New York Journal of Mathematics New York J. Math.

New York Journal of Mathematics

New York J. Math. 11(2005)247–290.

Nevanlinna–Pick interpolation for Schur–Agler class functions on domains with matrix polynomial

defining function in C

Joseph A. Ball and Vladimir Bolotnikov

Abstract. We consider a bitangential interpolation problem for operator- valued functions defined on a general class of domains in Cⁿ (including as particular cases, Cartan domains of types I, II and III) which satisfy a type of von Neumann inequality associated with the domain. The compact formu- lation of the interpolation conditions via a functional calculus with operator argument includes prescription of various combinations of functional values and of higher-order partial derivatives along left or right directions at a pre- scribed subset of the domain as particular examples. Using realization results for such functions in terms of unitary colligation and the defining polynomial for the domain, necessary and sufficient conditions for the problem to have a solution were established recently in Ambrozie and Eschmeier (preprint, 2002), and Ball and Bolotnikov, 2004. In this paper we present a parametrization of the set of all solutions in terms of a Redheffer linear fractional transformation acting on a free-parameter function from the class subject to no interpolation conditions. In the finite-dimensional case when functions are matrix-valued, the matrix of the linear fractional transformation is given explicitly in terms of the interpolation data.

Contents

1. Introduction 248

2. Right and left evaluation with operator argument 255

3. The solvability criterion 259

4. Solutions to the interpolation problem and unitary extensions 262 5. The associated universal unitary colligation 272

6. Explicit formulas 275

7. Nevanlinna–Pick interpolation problem 278

8. Some further examples 285

References 287

Received February 22, 2005.

Mathematics Subject Classification. 47A57.

Key words and phrases. Operator valued functions, Schur–Agler class, Nevanlinna–Pick interpolation.

ISSN 1076-9803/05

247

1. Introduction

In this paper we pursue our work on interpolation theory for Schur–Agler func- tions that are a far-reaching operator-valued multivariable analogue of classical Schur functions (that is, analytic and mapping the unit diskDinto the closed unit disk D. The operator-valued Schur class S(E,E_∗) consists, by definition, of ana- lytic functions F on D with values F(z) equal to contraction operators between two Hilbert spacesE and E∗. In what follows, the symbolL(E,E∗) stands for the algebra of bounded linear operators mapping E into E∗. The class S(E,E∗) ad- mits several remarkable characterizations. We mention in particular that any such functionF(z) can be realized in the form

F(z) =D+zC(I−zA)⁻¹B where the connecting operator (orcolligation)

U = A B

C D

:

H E

→ H

E_∗

is unitary, and whereHis some auxiliary Hilbert space (theinternal spacefor the colligation). From the point of view of system theory,F(z) =D+zC(I−zA)⁻¹B is thetransfer functionof the linear system

Σ = Σ(U) :

x(n+ 1) =Ax(n) +Bu(n) y(n) =Cx(n) +Du(n)

in the sense that any solution (u, x, y) of Σ defined on the nonnegative integersZ⁺ withx(0) = 0 satisfies

y(z) =F(z)·u(z).

Here in general we denote by x(z) =

∞ n=0

x(n)zⁿ

the Z-transformof the sequence {x(n)}^∞_n=0. It is also well-known that the Schur class functions satisfy a von Neumann inequality: F ∈ S(E,E_∗) andT ∈ L(H)a contraction operator =⇒ F(rT) ≤ 1 for all r <1. Here F(rT) can be defined, e.g., by

F(rT) = ∞ n=0

rⁿF_n⊗Tⁿ ∈ L(E ⊗ H,E∗⊗ H) if F(z) = ∞ n=0

F_nzⁿ.

