Ando dilations and inequalities on noncommutative domains

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by  Gelu Popescu

ANDOˆ DILATIONS AND INEQUALITIES ON NONCOMMUTATIVE DOMAINS
GELUPOPESCU
7 Acobmstmruatcatti.veWvearoibettiaeisnVinJteinrtwBi(nHin)gn,diwlahtiicohngtehneeorraelmizsefSorarnaosnocno[m18m]uatnadtivSez.r-eNgauglayr–Fdooima¸sai[n2s0]Dcfomanmduntaonnt-
1
liftingtheorem for commuting contractions. We present several applications including a new proof for
0 thecommutantliftingtheoremforpureelementsinthedomainDf (resp. varietyVJ)aswellasaSchur
2 typerepresentationfortheunitballoftheHardyalgebraassociatedwiththevarietyVJ.
n WeprovideAndoˆtype dilationsandinequalitiesforbi-domainsDf ×cDf whichconsistofallpairs
a (X,Y)oftuplesX:=(X1,...,Xn1)∈Df andY:=(Y1,...,Yn2)∈Dgwhichcommute,i.e. eachentry
J of Xcommutes with each entry of Y. The results arenew even when n1 =n2 =1. In this particular
3 case, we obtain extensions of Andoˆ’s results [2] and Agler-McCarthy’s inequality [1] for commuting
contractions tolargerclassesofcommutingoperators.
] Alltheresultsareextendedtobi-varietiesVJ1×cVJ2,whereVJ1 andVJ2 arenoncommutativevari-
A eties generated byWOT-closedtwo-sidedidealsinnoncommutative Hardyalgebras. Thecommutative
F caseaswellasthematrixcasewhenn1=n2=1arealsodiscussed.
.
h
t
a
m
Introduction
[

