ′ ( z ) | dA ( z ) (3.12) = Z D h P [ µ T ]( w ) f ′ ( w ) , f ′ ( w ) i E dA ( w ) , which completes the proof of the second part. The first part now follows fromLittlewood’s Theorem [27, Chapter 1]. (cid:3) We present below a proof of the special case q = 2 of Theorem 2.4, which exploitsthe theory of vector-valued Dirichlet spaces [25]. Proof of Theorem 2.4 (Case q = 2 ). For j = 1 ,
2, let H ( j )2 be the Dirichlet spaceassociated with T j = ( V j , E j ) and let M ( j ) z, be the operator of multiplication by z on H ( j )2 . In view of Proposition 3.1, it suffices to check that M (1) z, is unitarilyequivalent to M (2) z, if and only if (2.3) holds for every n ∈ N . By the precedingproposition and [25, Theorem 4.2], the multiplication operator M (1) z, on D ( µ T ) isunitarily equivalent to the multiplication operator M (2) z, on D ( µ T ) if and only ifthere exists a unitary map U : E → E such that µ T ( A ) = U ∗ µ T ( A ) U for everyBorel subset A of the unit circle T (see (3.10)). However, by (3.10), this happens ifand only if the diagonal matrices ⊕ v ∈ V (1) ≺ α v I β v, and ⊕ v ∈ V (2) ≺ α v I β v, are unitarilyequivalent, where I m denotes the m × m identity matrix and α v := 1 n v + 2 , β v,j := card( Chi ( v )) − j = 1 , . IRICHLET SPACES ASSOCIATED WITH DIRECTED TREES 13
The later one holds if and only if ⊕ v ∈ V (1) ≺ α v I β v, and ⊕ v ∈ V (2) ≺ α v I β v, have the sameeigenvalues counted with multiplicity. However, P v ∈ V ( j ) ≺ ∩G (1) n (cid:16) card( Chi ( v )) − (cid:17) isthe multiplicity of the eigenvalue n +2 for j = 1 , . The desired equivalence is nowimmediate. (cid:3) Bergman Spaces Associated with Directed Trees
It is evident from the proof of Proposition 3.1 that one can also associate afunctional Hilbert space, say, H – q with the Cauchy dual S λ ′ ,q of the Dirichlet shift S λ,q . In particular, H – q is a reproducing kernel Hilbert space associated with thekernel κ H – q given by κ H – q ( z, w ) = ∞ X n =0 ( q ) n (1) n z n w n P h e root i + X v ∈ V ≺ ∞ X n =0 ( n v + q + 1) n ( n v + 2) n z n w n P l ( Chi ( v )) ⊖h λ v i ( z, w ∈ D ) . This has been recorded in [15, Proposition 5.1.8]. It is equally clear that S λ ′ ,q isunitarily equivalent to the operator of multiplication by z on H – q . Definition 4.1.
Let q be an integer bigger than 1. Let T = ( V, E ) be a leafless,locally finite rooted directed tree. We refer to H – q as the Bergman space associatedwith T .Remark . Note that in case T is a rooted directed tree without any branchingvertex, H – q is nothing but the classical Bergman space B q ( q > ∞ X n =0 ( q ) n (1) n z n w n ( z, w ∈ D )(refer to [21] for the basic theory of classical Bergman spaces). Lemma 4.3.