Multivariable generalizations of these and many other related results have been obtained recently in the following way: letQbe ap×qmatrix-valued polynomial

Q(z) =





q¹¹(z) . . . q¹q(z)

... ...

qp1(z) . . . qpq(z)



: Cⁿ→C^p×q (1.1)

and letDQ∈Cⁿ be the domain defined by

D_Q={z∈Cⁿ:Q(z)_C^p×q <1}. (1.2)

(Here · _C^p×q refers to the induced operator norm arising by considering a p×q matrix M as an operator from C^q into C^p.) Now we recall the Schur–Agler

classSAQ(E,E∗) that consists, by definition, ofL(E,E∗)-valued functionsS(z) = S(z1, . . . , z_n) analytic onDQ and such that

S(T1, . . . , T_n) ≤1

for any collection of n commuting operators (T₁, . . . , T_n) on a Hilbert space K, subject to

Q(T₁, . . . , T_n)<1.

By [9, Lemma 1], the Taylor joint spectrum of the commutingn-tuple (T1, . . . , T_n) is contained inDQ wheneverQ(T1, . . . , T_n)<1, and henceS(T1, . . . , T_n) is well- defined by the Taylor functional calculus (see [28]) for anyL(E,E∗)-valued function S which is analytic onDQ. Upon usingK=CandT_j =z_j forj= 1, . . . , nwhere (z₁, . . . , z_n) is a point inDQ we conclude that anyL(E,E∗) function is contractive valued, and thus, the classSAQ(E,E∗) is the subclass of theSchur classSDQ(E,E∗) of contractive valued functions analytic onD_Q. By the von Neumann result, in the case when Q(z) = z, these classes coincide; in general, SA_Q(E,E_∗) is a proper subclass ofS_DQ(E,E_∗).

Special choices of Q(z) =



 z1

. .. z_n



 and Q(z) =

z1 z2 . . . z_n

lead to the unit polydiskDQ=Dⁿ and the unit ballDQ=Bⁿ ofCⁿ, respectively.

The classes SAQ(E,E∗) for these two generic cases have been known for a while.

The polydisk setting was first presented by J. Agler in [2] and then extended to the operator valued case in [19, 22]; see also [3, 15, 20]. The Schur–Agler func- tions on the unit ball appeared in [30] and later in [1,47, 40] in connection with complete Nevanlinna–Pick kernels and in [13,46] in connection with the study of dilation theory for commutative row contractions; we refer to [23] for a thorough review of the operator-valued case. The general setting introduced above unifies these two generic settings and besides, covers some other interesting cases including Cartan domains of the first three types, their cartesian products and their inter- sections. General domainsD_Qand classesSA_Q(E,E_∗) (forE=E_∗=C) have been introduced in [9]. The operator-valued version of this class has appeared in [8], [17].

The following theorem gives several equivalent characterizations of the class SA_Q(E,E_∗); the proof can be found in [8, 17]; the proof for the scalar-valued case (whereE=E_∗=C) can be found in [9] in a somewhat different form.

Theorem 1.1. Let S be aL(E,E∗)-valued function defined on DQ. The following statements are equivalent:

(1) S belongs toSAQ(E,E∗).

(2) There exist an auxiliary Hilbert spaceHand a function H(z) =

H1(z) . . . H_p(z) (1.3)

analytic onD_Q with values in L(C^p⊗ H, E_∗) so that

I_E∗ −S(z)S(w)^∗ =H(z) (I_Cp⊗H−Q(z)Q(w)^∗)H(w)^∗. (1.4)

(3) There exist an auxiliary Hilbert spaceHand a function G(z) =



 G1(z)

... G_q(z)



 (1.5)

analytic onDQ with values in L(C^q⊗ H,E)so that

I_E−S(z)^∗S(w) =G(z)^∗(I_Cq⊗H−Q(z)^∗Q(w))G(w).

(1.6)

(4) There exist an auxiliary Hilbert space H and analytic functions H(z) and G(z)as in(1.3)and(1.5), so that relations (1.4) and(1.6)hold along with

S(z)−S(w) =H(z) (Q(z)−Q(w))G(w) (z, w∈ D_Q).