Extending von Neumann [23] inequality for one contraction and Sz.-Nagy proof using dilation theory
1
[21], Andˆo [2] proveda dilation result that implies his celebrated inequality which says that if T and T
v 1 2
8 are commuting contractions on a Hilbert space, then for any polynomial p in two variables,
5
p(T ,T ) p ,
7 k 1 2 k≤k kD2
0 where D2 is the bidisk in C2. For a nice survey and further generalizations of these inequalities we
0
refer to Pisier’s book [9] (see also [23], [21], [7], [22], [10], [8], [4], [5], [1], and [6]). Inspired by the
.
1 work of Agler-McCarthy [1] and Das-Sarkar [6] on distinguished varieties and Ando’s inequality for two
0 commuting contractions, we found, in a very recent paper [16], analogues of Andˆo’s results for the
7
elements of the bi-ball P which consists of all pairs (X,Y) of row contractions X:= (X ,...,X )
1 n1,n2 1 n1
andY :=(Y ,...,Y )whichcommute,i.e. eachentryofXcommuteswitheachentryofY. Theresults
v: were obtaine1d in a mn2ore general setting, namely, when X and Y belong to noncommutative varieties
1
i V
X and determined by row contractions subject to constraints such as
2
V
r q(X ,...,X )=0 and r(Y ,...,Y )=0, q ,r ,
a 1 n1 1 n2 ∈P ∈R
respectively,where and aresetsofnoncommutativepolynomials. Thisledtooneofthe mainresults
P R
of the paper, an Andˆo type inequality on noncommutative varieties, which, in the particular case when
n = n = 1 and T and T are commuting contractive matrices with spectrum in the open unit disk
1 2 1 2
D:= z C: z <1 , takes the form
{ ∈ | | }
p(T ,T ) min p(B I ,ϕ (B )) , p(ϕ (B ),B I ) ,
k 1 2 k≤ {k 1⊗ Cd1 1 1 k k 2 2 2⊗ Cd2 k}
where (B I ,ϕ (B )) and (ϕ (B ),B I ) are analytic dilations of (T ,T ) while B and B are
1⊗ Cd1 1 1 2 2 2⊗ Cd2 1 2 1 2
the universal models associated with T and T , respectively. In this setting, the inequality is sharper
1 2
than Andˆo’s inequality and Agler-McCarthy’s inequality [1]. We obtained more general inequalities for
arbitrary commuting contractive matrices and improve Andˆo’s inequality for commuting contractions
Date:November14,2016.
2010 Mathematics Subject Classification. Primary: 47A13; 47A20;Secondary: 47A63;47A45; 46L07.
Key words and phrases. Andoˆ’s dilation; Andoˆ’s inequality; Commutant lifting; Fock space; Noncommutative bi-
domain;Noncommutative variety;Poissontransform;Schurrepresentation.
ResearchsupportedinpartbyNSFgrantDMS1500922.
1
2 GELUPOPESCU
when at least one of them is of class . In this setting, it would be interesting to find good analogues
0
C
for distinguished varieties in the sense of [1].
Let F+ be the unital free semigroup on n generators g ,...,g and the identity g . The length of
n 1 n 0
α F+ is defined by α := 0 if α = g and α := k if α = g g , where i ,...,i 1,...,n . If
∈ n | | 0 | | i1··· ik 1 k ∈ { }
Z:= Z ,...,Z is ann-tuple ofnoncommutativeindeterminates, we use the notationZ :=Z Z
h 1 ni α i1··· ik
and Z := 1. We denote by C Z the complex algebra of all polynomials in Z ,...,Z . A polynomial
g0 h i 1 n
f := a Z in C Z is called positive regular if the coefficients satisfy the conditions: a 0 for
any α αF∈+F+n, aα α=0, anhdia >0 if i=1,...,n. Define the noncommutative regular domain α ≥
P∈ n g0 gi
( ):= X:=(X ,...,X ) B( )n : a X X I
Df H  1 n ∈ H α α α∗ ≤ H
 |αX|≥1 
and the noncommutative ellipsoid ( ) ( ) by setting
 f f 
E H ⊇D H
( ):= X:=(X ,...,X ) B( )n : a X X I ,
Ef H  1 n ∈ H β β β∗ ≤ H
 |βX|=1 
where B( ) stands for the algebra of all bounded linear operators on a Hilbert space . Given n ,n
  1 2
H H ∈
N:= 1,2,... andΩj B( )nj,j =1,2,wedenotebyΩ1 cΩ2thesetofallpairs(X,Y) Ω1 Ω2with
{ } ⊆ H × ∈ ×
thepropertythattheentriesofX:=(X ,...,X )arecommutingwiththeentriesofY :=(Y ,...,Y ).
1 n1 1 n2
Themaingoalofthepresentpaperistoextendtheresultsfrom[16]forbi-ballsandobtainAndˆotype
dilations and inequalities for bi-domains and noncommutative varieties:
( ) ( ) and ( ) ( ),
Df H ×cDg H VJ1 H ×cVJ2 H
where f C Z and g C Z are positive regular noncommutative polynomials while and are
∈ h i ∈ h ′i VJ1 VJ2
varieties generated by WOT-closed two-sided ideals in certain noncommutative Hardy algebras.
InSection2,weobtainanintertwiningdilationtheoremforbi-domainswhichgeneralizesSarason[18]
andSz.-Nagy–Foia¸s[20]commutantliftingtheoremforcommutingcontractionsintheframeworkofnon-
commutative regulardomains and Poissonkernels on weightedFock spaces (see [15]). As a consequence,
we obtain a new proof for the commutant lifting theorem for pure elements in .
f
D
Theseresultsareextended,inSection3,tononcommutativevarieties whicharegeneratedby
J f
V ⊆D
WOT-closed two-sided ideals J in the Hardy algebra F ( ), a noncommutative multivariable version
n∞ Df
of the classical Hardy algebra H (D). More precisely, the noncommutative variety ( ) is defined as
∞ J
V H
the set of all pure n-tuples X:=(X ,...,X ) ( ) with the property that
1 n f