Let H – q be a Bergman space associated with T = ( V, E ) and let M z, – q be the operator of multiplication by z on H – q . Then for f ( z ) = P ∞ n =0 f n z n in H – q with f n = a n e root + P v ∈ V ≺ b n,v ∈ ker M ∗ z, – q , a n ∈ C and b n,v ∈ l ( Chi ( v )) ⊖ h λ v i for every n ∈ N , we have the following: (i) k f k H – q = P ∞ n =0 (cid:16) | a n | n ( q ) n + P v ∈ V ≺ k b n,v k l ( V ) ( n v +2) n ( n v + q +1) n (cid:17) . (ii) if q > , then k f k H – q = Z D h dν T ( z ) f ( z ) , f ( z ) i = ∞ X n =0 Z D | z n | h dν T ( z ) f n , f n i , where dν T ( z ) (cid:16) ae root + X v ∈ V ≺ b v (cid:17) := (cid:16) aw ( z ) e root + X v ∈ V ≺ w n v +1 ( z ) b v (cid:17) dA ( z ) (4.1) with dA denoting the normalized area measure on unit disc D and w l ( z ) = ( l + 1) · · · ( l + q − q − X i =1 | z | i + l − Y j = i q − ( j − i ) ( z ∈ D , l ∈ N ) . In particular, H – q equals L a ( ν T ) , that is, the closure of E -valued analytic polyno-mials in L ( ν T ) , with equality of norms. IRICHLET SPACES ASSOCIATED WITH DIRECTED TREES 14
Proof.
The first part can be obtained along the lines of the proof of Corollary 3.5.The first equality in the second part may be deduced from (i) and [5, Corollary 3.8]while the second one is an immediate consequence of the fact that Z D z n z m w ( | z | ) dA ( z ) = δ mn ( m, n ∈ N )for any continuous function w : [0 , → (0 , ∞ ) . (cid:3) The Bergman spaces H – q can be realized as Hilbert modules over the polynomialring C [ z ] with module action given by( p, f ) ∈ C [ z ] × H – q p ( z ) f ∈ H – q . A family of Hilbert modules H ( η,Y ) N (vector-valued analogs of the reproducing kernelHilbert spaces associated with the kernel − zw ) q , q >
0) have been introduced in[24] in the context of classification of homogeneous operators within the class ofCowen-Douglas operators (see the discussion prior to [24, Theorem 3.1] for theexact description of the spaces H ( η,Y ) N ). Recall that a bounded linear operator T on H is homogeneous if for every automorphism φ of the open unit disc D , φ ( T )is unitarily equivalent to T . Note that the assumption that φ ( T ) is well-definedas a bounded linear operator on H is a part of the definition of the homogeneousoperator. Also, the multiplication operator M z, – q on the classical Bergman space B q provides an example of homogeneous operator [9, Theorem 5.2].We say that the Hilbert module H – q over C [ z ] is homogeneous if M z, – q is homo-geneous. The natural question arises here is that for which directed trees T , H – q is a homogeneous Hilbert module ? The second question arises is whether H – q and H ( η,Y ) N are unitarily equivalent as Hilbert modules over C [ z ] ? Both these questionscan be answered with the help of [24, Theorem 4.1] and the following general fact. Proposition 4.4.
Let H – q be a Bergman space associated with the directed tree T = ( V, E ) of finite branching index. Then the following statements are equivalent: (i) H – q is a homogeneous Hilbert module. (ii) T is isomorphic to N . Proof.