(1.7)

(5) There is a unitary operator U=

A B

C D

:

C^p⊗ H E

→

C^q⊗ H E∗

(1.8)

such that

S(z) =D+C(I_Cp⊗H−Q(z)A)⁻¹Q(z)B for allz∈Ω.

(1.9)

Moreover, if S is of the form (1.9), then it holds that

I_E∗ −S(z)S(w)^∗=C(I−Q(z)A)⁻¹(I−Q(z)Q(w)^∗) (I−A^∗Q(w)^∗)⁻¹C^∗, S(z)−S(w) =C(I−Q(z)A)⁻¹(Q(z)−Q(w)) (I−AQ(w))⁻¹B, I_E −S(z)^∗S(w) =B^∗(I−Q(z)^∗A^∗)⁻¹(I−Q(z)^∗Q(w)) (I−AQ(w))⁻¹B.

Hence the representations (1.4),(1.6)and (1.7) are valid with

H(z) =C(I−Q(z)A)⁻¹ and G(z) = (I−AQ(z))⁻¹B.

(1.10)

The representation (1.9) is calleda unitary realizationof S ∈ SAQ(E,E∗) and can be viewed as a realization of S as the transfer functionof a certain type of multidimensional system; see Section4.

Remark 1.2. In formulas (1.9) and (1.10) we abused notations and used Q(z) instead ofQ(z)⊗I_H.

LetHDQ(E,K) be the set of allL(E,K)-valued functionsF which are analytic on DQ. GivenF ∈ HDQ(E,K) andT = (T1, . . . , T_n) ann-tuple of commuting bounded operators onKfor which the Taylor joint spectrumσTaylor(T) is contained inDQ, one can use the Taylor functional calculus (details below in Section 2) to define a left evaluation map F → F^∧L(T) ∈ L(E,K). Similarly, if F ∈ HDQ(K,E∗) and T = (T₁, . . . , T_n) is an n-tuple of commuting bounded operators on K with σ_Taylor(T) ⊂ D_Q, one can use the Taylor functional calculus to define a right evaluation mapF →F^∧R(T)∈ L(K,E∗).

LetKandK be two Hilbert spaces and let

T = (T1, . . . , T_n) and T= (T₁, . . . , T_n) (1.11)

be commutativen-tuples of operatorsT_j∈ L(K) andT_j ∈ L(K) such that σ_Taylor(T)⊂ DQ and σ_Taylor(T)⊂ DQ.

(1.12)

We shall consider bitangential interpolation problems with the data sets consisting of two Hilbert spaces K andK, two commutativen-tuples of the form (1.11) and satisfying (1.12), and bounded operators

X_L:E∗→ K, Y_L:E → K, X_R:K → E∗, Y_R:K→ E. Given this data set

D={T, T, X_L, Y_L, X_R, Y_R}, (1.13)

the formal statement of the associated bitangential interpolation problem is:

Problem 1.3. Find necessary and sufficient conditions for existence of a function S∈ SAQ(E,E∗)such that

(X_LS)^∧L(T) =Y_L and (SY_R)^∧R(T) =X_R. (1.14)

To formulate the solution criterion we need some additional notation. Define operators

E1=





 I_K

0 ... 0





, E2=





 0 I_K

... 0





, . . . , E_p=





 0 ... 0 I_K





, (1.15)

E₁=





 I_K

0 ... 0





, E₂ =





 0 I_K

... 0





, . . . , E_q =





 0

... 0 I_K





, (1.16)

Qj·(T) =



 qj1(T)

... qjq(T)



, Q·k(T) =





q1k(T)^∗ ... qpk(T)^∗



 (1.17)

and the operators

M_j=M_j(T) =

E_j 0 0 Qj·(T)

for j= 1, . . . , p, (1.18)

N_k=N_k(T) =

Q·k(T) 0 0 E_k

for k= 1, . . . , q.