∈D H
ϕ(X ,...,X )=0 for any ϕ J,
1 n
∈
where ϕ(X ,...,X ) is defined using the F ( )-functional calculus for pure elements in ( ) (see
1 n n∞ Df Df H
[15]). Eachvariety isassociatedwithcertainuniversalmodelsB=(B ,...,B )andC=(C ,...,C )
J 1 n 1 n
V
ofconstrainedcreationoperatorsactingonasubspace ofthefullFockspacewithngeneratorsF2(H ).
J n
N
The noncommutative Hardy algebras F ( ) and R ( ) are the WOT-closed algebras generated by
n∞ VJ n∞ VJ
I,B ,...,B andI,C ,...,C ,respectively. Usingourintertwiningdilationtheoremonnoncommutative
1 n 1 n
varieties, we obtain a Schur [19] type representationfor the unit ball of ( )¯B( , ).
R∞n VJ ⊗ H′ H
In Section 4, we obtain Andˆo type dilations and inequalities for noncommutative varieties
( ) ( ),
VJ1 H ×cVJ2 H
where ( ) ( ) and ( ) ( ). We provethatanypair(T ,T )in ( ) ( ) has
VJ1 H ⊆Df H VJ2 H ⊆Dg H 1 2 VJ1 H ×cVJ2 H
analytic dilations
(B I ,ϕ (C )) and (ϕ (C ),B I )
1 ℓ2 1 1 2 2 2 ℓ2
⊗ ⊗
whereϕ (C )andϕ (C )aresomemulti-analyticoperatorswithrespecttotheuniversalmodelsB and
1 1 2 2 1
B of the varieties and , respectively. As a consequence, we show that the inequality
2 VJ1 VJ2
[p (T ,T )] min [p (B I ,ϕ (C ))] , [p (ϕ (C ),B I )]
k rs 1 2 kk≤ {k rs 1⊗ ℓ2 1 1 kk k rs 2 2 2⊗ ℓ2 kk}
ANDOˆ DILATIONS AND INEQUALITIES ON NONCOMMUTATIVE DOMAINS 3
holds for any [p ] M (C Z,Z ) and k N. Here, C Z,Z denotes the complex algebra of all
rs k k ′ ′
∈ h i ∈ h i
polynomials in noncommutative indeterminates Z := Z ,...,Z and Z := Z ,...,Z , where we
h 1 n1i ′ 1′ n′2
assume that Z Z =Z Z for any i 1,...,n and j 1,...,n .
i j′ j′ i ∈{ 1} ∈{ 2} (cid:10) (cid:11)
On the other hand, we prove that the abstract bi-domain
:= ( ) ( ): is a Hilbert space
f c g f c g
D × E {D H × E H H }
has a universal analytic model (W I ,ψ(Λ )), where the tuples W = (W ,...,W ) and Λ =
1⊗ ℓ2 1 1 1,1 1,n1 1
(Λ ,...,Λ ) are the weighted left and right creation operators on the full Fock space F2(H ),
1,1 1,n1 n1
respectively, associated with the regular domain . More precisely, we show that the inequality
f
D
[prs(T1,T2)]k [prs(W1 Iℓ2,ψ(Λ1))]k , [prs]k Mk(C Z,Z′ ),
k k≤k ⊗ k ∈ h i
holds for any (T ,T ) ( ) ( ) and any k N. A similar result holds for the abstract variety
1 2 f c g
∈D H × E H ∈
.
J c g
V × E
We will see, in Section 4, that all the results of the present paper concerning Andˆo type dilations and
inequalities can be written in the commutative multivariable setting in terms of analytic multipliers of
certain Hilbert spaces of holomorphic functions. These results are new even when n = n = 1. In this
1 2
particularcase,we obtainextensionsofAndˆo’s resultsforcommutingcontractions[2],Agler-McCarthy’s
inequality [1], and Das-Sarkar extension [6], to larger classes of commuting operators.
Finally, we would like to thank the referee for helpful comments and suggestions on the paper.
1. Preliminaries on noncommutative regular domains and universal models
Inthissection,werecallfrom[15]basicfactsconcerningthenoncommutativeregulardomains ( )
f
D H ⊂
B( )n generatedby positive regularformalpowerseries,their universalmodels,andthe Hardyalgebras
H
they generate. We mention that commutative domains generated by positive regular polynomials were
first introduced in [17] and further elaborated in [3] and in a series of papers by the author (see [15] and
the references there in).
Let H be an n-dimensional complex Hilbert space with orthonormal basis e , e , ...,e , where
n 1 2 n
n 1,2,... . We consider the full Fock space of H defined by
n
∈{ }
F2(Hn):= Hn⊗k,
k 0
M≥
where H 0 := C1 and H k is the (Hilbert) tensor product of k copies of H . Define the left creation
n⊗ n⊗ n
operators S :F2(H ) F2(H ), i=1,...,n, by
i n n
→
S ϕ:=e ϕ, ϕ F2(H ),
i i n
⊗ ∈
and the right creation operators R : F2(H ) F2(H ) by R ϕ := ϕ e , ϕ F2(H ). The noncom-
i n n i i n
→ ⊗ ∈
mutativeanalyticToeplitzalgebraF anditsnormclosedversion,thenoncommutativediscalgebra ,
n∞ An
were introduced by the author (see [10], [11], [12]) in connection with a multivariable noncommutative
von Neumann inequality. F is the algebra of left multipliers of F2(H ) and can be identified with
n∞ n
the weakly closed (or w -closed) algebra generated by the left creation operators S ,...,S acting on
∗ 1 n
F2(H ), and the identity. The noncommutative disc algebra is the normclosedalgebrageneratedby
n n
A
S ,...,S , and the identity.
1 n
A formal power series f := a Z in noncommutative indeterminates Z ,...,Z is called posi-
tiveregularifthecoefficientssatisαfy∈Ft+nhecαonαditions: a 0foranyα F+, a =01,a >0nifi=1,...,n,
P α ≥ ∈ n g0 gi
1/2k
and limsup a 2 < . If X := (X ,...,X ) B( )n, we set X := X X if
k→∞ |α|=k| α| ∞ 1 n ∈ H α i1··· ik
α=gi1···gik, wh(cid:16)ePre i1,...,ik(cid:17)∈{1,...,n}, and Xg0 :=IH. Define the noncommutative regular domain
( ):= X:=(X ,...,X ) B( )n : a X X I ,
Df H  1 n ∈ H α α α∗ ≤ H
 |αX|≥1 
 