Let φ be an automorphism of the open unit disc D . Since M z, – q is of spec-tral radius 1 (Lemma 2.6) and since κ ( · , w ) g provides eigenvector for M ∗ z, – q withcorresponding eigenvalue w ∈ D , the spectrum of M z, – q equals the closed unit disc D . In particular, M φ,q := φ ( M z, – q ) makes sense.In case T is isomorphic to N , the implication (ii) = ⇒ (i) is immediate from [9,Theorem 5.2].To see that (i) implies (ii), suppose there exists a unitary U : H – q → H – q such that U M z, – q = M φ,q U. Let { g j : j = 1 , · · · , d } be an orthonormal basis for E := ker M ∗ z, – q , where g := e root and d is finite. One may conclude from (3.3) and(3.7) that for 1 i = j d, the sequences { z n g i } n ∈ N and { z n g j } n ∈ N are mutuallyorthogonal. This yields the decomposition H – q = ⊕ dj =1 H j , where H j = _ { z n g j : n ∈ N } ( j = 1 , · · · , d ) . Since H , · · · , H d are z -invariant subspaces of H – q , M z, – q = ⊕ dj =1 M ( j ) z, – q , where M ( j ) z, – q = M z, – q | H j for j = 1 , · · · , d. On the other hand, by [17, Corollary 5.6],the Hilbert space adjoint of M z, – q belongs to the Cowen-Douglas class B d ( D ) (thereader is referred to [19] for the definition of Cowen-Douglas class B d ( · )). Also,the Hilbert space adjoint of M ( j ) z, – q belongs to B ( D ) for every j = 1 , · · · , d. Sincethe classification of homogeneous operators in B d ( D ) is the same as that of the IRICHLET SPACES ASSOCIATED WITH DIRECTED TREES 15 corresponding homogeneous holomorphic Hermitian bundles defined on D [11, Sec-tion 1], we may conclude from [24, Corollary 2.1] that M ( j ) z, – q must be homogeneousfor every j = 1 , · · · , d. However, by [9, List 3.1], this is possible only if for every j = 1 , · · · , d, there exist a j > M ( j ) z, – q is unitarily equivalent to the opera-tor M ( j ) z of multiplication by z on a reproducing kernel Hilbert space H j associatedwith the kernel − zw ) aj ( z, w ∈ D ) . For j = 1 , · · · , d , let U j : H j → H j be theunitary operator such that M ( j ) z, – q U j = U j M ( j ) z . Note that U j maps ker( M ( j ) z ) ∗ onto ker( M ( j ) z, – q ) ∗ . Note that g := U j (1 / k k ) ∈ ker( M ( j ) z, – q ) ∗ . If d = 1 then clearly T is isomorphic to N . Suppose the contrary.Then g = e root for some j = 1 , · · · , d. Thus g = P w ∈ Chi ( v ) α w e w ∈ l ( Chi ( v )) ⊖ h λ v i for some v ∈ V ≺ . Hence, for any integer k > , by Lemma 2.6, k ( M ( j ) z, – q ) k g k = X w ∈ Chi ( v ) | α w | ( n w + 1) k ( n w + q ) k = ( n v + 2) k ( n v + q + 1) k . On the other hand, k ( M ( j ) z, – q ) k g k = k ( M ( j ) z ) k (1 / k k ) k = (1) k ( a j ) k , After combining last two equations for k = 1 ,
2, we obtain n v + 1 = 0, which isabsurd. Hence d = 1, and we obtain (ii). (cid:3) The proof of the preceding proposition actually yields the following general fact.
Corollary 4.5.
Let S λ be a left-invertible, homogeneous weighted shift on T withfinite dimensional cokernel E := ker S ∗ λ . If { g j : j = 1 , · · · , d } is an orthonormalbasis of E such that for i = j d, the sequences { S nλ g i } n ∈ N and { S nλ g j } n ∈ N aremutually orthogonal, then there exists a > such that S λ is unitarily equivalent tothe operator of multiplication by z on a reproducing kernel Hilbert space associatedwith the kernel − zw ) a ( z, w ∈ D ) . We do not know whether a rooted directed tree, which is non-isomorphic to N ,supports a homogeneous weighted shift.5. Classification of Dirichlet Shifts
For the sake of convenience, we reproduce the statement of Theorem 2.4 fromSection 2.
Theorem 5.1.
Let q be an integer bigger than . For j = 1 , , let S ( j ) λ,q be theDirichlet shift on rooted directed tree T j = ( V j , E j ) with root j , let G ( j ) n := { v ∈ V j : v ∈ Chi h n i ( root j ) } ( n ∈ N ) , and E j = ker( S ( j ) λ,q ) ∗ . Then S (1) λ,q is unitarily equivalentto S (2) λ,q if and only if for every n ∈ N , X v ∈ V (1) ≺ ∩G (1) n (cid:16) card ( Chi ( v )) − (cid:17) = X v ∈ V (1) ≺ ∩G (2) n (cid:16) card ( Chi ( v )) − (cid:17) . (5.1) Proof.