(1.19)

Theorem 1.4. There is a function S ∈ SA_Q(E,E_∗) satisfying interpolation con- ditions (1.14)if and only if there exists a positive semidefinite operator

P∈ L((C^p⊗ K)⊕(C^q⊗ K)) subject to the Stein identity

p j=1

M_j^∗P M_j− q k=1

N_k^∗P N_k=X^∗X−Y^∗Y (1.20)

whereM_j andN_k are the operators defined via formulas (1.15)–(1.19)and where X=

X_L^∗ X_R

and Y =

Y_L^∗ Y_R . (1.21)

The special case where only a set of left tangential interpolation conditions (X_LS)^∧L(T) =Y_L or only right tangential interpolation conditions (SY_k)^∧R(T) = X_R is considered corresponds to the special case of Problem1.3 where one takes K={0}(respectively,K={0}). As a corollary to Theorem1.4we therefore have the following.

Corollary 1.5. (1) Suppose that DL = {T, X_L, Y_L} is the data set for a left tangential interpolation problem (i.e., DL is as in (1.13) with K = {0}).

Then there exists anS∈ SAQ(E,E∗)satisfying the interpolation condition (X_LS)^∧L(T) =Y_L

if and only if there exists a positive semidefinite solution P= [P_ij]^p_i,j=1∈ L(K ⊗C^p)

to the Stein equation p

j=1

P_jj− q k=1

p i,j=1

qik(T)P_ijqjk(T)^∗=X_LX_L^∗−Y_lY_L^∗.

(2) Suppose thatDR={T;, Y_R, X_R}is the data set for a right tangential inter- polation problem(i.e., DR is as in (1.13)with K={0}). Then there exists anS∈ SA_Q(E,E_∗)satisfying the interpolation condition

(SY_R)^∧R(T) =X_R

if and only if there exists a positive semidefinite solution P= [P_ij ]^q_i,j=1∈ L(K⊗C^q)

of the Stein equation q

j=1

P_jj − p k=1

q i,j=1

qki(T)^∗P_ijqjk(T) =Y_R^∗Y_R−X_R^∗X_R.

In the special case in Corollary1.5whereK=⊕ω∈ΩE∗for some subset Ω ofDQ

and one takes

T = diag_ω∈Ω[ωI_E∗], X_L = col_ω∈Ω[I_E∗], Y_L= col_ω∈Ω[F(ω)]

for some given function F: Ω → L(E,E_∗), the left interpolation condition with operator argument (X_LF)^∧L(T) =Y_Lgives rise to full-operator-value interpolation along the subset Ω of D_Q. The interpolation problem then is: given F: Ω → L(E,E_∗), findS∈ SA_Q(E,E_∗)so that

S(ω) =F(ω) for all ω∈Ω⊂ D_Q.

This case of part (1) of Corollary1.5can already be found in [17]. We note that a more general version of this problem, where the matrix polynomialQ(z) is replaced by a continuumz →Qλ(z) of matrix-valued analytic functions indexed by λin a separable compact Hausdorff space Λ, has been worked out in [7].

LetPbe any operator satisfying the conditions in Theorem1.4. Let us represent its block entries explicitly as

P =

P_L P_LR P_LR^∗ P_R

(1.22)

where

P_L=





Ψ₁₁ . . . Ψ_1p ... ... Ψ_p1 . . . Ψ_pp



, P_R=





Φ₁₁ . . . Φ_1q ... ... Φ_q1 . . . Φ_qq



, (1.23)

P_LR=





Λ11 . . . Λ1q

... ... Λ_p1 . . . Λ_pq



, (1.24)

with 





Ψ_j∈ L(K) forj, = 1, . . . , p, Φ_j∈ L(K) forj, = 1, . . . , q,

Λ_j∈ L(K,K) forj= 1, . . . , p; = 1, . . . , q.

(1.25)

It turns out that for every positive semidefinite P satisfying (1.20), there is a solutionS of the bitangential interpolation Problem1.3such that, for some choice of associated functionsH(z) andG(z) of the form (1.3) and (1.5) in representations (1.4), (1.6), (1.7), it holds that

(X_LH_j)^∧L(T)

(X_LH)^∧L(T)_∗

= Ψ_jforj, = 1, . . . , p, (1.26)

(X_LH_j)^∧L(T) (GY_R)^∧R(T) = Λ_jforj= 1, . . . , p; = 1, . . . , q, (1.27)

(G_jY_R)^∧R(T)_∗

(GY_R)^∧R(T) = Φ_jforj, = 1, . . . q.