4 GELUPOPESCU
where the convergenceof the series is in the weak operatortopology. The power series 1 f is invertible
−
with its inverse g = b X , b C, satisfies the relation
α∈F+n α α α ∈
P
α
∞ | |
g =1+f +f2+ =1+ a a X .
···  γ1··· γj α
mX=1|αX|=mXj=1 |γ1|γ≥11·X·,·.γ..j,|=γαj|≥1 
 
Consequently, we have
α
| |
(1.1) b =1 and b = a a if α 1.
g0 α γ1··· γj | |≥
Xj=1 |γ1|γ≥11·X·,·.γ..j,|=γαj|≥1
Theweightedleft creationoperatorsW :F2(H ) F2(H ),i=1,...,n,associatedwiththenoncom-
i n n
→
mutative domain are defined by setting W :=S D , where S ,...,S are the left creation operators
f i i i 1 n
D
on the full Fock space F2(H ) and each diagonal operator D :F2(H ) F2(H ), is given by
n i n n
→
b
D e := α e , α F+.
i α sbgiα α ∈ n
Note that
Wβeγ = pbbβγγeβγ and Wβ∗eα =0√√bbαγeγ iofthαe=rwβisγe

for any α,β F+. We provepin [15] that W := (W ,...,W ) is a pure n-tuple in (F2(H )),
∈ n 1  n Df n
i.e. Φf,W(I) ≤ I and Φkf,W(I) → 0 in the strong operator topology, as k → ∞, where Φf,W(Y) :=
a W YW for Y B(F2(H )) and the convergence is in the weak operator topology. The n-tuple
α α α∗ ∈ n
α 1
W|P|≥plays the role of universal model for the noncommutative domain .
f
D
We also define the weighted right creation operators Λ : F2(H ) F2(H ) by setting Λ := R G ,
i n n i i i
→
i = 1,...,n, where R ,...,R are the right creation operators on the full Fock space F2(H ) and each
1 n n
diagonal operator G is defined by
i
b
G e := α e , α F+,
i α sbαgi α ∈ n
where the coefficients b , α F+, are given by relation (1.1). In this case, we have
α ∈ n
Λβeγ = bγ eγβ˜ and Λ∗βeα =√√bbαγeγ if α=γβ˜
pbγβ˜ 0 otherwise
q
for any α,β F+, where β˜ denotes the reverse of β =g g , i.e. β˜ := g g . We remark
∈ n i1··· ik ik··· i1
that if f := a Z is a positive regular power series, then so is f˜ := a Z . Moreover,
α 1 α α α 1 α˜ α
Λ := (Λ ,...,Λ| |)≥ (F2(H )) and W = U Λ U, where U B(F2(H )) is| t|≥he unitary operator
1 P n ∈ Df˜ n i ∗ i ∈ n P
defined by Ue :=e , α F+. Throughout this paper, we will refer to the n-tuples W:=(W ,...,W )
α α˜ ∈ n 1 n
and Λ:=(Λ ,...,Λ ) as the weighted creation operators associated with the regular domain .
1 n f
D
In[15],we introducedthe domainalgebra ( )associatedwiththe noncommutativedomain to
n f f
A D D
bethenormclosureofallpolynomialsintheweightedleftcreationoperatorsW ,...,W andtheidentity.
1 n
Usingtheweightedrightcreationoperatorsassociatedwith ,onecandefinethecorrespondingdomain
f
D
algebra ( ). The Hardy algebra F ( ) (resp. R ( )) is the w - (or WOT-, SOT-) closure of
Rn Df n∞ Df n∞ Df ∗
all polynomials in W ,...,W (resp. Λ ,...,Λ ) and the identity. We proved that F ( ) =R ( )
1 n 1 n n∞ Df ′ n∞ Df
and R ( ) =F ( ), where stands for the commutant.
n∞ Df ′ n∞ Df ′
Now, we recall([13], [15]) some basic facts concerning the noncommutative Poissonkernelsassociated
with the regular domains. Let T := (T ,...,T ) be an n-tuple of operators in the noncommutative
1 n
ANDOˆ DILATIONS AND INEQUALITIES ON NONCOMMUTATIVE DOMAINS 5
domain ( ), i.e. a T T I . Define the positive linear mapping
Df H α α α∗ ≤ H
α 1
| |≥
P
Φf,T :B(H)→B(H) by Φf,T(X)= aαTαXTα∗,
|αX|≥1
where the convergence is in the weak operator topology. We use the notation Φm for the composition
f,T
Φf,T Φf,T of Φf,T by itself m times. Since Φf,T(I) I and Φf,T() is a positive linear map,
◦···◦ ≤ ·
it is easy to see that Φm (I) is a decreasing sequence of positive operators and, consequently,
{ f,T }∞m=1
Qf,T :=SOT-mlim Φmf,T(I) exists. We say that T is a pure n-tuple in Df(H) if SOT-mlim Φmf,T(I)=0.
Note that, for an→y∞T:=(T ,...,T ) ( ) and 0 r <1, the n-tuple rT:=(rT ,..→.,∞rT ) ( )
1 n f 1 n f
∈D H ≤ ∈D H
is pure. Indeed, it is enough to see that Φm (I) rmΦm (I) rmI for any m N. Note also
f,rT ≤ f,T ≤ ∈
that if kΦf,T(I)k < 1, then T is pure. This is due to the fact that kΦmf,T(I)k ≤ kΦf,T(I)km. We
define the noncommutative Poisson kernel associated with the n-tuple T ( ) to be the operator
f
∈ D H
Kf,T : F2(Hn) T defined by
H→ ⊗D
Kf,Th= bαeα⊗∆f,TTα∗h, h∈H,
αX∈F+np
where ∆f,T := (I Φf,T(I))1/2 is the defect operator associated with T and T := ∆f,T( ) is the
− D H
correspondingdefectspace. TheoperatorKf,T isacontractionsatisfyingrelationKf∗,TKf,T =IH−Qf,T
and
(1.2) Kf,TTi∗ =(Wi∗⊗IDT)Kf,T, i=1,...,n,
where W := (W ,...,W ) is the universal model associated with the noncommutative regular domain
1 n
f. Moreover,Kf,T is an isometry if and only if T is pure element of f( ).
D D H
2. Intertwining dilation theorem on noncommutative bi-domains
In this section, we obtain an intertwining dilation theorem which generalizes Sarason and Sz.-Nagy–
Foia¸scommutantliftingtheoremforcommutingcontractionsintheframeworkofnoncommutativeregular
domains and Poisson kernels on weighted Fock spaces. As a consequence, we obtain a new proof for the
commutant lifting theorem for pure elements in . More applications of this result will be considered
f
D
in the next sections.
Unless otherwise specified, we assume, throughout this paper, that f and g are two positive regular
polynomialsinnoncommutativeindeterminatesZ:= Z ,...,Z andZ := Z ,...,Z ,respectively,
h 1 n1i ′ 1′ n′2
of the form
(cid:10) (cid:11)
f := aαZα and g := cβZβ′.
α∈F+n1X,1≤|α|≤k1 β∈F+n2X,1≤|β|≤k2
Fix two tuples of operators T =(T ,...,T ) ( ) and T =(T ,...,T ) ( ) and let
1 1,1 1,n1 ∈Df H ′1 1′,1 1′,n1 ∈Df H′
T := (T ,...,T ), with T : , be such that T ( , ) and intertwines T with T ,
2 2,1 2,n2 2,j H′ → H 2 ∈ Dg H′ H 1 ′1
i.e.
T T =T T
2,j 1′,i 1,i 2,j
for any i 1,...,n andj 1,...,n . We denote by (T ,T ) the set of allintertwining tuples T
∈{ 1} ∈{ 2} I 1 ′1 2
of T and T . A straightforwardcalculation reveals that
1 ′1
∆2f,T1 +Φf,T1(∆2g,T2)=Φg,T2(∆2f,T′1)+∆2g,T2.
If the defect spaces DT1, DT′1, and DT2 are finite dimensional with dimensions d1 := dimDT1, d′1 :=
dimDT′1, and d2 :=dimDT2, and such that d1+m1d2 =m2d′1+d2, where
m :=card α F+ : 1 α k , j =1,2,
i { ∈ nj ≤| |≤ j}
6 GELUPOPESCU
then there are unitary extensions U :DT1 ⊕ α∈F+n1,1≤|α|≤k1DT2 → β∈F+n2,1≤|β|≤k2DT′1 ⊕DT2 of the
isometry
L L
(2.1) U∆T1h⊕ √aα∆T2T1∗,αh:= √cβ∆T′1T2∗,βh, h∈H.
α∈F+n1M,1≤|α|≤k1 β∈F+n2M,1≤|β|≤k2
 