Note that S (1) λ,q is unitarily equivalent to S (2) λ,q if and only if S (1) λ ′ ,q is unitarilyequivalent to S (2) λ ′ ,q . Hence, in view of Lemma 4.3, it suffices to check that the mul-tiplication operator M (1) z,q on L a ( ν T ) is unitarily equivalent to the multiplicationoperator M (2) z,q on L a ( ν T ) if and only if (5.1) holds for every n ∈ N . IRICHLET SPACES ASSOCIATED WITH DIRECTED TREES 16
Let U : L a ( ν T ) → L a ( ν T ) be a unitary map such that U M (1) z,q = M (2) z,q U. (5.2)Note that U maps ker( M (1) z,q ) ∗ unitarily onto ker( M (2) z,q ) ∗ . For g, h ∈ ker( M (1) z,q ) ∗ , define a signed finite measure µ g,h on the open unit disc D by µ g,h ( σ ) = Z σ (cid:10)(cid:0) U ∗ dν T ( z ) U − dν T ( z ) (cid:1) g, h (cid:11) for a Borel subset σ of D . Note that Z D z n z m (cid:10) U ∗ dν T ( z ) U g, h (cid:11) = h z n U g, z m U h i L a ( ν T ) (5.2) = h U z n g, U z m h i L a ( ν T ) = h z n g, z m h i L a ( ν T ) = Z D z n z m (cid:10) dν T ( z ) g, h (cid:11) . It follows that R D p ( z, z ) dµ g,h = 0 for any polynomial p in z and z . By Stone-Weierstrass and Riesz Representation Theorems, µ g,h is identically 0 . Since thisholds for arbitrary choices of g, h in ker( M (1) z,q ) ∗ , we conclude that U ∗ ν T ( σ ) U = ν T ( σ ) for every Borel subset σ of D . (5.3)It is now clear from (4.1) that for every Borel subset σ of D , a ∈ C , and b v ∈ l ( Chi ( v )) ⊖ h λ v i ( v ∈ V (1) ≺ ), ν T ( σ ) U (cid:16) ae root + X v ∈ V (1) ≺ b v (cid:17) = U Z σ (cid:16) aw ( z ) e root + X v ∈ V (1) ≺ w n v +1 ( z ) b v (cid:17) dA ( z ) . (5.4)Since U e root ∈ ker( M (2) z,q ) ∗ , there exist a ′ ∈ C , and b ′ v ∈ l ( Chi ( v )) ⊖ h λ v i for v ∈ V (2) ≺ such that U e root = a ′ e root + X v ∈ V (2) ≺ b ′ v . Letting a = 1 and b v = 0 for all v ∈ V (1) ≺ in (5.4), we obtain (cid:16) Z σ w ( z ) dA ( z ) (cid:17)(cid:16) a ′ e root + X v ∈ V (2) ≺ b ′ v (cid:17) = (cid:16) Z σ w ( z ) dA ( z ) (cid:17) U e root (5.4) = ν T ( σ ) U e root (4.1) = Z σ (cid:16) a ′ w ( z ) dA ( z ) (cid:17) e root + X v ∈ V (2) ≺ Z σ (cid:16) w n v +1 ( z ) dA ( z ) (cid:17) b ′ v for every Borel subset σ of D . Since for some Borel set σ, R σ w ( z ) dA ( z ) = R σ w l ( z ) dA ( z ) for any positive integer l, by comparing coefficients on both sides, weobtain b ′ v = 0 for every v ∈ V (2) ≺ , and hence U e root = a ′ e root with | a ′ | = 1 . Thisshows that U ( h e root i ) = h e root i . IRICHLET SPACES ASSOCIATED WITH DIRECTED TREES 17