(1.28)

Furthermore, it turns out that conversely, for every solutionS of Problem1.3with representations (1.4), (1.6), (1.7) (existence of these representations is guaranteed by Theorem1.1), the operatorPdefined via (1.22)–(1.24) and (1.26)–(1.28) satisfies conditions of Theorem1.4. These observations suggested the following modification of Problem1.3with the data set

D={T, T, X_L, Y_L, X_R, Y_R, Ψ_j, Φ_j,Λ_j}. (1.29)

Problem 1.6. Given the data D as in (1.29), find all functions S ∈ SA_Q(E,E_∗) satisfying interpolation conditions(1.14)and such that for some choice of associated functionsH_j andGin the representations(1.4),(1.6),(1.7), the equalities (1.26)–

(1.28) hold.

In contrast to Problem1.3, the solvability criterion for Problem1.6can be given explicitly in terms of the interpolation data.

Theorem 1.7. Problem 1.6 has a solution if and only if the operator P given by (1.22)–(1.24)is positive semidefinite and satisfies the Stein identity (1.20).

Moreover, there exists defect subspaces∆ and∆_∗and an operator-valued function z→Σ(z) =

Σ11(z) Σ12(z) Σ21(z) Σ22(z)

: E

∆_∗

→ E∗

∆

forz∈ D_Q of the form

Σ(z) =

U₂₂ U₂₃ U32 0

+

U₂₁ U31

(I_∆

∗−Q(z)U₁₁)⁻¹Q(z)

U₁₂ U₁₃

with

U⁰=



U11 U12 U13

U21 U22 U23

U31 U32 0



:



H E

∆_∗



→



H E∗

∆





unitary and completely determined by the interpolation data set D(see (1.13))so that S is a solution of Problem1.6 if and only ifS has the form

S(z) = Σ11(z) + Σ12(z)

I_∆

∗− T(z)Σ22(z) ₋1

T(z)Σ21(z) for a free-parameter functionT(z)∈ SAQ(∆, ∆_∗).

As a corollary of Theorem 1.7we get a description (albeit less satisfactory) of the set of all solutions of a Problem1.3.

Corollary 1.8. Suppose that D is an interpolation data set for Problem 1.3 as in (1.13). Given any positive semidefinite solution P of the Stein identity (1.20), define operators Ψ^P_j∈ L(K)forj, = 1, . . . , p,Φ^P_ij ∈ L(K)forj, = 1, . . . , p, and Λ^p_j∈ L(K,K)forj= 1, . . . , pand= 1, . . . , q, by (1.22),(1.23)and (1.24). Let

Σ^P(z) =

Σ^P₁₁(z) Σ^P₁₂(z) Σ^P₂₁(z) Σ^P₂₂(z)

: E

∆^P_∗

→ E_∗

∆^P

be the linear-fractional coefficient matrix function generated from the expanded in- terpolation data set

D^P =

T, T, X_L, Y_L, X_R, Y_R,Ψ^P_j,Φ^P_j,Λ^P_j

associated with a Problem 1.6 as in Theorem 1.7. Then S is a solution of Prob- lem1.3 if and only ifS has the form

S(z) = Σ^P₁₁(z) + Σ^P₁₂(z)

I_∆P

∗ − T^P(z)Σ^P₂₂(z) ₋1

T^P(z)Σ^P₂₁(z)

for some choice P of positive semidefinite solution of (1.20) and some choice of free-parameter function T^P(z)∈ SA_Q(∆^P,∆^P_∗).