We denote by T the set of all unitary extensions of the isometry given by relation (2.1). In case the
U
above-mentioned dimensional conditions are not satisfied, then let be an infinite dimensional Hilbert
K
space and note that the operator defined by
(2.2) U∆T1h⊕ [√aα∆T2T1∗,αh⊕0]:= √cβ∆T′1T2∗,βh⊕0
α∈F+n1M,1≤|α|≤k1 β∈F+n2M,1≤|β|≤k2
   
is an isometry which can be extended to a unitary operator
U : T ( T ) T′ ( T ).
D 1 ⊕ D 2 ⊕K → D 1 ⊕ D 2 ⊕K
α∈F+n1M,1≤|α|≤k1 β∈F+n2M,1≤|β|≤k2
In this setting, we denote by the set of all unitary extensions of the isometry defined by (2.2). Let
TK
U
A B
U = be the operator matrix representation of U , where
C D ∈UTK
(cid:20) (cid:21)
A:DT1 → DT′1,
β∈F+n2M,1≤|β|≤k2
B : ( T ) T′,
D 2 ⊕K → D 1
(2.3) α∈F+n1M,1≤|α|≤k1 β∈F+n2M,1≤|β|≤k2
C : T T ,and
D 1 →D 2 ⊕K
D : ( T ) T .
D 2 ⊕K →D 2 ⊕K
α∈F+n1M,1≤|α|≤k1
Given an operator Z : and n N, we introduce the ampliation
N →M ∈
Z 0
··· n n
diagn(Z):=... ... ...: N → M.
0 Z Ms=1 Ms=1
 ··· 
 
In what follows, we consider the lexicographic order for the free semigroup F+ , that is
n1
g <g < <g <g g < <g g < <g g < <g g <
0 1 ··· n1 1 1 ··· 1 n1 ··· n1 1 ··· n1 n1 ···
andsoon. ForthedirectproductF+ F+ ofpcopiesofF+ ,wesaythat(α ,...,α )<(β ,...,β )
n1×···× n1 n1 p 1 p 1
if α < β or there is i 2,...,p such that α = β ,...,α = β , and α < β . We also use the
p p p p i i i 1 i 1
∈ { } − −
Y
(αp,...,α1)
.
operatorcolumnnotation . ,wheretheentriesY arearrangedintheorder
 .  (αp,...,α1)
α F+ ,1 α k
 i ∈ n1 ≤| i|≤ 
mentioned above. For simplicity, [X ,...,X ] denotes either the n-tuple (X ,...,X ) B( )n or the
1 n 1 n
∈ H
operator row matrix [X X ] acting from (n), the direct sum of n copies of the Hilbert space , to
1 n
··· H H
.
H
ANDOˆ DILATIONS AND INEQUALITIES ON NONCOMMUTATIVE DOMAINS 7
Lemma 2.1. If X := [√a T : α F+ ,1 α k ] and m := card α F+ : 1 α k ,
1 α 1,α ∈ n1 ≤ | | ≤ 1 1 { ∈ n1 ≤ | | ≤ 1}
then
diagm1 Ddiagm1 ···Ddiagm1 ∆T2 X∗1··· X∗1 X∗1
(cid:16) (cid:16) (cid:16)b (cid:17) (cid:17) (cid:17) √aα1···aαp(T1,αp···T1,α1)∗
.
=diagm1 Ddiagm1 ···Ddiagm1 ∆T2 ···  .. ,
(cid:16) (cid:16) (cid:16) (cid:17) (cid:17)(cid:17) α F+ ,1 α k
b  i ∈ n1 ≤| i|≤ 1 
 