Since U maps ker( M (1) z,q ) ∗ unitarily onto ker( M (2) z,q ) ∗ , we obtain U (cid:16) M v ∈ V (1) ≺ l ( Chi ( v )) ⊖ h λ v i ) (cid:17) = M v ∈ V (2) ≺ l ( Chi ( v )) ⊖ h λ v i . (5.5)Next, since U b v ∈ ker( M (2) z,q ) ∗ for a fixed v ∈ V (1) ≺ , there exist b ′ v ∈ l ( Chi ( v )) ⊖h λ v i for v ∈ V (2) ≺ such that U b v = X v ∈ V (2) ≺ b ′ v . As in the preceding paragraph, one may let a = 0 and b v = 0 for all v ∈ V (1) ≺ \ { v } in (5.4) to obtain (cid:16) Z σ w n v +1 ( z ) dA ( z ) (cid:17)(cid:16) X v ∈ V (2) ≺ b ′ v (cid:17) = X v ∈ V (2) ≺ Z σ (cid:16) w n v +1 ( z ) dA ( z ) (cid:17) b ′ v . Since for some Borel set σ, R σ w l ( z ) dA ( z ) = R σ w m ( z ) dA ( z ) for positive integers l = m, by comparing coefficients on both sides, we obtain b ′ v = 0 for every v ∈ V (2) ≺ ∩ G (2) n with n = n v , and hence U b v = P v ∈ V (2) ≺ ∩G nv b ′ v . Since v is arbitrary,by (5.5), U must map W ( n )1 injectively onto W ( n )2 for every n ∈ N , where W ( n ) j = M v ∈ V ( j ) ≺ ∩G ( j ) n l ( Chi ( v )) ⊖ h λ v i , n ∈ N , j = 1 , . Thus for every n ∈ N , X v ∈ V (1) ≺ ∩G (1) n (cid:16) card( Chi ( v )) − (cid:17) = X v ∈ V (2) ≺ ∩G (2) n (cid:16) card( Chi ( v )) − (cid:17) . Conversely, suppose (5.1) holds for every n ∈ N . For n ∈ N and j = 1 ,
2, let M j, = h e root j i , M j,n = M v ∈ V (1) ≺ ∩G ( j ) n l ( Chi ( v )) ⊖ h λ v i . By (5.1), we must have dim M ,n = dim M ,n for every n ∈ N . Let U = ⊕ n ∈ N U n ,where U n is a unitary from M ,n onto M ,n for n ∈ N . Further, we can choose U such that U e root = e root . Note that U is a unitary from ker( M (1) z,q ) ∗ ontoker( M (2) z,q ) ∗ . We verify that U satisfies (5.3). It suffices to check that U ∗ ν T ( σ ) U f = ν T ( σ ) f, (5.6)where f = αe root + βb v,n for α, β ∈ C and { b v,n } is an orthonormal basis for M ,n for integers n ≥ . Indeed, for a Borel subset σ of D ,U ∗ ν T ( σ ) U f = U ∗ ν T ( σ )( αe root + βU b v,n )= U ∗ Z σ (cid:16) αw ( z ) e root + βw n +1 ( z ) U b v,n (cid:17) dA ( z )= Z σ (cid:16) αw ( z ) e root + βw n +1 ( z ) b v,n (cid:17) dA ( z )= ν T ( σ ) f, which completes the verification of (5.6). One can now define ˜ U : L a ( ν T ) → L a ( ν T ) by ( ˜ U f )( z ) = U ( f ( z )) ( f ∈ L a ( ν T ) , z ∈ D ) . (5.7)It is easy to see using (5.3) that ˜ U is a unitary map such that ˜ U M (1) z,q = M (2) z,q ˜ U . (cid:3)
IRICHLET SPACES ASSOCIATED WITH DIRECTED TREES 18
Acknowledgment . The authors convey sincere thanks to Gadadhar Misra forsome fruitful conversations pertaining to the theory of homogeneous operators.In particular, they acknowledges his help for providing an exact reference for afact essential in the proof of Proposition 4.4. Further, the last author expresses hisgratitude to the Department of Mathematics, IIT Kanpur for their warm hospitalityduring the preparation of this paper.
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