The paper is organized as follows. Section2 reviews material from [49, 51] (see also [28]) on Vasilescu’s adaptation based on the Martinelli kernel (see [51]) of the Taylor functional calculus (see [49,50]) to formulate and develop the basic proper- ties of the left and right point evaluation operatorsF →F^∧L(T) andF →F^∧R(T) needed in the very formulation of the bitangential interpolation problem. Section3 derives the necessity direction of the solvability criterion in Theorem1.7. Section4 discusses the connections with multidimensional system theory and delineates how solutions of Problem 1.6 are in correspondence with the characteristic functions of unitary colligations arising as unitary extensions of a certain partially defined isometry uniquely specified by the interpolation data. Section 5 sets up the uni- versal unitary colligation completely determined by the interpolation data which leads to the linear fractional parametrization for the set of all solutions of Prob- lem1.6asserted in the second part of Theorem1.7; the ideas here adapt the earlier work of [11,12] done for the classical one-variable setting. Section7makes explicit how the bitangential interpolation problems covered in [17] can be seen as exam- ples of Problems 1.3 and1.6 here and considers the special case of a bitangential Nevanlinna–Pick interpolation problem involving only finitely many interpolation

nodes. In the latter case the coefficients of the Redheffer linear fractional map parametrizing the set of all solutions can be described more explicitly. For some particular choices of Q similar formulas were obtained in [26], [5] and [14], [27].

The results and techniques used in Sections4–7are an adaptation of the approach carried out for closely related multivariable interpolation problems for particular cases of the domainsDQ in [14,16,15].

2. Right and left evaluation with operator argument

We begin with a review of the elements of the Taylor functional calculus for a tuple of commuting Hilbert space operators, as worked out in explicit form by use of an adaptation of the Bochner–Martinelli kernel by Vasilescu. A good survey of the general topic of joint spectra and functional calculus for operator tuples is [28];

more specific information can be found in [51,52,43].

Denote by Λ[e] =⊕ⁿ_k=0Λ^k[e] the exterior algebra overConngeneratorse1,. . ., e_n. The linear space Λ[e] becomes a Hilbert space if we declare the collection

{e_i1∧ · · · ∧e_ik: 1≤i1<· · ·< i_k≤n}

to be an orthonormal basis for Λ^k[e] for eachk = 1, . . . , n. Note that we identify Λ⁰[e] withC. Fori= 1, . . . , nletE_ibe the operator defined on Λ[e] byE_i:ξ→e_i∧ξ forξ∈Λ[e]. Then one can check that

E_iE_j+E_jE_i= 0, E_i²= 0, E_i^∗E_j+E_jE_i^∗=δ_i,jI_Λ[e] whereδ_i,j is the Kronecker delta. Therefore

E_iE_i^∗E_i =E_i(I−E_iE_i^∗) =E_i−E_i²E^∗_i =E_i,

so eachE_iis a partial isometry. Now letT = (T₁, . . . , T_n) be a commutingn-tuple of operators on a Hilbert spaceK. OnK ⊗Λ[e] define the operator

D_T =T1⊗E1+· · ·+T_n⊗E_n.

One can check thatD²_T = 0, i.e., that RanD_T ⊂KerD_T. We say thatTis invertible in the sense of Taylor if we have the equality RanD_T = KerD_T, or equivalently (see [51, Lemma 2.1]), ifR_T :=D_T+D_T^∗ is invertible(as an operator onK ⊗Λ[e]).

We define theTaylor spectrum ofT to be the set of allλ= (λ1, . . . , λ_n)∈Cⁿ for which then-tuple

λ−T := (λ₁I_K−T₁, . . . , λ_nI_K−T_n) is not invertible in the sense of Taylor.

Now suppose that z → β(z) = (β₁(z), . . . , β_n(z)) is an n-tuple of L(K)-valued holomorphic functions on an open set Ω⊂Cⁿ. (We shall eventually restrict to the case

β(z) =z−T := (z1I_K−T1, . . . , z_nI_K−T_n) (2.1)

for ann-tuple of operatorsT = (T₁, . . . , T_n) inL(K), so the reader should keep this example in mind.) Define an operatorD_β:C^∞(Ω,Λ(K))→C^∞(Ω,Λ(K)) by

(D_βf)(z) :=D_β(z)f(z) forz∈Ω.