∆T
where diagm1 appears p times on each side of the equality, and ∆T2 := 02 :H→DT2 ⊕K.
(cid:20) (cid:21)
b
Proof. Let D =[D(α) : α∈F+n1,1≤|α|≤k1] with D(α) ∈B(DT2 ⊕K) and note that
Ddiagm1 ∆T2 X∗1 = D(α1)∆T2√aα1T1∗,α1
(cid:16) (cid:17) α1∈F+n1X,1≤|α1|≤k1
b b
and
α1∈F+n1 D(α1)∆T2√aα1√aα2(T1,α2T1,α1)∗
diagm1 Ddiagm1 ∆T2 X∗1 X∗1 =P1≤|α1|≤k1 b ... .
(cid:16) (cid:16) (cid:17) (cid:17)  α F+ ,1 α k 
b  2 ∈ n1 ≤| 2|≤ 1 
 
An inductive argument shows that
diagm1 Ddiagm1 ···Ddiagm1 ∆T2 X∗1··· X∗1 X∗1
(cid:16) (cid:16) (cid:16) (cid:17) (cid:17) (cid:17)
α1,...,αp−1∈F+n1 D(αp−1)·b··D(α1)∆T2√aα1···aαp−1aαp(T1,αpαp−1···α1)∗
1≤|αi|≤k1
=P . ,
b ..
 α F+ ,1 α k 
 p ∈ n1 ≤| p|≤ 1 
 
where diag appears p times. On the other hand, one can easily prove by induction that
m1
√aα1···aαp(T1,αp···T1,α1)∗
.
diagm1 Ddiagm1 ···Ddiagm1 ∆T2 ···  .. 
(cid:16) (cid:16) (cid:16) (cid:17) (cid:17)(cid:17) α F+ ,1 α p
b  i ∈ n1 ≤| i|≤ 
 
=diagm1 [D(αp−1)···D(α1)∆T2 : α1,...,αp−1 ∈F+n1,1≤|αi|≤k1]
(cid:16) √aα1···aαp−1baα.p(T1,αpαp−1···α1)∗ (cid:17)
.
 . 
α ,...,α F+ ,1 α k .
× 1 p−1 ∈ ..n1 ≤| i|≤ 1
 . 
 
 α F+ ,1 α k 
 p ∈ n1 ≤| p|≤ 1 
 
The proof is complete. (cid:3)
A B
Lemma 2.2. Let T (T ,T ) and let U = be the matrix representation of a unitary
2 ∈ I 1 ′1 C D
(cid:20) (cid:21)
extension U (see relation (2.3)). If
TK
∈U
X :=[√a T : α F+ ,1 α k ],
1 α 1,α ∈ n1 ≤| |≤ 1
m := card α F+ : 1 α k , j =1,2,
j { ∈ nj ≤| |≤ j}
8 GELUPOPESCU
and T is a pure element in ( ), then
1 f
D H
√cβT2∗,β
.
diagm2(∆T′1) .. 
β F+ ,1 β k
 ∈ n2 ≤| |≤ 2
 
∞
=A∆T1h+B diagm1 Ddiagm1 ···Ddiagm1(C∆T1)X∗1··· X∗1 X∗1h,
p=0
X (cid:0) (cid:0) (cid:1) (cid:1)
for any h , where diag appears p+1 times in the general term of the series.
∈H m1
Proof. Due to relation (2.2), we have
∆T2√a0αT1∗,αh ∆T′1√cβT2∗,β
(2.4) A∆T1h+B (cid:20) .. (cid:21) = ... 
.
  β F+ ,1 β k
α∈F+n1,1≤|α|≤k1  ∈ n2 ≤| |≤ 2
 
and
∆T2√aαT1∗,αh
(2.5) C∆T h+D (cid:20) 0. (cid:21) = ∆T2h
1 . 0
.
  (cid:20) (cid:21)
α F+ ,1 α k 
 ∈ n1 ≤| |≤ 1
 
∆T
for any h . Since ∆T := 2 : T , we can rewrite relations (2.4) and (2.5) as
∈H 2 0 H→D 2 ⊕K
(cid:20) (cid:21)
(2.6) b A∆T1 +Bdiagm1(∆T2)X∗1 =diagm2(∆T′1)X∗2,
where X2 :=[√cβT2,β : β ∈F+n2,1≤|β|≤k2],band
(2.7) C∆T1 +Ddiagm1(∆T2)X∗1 =∆T2,
respectively. Note that using relation (2.7) we deduce that
b b
(2.8) diagm1 ∆T2 X∗1 =diagm1(C∆T1)X∗1+diagm1 Ddiagm1(∆T2)X∗1 X∗1,
(cid:16) (cid:17) (cid:16) (cid:17)
which combined with relation (2.6) yields
b b
diagm2(∆T′1)X∗2 =A∆T1 +Bdiagm1(C∆T1)X∗1+Bdiagm1 Ddiagm1(∆T2)X∗1 X∗1.
(cid:16) (cid:17)
Continuingtouserelation(2.8)inthelatterrelationandthe resultingones,aninductionargumentleads
b
to the identity
diagm2(∆T′1)X∗2 =A∆T1 +Bdiagm1(C∆T1)X∗1
m
(2.9) +B diagm1 Ddiagm1 ···Ddiagm1(C∆T1)X∗1··· X∗1 X∗1
p=1
X (cid:0) (cid:0) (cid:1) (cid:1)
+Bdiagm1 Ddiagm1 ···Ddiagm1 ∆T2 X∗1··· X∗1 X∗1,
wherediag appearsp+1times inthe general(cid:16)termofth(cid:16)e sumabovea(cid:16)ndm(cid:17)+2 tim(cid:17)es in(cid:17)the lastterm.
m1 b
Since ∆T and D are contractions and due to Lemma 2.1, one can easily see that
2
Bdiagm1 Ddiagm1 ···Ddiagm1 ∆T2 T∗1··· T∗1 T∗1h
(cid:13) (cid:16) (cid:16) (cid:16) (cid:17) (cid:17) (cid:17) (cid:13) 1/2
(cid:13)(cid:13) b (cid:13)(cid:13)
≤kBk k√aα1···aαm+2(T1,αm+2···T1,α1)∗hk2 = Φmf,T+12(I)h,h
α1,...,αXm+2∈F+n1  D E
 1≤|αi|≤k1 
 