HenceR_β is then defined by

(R_βf)(z) = (D_β(z)+D^∗_β(z))f(z).

In addition we need the so-calledDolbeault complex (see [39, page 268]), i.e., the exterior algebra Λ[∂_z] generated by the indeterminantsdz = (dz₁, . . . , dz_n). The operator∂ onC^∞(Ω)⊗Λ[∂_z] is then given by

∂:ξ→∂f∧dz_i1∧ · · · ∧dz_ik if ξ=f dz_i1∧ · · · ∧dz_ik where

∂f := ∂f

∂z1

dz1+· · ·+ ∂f

∂z_ndz_n.

For an n-tuple β = (β1, . . . , β_n) of functions in HΩ(K,K), we declare the Taylor spectrum ofβ inΩ to be

σ_Taylor^Ω (β,K) :=

λ∈Ω : Ker(D_β(λ))= Ran(D_β(λ)) . Then theMartinelli kernel associated withβ is defined by

M(β)(z) :=R⁻¹

β(z)

∂_zR⁻¹

β(z)

_n−1

E|K⊗Λ⁰[e]:K ⊗Λ⁰[e]→ K ⊗Λ⁰[e]⊗Λⁿ⁻¹[dz].

for all z /∈ σ^Ω_Taylor(β,K). If we identify K ⊗Λ⁰[e] with K, then we view M(β)(z) simply as an element ofL(K)⊗Λⁿ⁻¹[dz].

We now specialize to the case when β(z) is of the form (2.1) for an n-tuple of operators T = (T₁, . . . , T_n) ∈ L(K)ⁿ. Assume that σ_Taylor(T) ⊂Ω. One can use the Martinelli kernel associated withz−T to define a functional calculus forT for functionsf ∈ H(Ω,C) as follows (see [51]). Choose an open subset Ω with smooth boundary∂Ω so that

σTaylor(T)⊂Ω⊂Ω⊂Ω.

(2.2)

Note that by definitionσ^D_Taylor^Q (z−T) = σTaylor(T) ⊂Ω and henceM(z−T) is defined on ∂Ω. Then, for f a scalar-valued holomorphic function on Ω, we can definef(T) via

f(T) = 1 (2πi)ⁿ

∂Ω

M(z−T)·f(z)∧dz.

For further details, we refer to [51, 28]. The definition of the Taylor spectrum originates in [49] and an equivalent formulation of the functional calculus using more homological algebra machinery can be found in [50].

ForF ∈ HΩ(E,E∗), we can define a functional calculus F →F(T)∈ L(E ⊗ K,E∗⊗ K) by

F(T) = 1 (2πi)ⁿ

∂Ω

F(z)⊗M(z−T)∧dz.

This is the functional calculus needed to define the Schur–Agler class above.

To formulate the general bitangential interpolation problem, we need to intro- duce left and right operator evaluation defined as follows. Suppose first that we are given a functionF ∈ HΩ(E,K) (i.e.,F is holomorphic on Ω with values inL(E,K)) together with a commutingn-tupleT = (T1, . . . , T_n)∈ L(K)ⁿwithσTaylor(T)⊂Ω.

We then definethe left evaluation of F with operator argumentT by F^∧L(T) = 1

(2πi)ⁿ

∂Ω

M(z−T)·F(z)∧dz (2.3)

with Ω chosen as in (2.2). Similarly, ifF ∈ HΩ(K,E∗) andT = (T₁, . . . , T_n)∈ L(K)ⁿ with σ_Taylor(T)⊂Ω, definethe right evaluation of F with operator argu- ment T by

F^∧R(T) = 1 (2πi)ⁿ

∂Ω

F(z)·M(z−T)∧dz.

(2.4)

We need the following general result of Fubini type concerning this left and right functional calculus with operator argument.