ANDOˆ DILATIONS AND INEQUALITIES ON NONCOMMUTATIVE DOMAINS 9
foranyh . Since T ispurein ( ),wehavelim Φm+2(I)h=0foranyh . Consequently,
∈H 1 Df H m→∞ f,T1 ∈H
relation (2.9) implies
∞
diagm2(∆T′1)X∗2h=A∆T1h+B diagm1 Ddiagm1 ···Ddiagm1(C∆T1)X∗1··· X∗1 X∗1h,
p=0
X (cid:0) (cid:0) (cid:1) (cid:1)
for any h , where diag appears p + 1 times in the general term of the series. The proof is
complete. ∈ H m1 (cid:3)
We recall ([12], [15]) a few facts concerning multi-analytic operators on Fock spaces. We say that a
bounded linear operator M acting from F2(H ) to F2(H ) is multi-analytic with respect to
n n ′
⊗K ⊗K
the universal model W := (W ,...,W ) if M(W I ) = (W I )M for any i = 1,...,n. We can
1 n i i ′
associate with M a unique formal Fourier expansio⊗n K Λ ⊗ θK where θ B( , ). We know
uthnaiftoMrm=noSrmO.TM-liomrero→v1er,t∞kh=e0set|αo|f=akllrm|αu|Λltαi-⊗anθa(lαy)tiwchoPperαeer∈,aFtf+noorrseαian⊗chBr((αF∈)2([0H,1)), th(αe,)Fs∈e2r(iHesKc)oKnv′er)gceosinincidthees
P P n ⊗K n ⊗K′
with ( )¯B( , ), the WOT-closed operator space generated by the spatial tensor product.
R∞n Df ⊗ K K′
Let , , and be Hilbert spaces and consider
′
H H E
′
H H
U = A B : ⊕ β∈F+n2,L1≤|β|≤k2
C D →
(cid:20) (cid:21) E ⊕
α∈F+n1,L1≤|α|≤k1 E
tobeaunitaryoperator. SettingD =[D : α F+ ,1 α 1]: ,weassociate
(α) ∈ n1 ≤| |≤ E →E
with U and any r [0,1) the operator ϕ (rΛ ) defined by α∈F+n1,L1≤|α|≤k1
∗ U∗ 1
∈
1
−
ϕU∗(rΛ1):=IF2(Hn1)⊗A∗+ IF2(Hn1)⊗C∗ IF2(Hn1)⊗E − r|α|√aαΛ1,α˜⊗D(∗α)
(cid:16) (cid:17) αX∈F+n1 
 1≤|α|≤k1 
 
× √aαΛ1,α˜ ⊗IH : α∈F+n1,1≤|α|≤k1 IF2(Hn1)⊗B∗ ,
whereΛ :=[Λ ,...,Λ ]isthe(cid:2)tupleofweightedrightcreationoperato(cid:3)r(cid:16)sonF2(H )as(cid:17)sociatedwith
1 1,1 1,n1 n1
the regular domain . In what follows, we use the notations: A := I A, B := I B,
Df F2(Hn1) ⊗ F2(Hn1)⊗
C:=IF2(Hn1)⊗C, D:=IF2(Hn1)⊗D, and Γ(r):= √aαr|α|Λ1,α˜ ⊗IE : α∈F+n1,1≤|α|≤k1 .
Lemma 2.3. The strong operator topology limit ϕ (cid:2)(Λ ):=SOT-lim ϕ (rΛ ) exists and(cid:3)defines a
U∗ 1 r 1 U∗ 1
contractive multi-analytic operator with respect to W having the row m→atrix representation
1
ϕ (Λ )=[ϕ (Λ ): β F+ ,1 β k ],
U∗ 1 (β) 1 ∈ n2 ≤| |≤ 2
with ϕ (Λ ) ( )¯B( , ), where ( ) is the noncommutative Hardy algebra generated by
(β) 1 ∈R∞n1 Df ⊗ H′ H R∞n1 Df
the weighted right creation operators Λ ,...,Λ and the identity.
1,1 1,n1
Proof. Since D and Γ(r) are contractions, we have (cid:13) r|α|√aαΛ1,α˜ ⊗D(∗α)(cid:13) ≤ r < 1. Con-
(cid:13)(cid:13)α∈F+n1,P1≤|α|≤k1 (cid:13)(cid:13)
sequently, the operator ϕ (rΛ ) makes sense and(cid:13)ϕ (rΛ ) = [ϕ (rΛ ) : β F+(cid:13),1 β k ]
U∗ 1 (cid:13) U∗ 1 (β) 1 ∈ n(cid:13)2 ≤ | | ≤ 2
withϕ (rΛ ) ( )¯B( , ),where ( (cid:13))is the noncommutativedisk alge(cid:13)brageneratedby
(β) 1 ∈Rn1 Df ⊗ H′ H Rn1 Df
Λ ,...,Λ and the identity. Consequently,
1,1 1,n1
(2.10) ϕ (rΛ )= ∞ rγ Λ Θ(β)
(β) 1 | | 1,γ ⊗ (γ)
kX=0γ∈F+nX1,|γ|=k
10 GELUPOPESCU
for some operators Θ(β) B( , ), where the convergence is in the operator norm topology. On the
(γ) ∈ H′ H
A C
other hand, since ∗ ∗ is a unitary operator, standard calculations (see e.g. [14]) show that
B D
∗ ∗
(cid:20) (cid:21)
I ϕ (rΛ )ϕ (rΛ )
U∗ 1 U∗ 1 ∗
−
=C∗(I −Γ(r)D∗)−1I − r2|α|aαΛ1,α˜Λ∗1,α˜⊗I(I −DΓ(r)∗)−1C.
α∈F+n1X,1≤|α|≤k1
  
This shows that ϕ (rΛ ) is a contraction for any r [0,1) having the row matrix representation
U∗ 1
∈
ϕ (rΛ ) = [ϕ (rΛ ) : β F+ ,1 β k ] with ϕ (rΛ ) ( )¯B( , ). Since
U∗ 1 (β) 1 ∈ n2 ≤ | | ≤ 2 (β) 1 ∈ Rn1 Df ⊗ H′ H
ϕ (rΛ ) has the Fourier representation (2.10), one can see that ϕ (Λ ) := SOT-lim ϕ (rΛ )
(β) 1 (β) 1 r 1 (β) 1
exists, ϕ (Λ ) ¯B( , ), and ϕ (Λ ) 1. The proof is complete. → (cid:3)
(β) 1 ∈R∞n1⊗ H′ H k U∗ 1 k≤
We recall that (T ,T ) is the set of all tuples T := (T ,...,T ), with T : , such
I 1 ′1 2 2,1 2,n2 2,j H′ → H
that T ( , ) intertwines T with T , i.e. T T = T T for any i 1,...,n and
2 ∈ Dg H′ H 1 ′1 2,j 1′,i 1,i 2,j ∈ { 1}
j 1,...,n . The main result of this section is the following intertwining dilation theorem for the
2
∈ { }
elements of (T ,T ).
I 1 ′1
Theorem 2.4. Let T := (T ,...,T ) ( ) and T := (T ,...,T ) ( ), and let
1 1,1 1,n1 ∈ Df H ′1 1′,1 1′,n1 ∈ Df H′
T := (T ,...,T ) ( , ) be such that T (T ,T ). Let W := (W ,...,W ) and
2 2,1 2,n2 ∈ Dg H′ H 2 ∈ I 1 ′1 1 1,1 1,n1
Λ := (Λ ,...,Λ ) be the weighted creation operators associated with the noncommutative domain
1 1,1 1,n1
. If ϕ (Λ )=[ϕ (Λ ): β F+ ,1 β k ] is the contractive multi-analytic operator associated
Df U∗ 1 (β) 1 ∈ n2 ≤| |≤ 2
with U and T is a pure element of the noncommutative regular domain ( ), then the following
TK 1 f
∈U D H
relations hold:
1
Kf,T′1T2∗,β = √cβϕ(β)(Λ1)∗Kf,T1, β ∈F+n2,1≤|β|≤k2,
and
Kf,T1T1∗,i = W1∗,i⊗IDT1 Kf,T1, Kf,T′1(T1′,i)∗ = W1∗,i⊗IDT′1 Kf,T′1, i∈{1,...,n1},
where Kf,T1 and K(cid:0)f,T′1 are the(cid:1)noncommutative Poisson (cid:16)kernels associ(cid:17)ated with T1 and T′1, respectively.
A B
Proof. Fix U = and set
C D ∈UTK
(cid:20) (cid:21)
D :=[D(α) : α∈F+n1,1≤|α|≤1]: (DT2 ⊕K)→DT2 ⊕K.
α∈F+n1M,1≤|α|≤k1
In what follows, we use the notations: A:=I A, B:=I B, C:=I C,
F2(Hn1)⊗ F2(Hn1)⊗ F2(Hn1)⊗
Q:= aαΛ1,α˜⊗D(∗α) and Γ:= √aσΛ1,σ˜⊗IDT2⊕K :σ ∈F+n1,1≤|σ|≤k1 .
α∈F+n1X,1≤|α|≤k1 (cid:2) (cid:3)
As in Lemma 2.1, an induction argument over q shows that
C (D D )
∗ (γ1)··· (γq) ∗
.
(2.11) diagm1 diagm1··· diagm1(C∗)D∗ ···D∗ =diagm1 .. ,
γ F+ ,1 γ k
(cid:0) (cid:0) (cid:1) (cid:1)  i ∈ n1 ≤| i|≤ 1
where diag appears q+1 times on the left-hand side of the equality. 
m1
WeassociatewithU∗themulti-analyticoperatorϕU∗(Λ1)∈R∞n1(Df)⊗¯B β∈F+n2,1≤|β|≤k2DT′1,DT1 ,
as in Lemma 2.3, in the particular case when H = DT1, H′ = DT′1, and E(cid:16)=LDT2 ⊕K. Note that, fo(cid:17)r
each y ∈ β∈F+n2,1≤|β|≤k2DT′1 and α∈F+n1 with |α|=n, we have
L
∞
ϕ (Λ )(e y)=e A y+ C QqΓB (e y)
U∗ 1 α α ∗ ∗ ∗ α
⊗ ⊗ ⊗
q=0
X