Proposition 2.1. Let T = (T₁, . . . , T_n) ∈ L(K)ⁿ, T = (T₁, . . . , T_m ) ∈ L(K)^m and let(T, T) be a commuting(n+m)-tuple of operators on the Hilbert space K. Suppose that the functions

F : Ω→ L(E,K) and F: Ω → L(E,E)

are analytic on open sets Ω and Ω containing σ_Taylor(T) and σ_Taylor(T) respec- tively. Define an analytic function ofn+mvariables

(z, w) = (z₁, . . . , z_n, w₁, . . . , w_m) by

H(z, w) =F(z)F(w) : Ω×Ω → L(E,K).

Then

H^∧L(T, T) = (F^∧L(T)·F)^∧L(T).

(2.5)

Similarly, ifF takes values inL(E,E)andF takes values inL(K,E)and if we set H(z, w) =F(z)F(w), then

H^∧R(T, T) = (F·F^∧R(T))^∧R(T).

(2.6)

Proof. This result is a mild generalization of Theorem 3.8 in [52]. Alternatively, one can view it as a generalization of Proposition 12 in [43] (specialized to the Hilbert space case where one can take the generalized inverseV appearing there to be simply (R_z−T)⁻¹ — see the concluding remark (2) in [43]). In these references, the result is given for the case where E = E = K and the values of F and G are scalar operators. The same proof goes through for our setting, with proper attention to the order of writing of values ofF,F andM(z−T, w−T).

Remark 2.2. If Ω is a logarithmically convex Reinhardt domain (see [39, Section 2.3]), then 0 ∈ Ω and any function F ∈ HΩ(E,E_∗) is given by its power series expansion about the origin

F(z) =

j∈Nⁿ

F_jz^j F_j∈ L(E,E∗)

uniformly converging on compact subsets of Ω. Then the left and right evaluation maps (2.5) and (2.6) are given explicitly by

F^∧L(T) =

j∈Nⁿ

T^jF_j and F^∧R(T) =

j∈Nⁿ

F_jT^j,

for every choice of F ∈ HΩ(E,K) and F ∈ HΩ(K,E∗) and for any commuting n-tupleT = (T1, . . . , T_n)∈ L(K)ⁿ.

Similarly, if Ω = Ω1× · · · ×Ω_n is a polydomain (the Cartesian product of n 1-variable domains Ω1, . . . ,Ω_n), then we may write

F^∧L(T) = 1 (2πi)ⁿ

Ω_n

· · ·

Ω₁

(z₁I−T₁)⁻¹· · ·(z_nI−T_n)⁻¹F(z)dz₁· · ·dz_n, F^∧R(T) = 1

(2πi)ⁿ

Ω_n

· · ·

Ω₁

F(z)(z1I−T1)⁻¹· · ·(z_nI−T_n)⁻¹ dz1· · ·dz_n. We need to note the following elementary properties of evaluations (2.3) and (2.4).

Lemma 2.3. Let T andT be commuting n-tuples of the form(1.11)with Taylor spectrum contained inΩ. Then:

(1) For every constant functionW(z)≡W ∈ L(K,K), (W)^∧L(T) = (W)^∧R(T) =W.

(2.7)

(2) For everyF ∈ HΩ(E,K),F∈ HΩ(K,E∗),W ∈ L(E,E)andW ∈ L(E∗,E_∗), (F·W)^∧L(T) =F^∧L(T)·W and

W·F _∧R

(T) =W·F^∧R(T).

(2.8)

(3) For everyF ∈ HΩ(E,K),F∈ HΩ(K,E_∗)andj∈ {1, . . . , d}, (z_jF(z))^∧L(T) =T_j·F^∧L(T) and

z_jF(z)

_∧R

(T) =F^∧R(T)·T_j. (2.9)

(4) For every choice ofF∈ HΩ(E,K)and ofF∈ HΩ(E,E),

F·F _∧L

(T) = (F^∧L(T)·F)^∧L(T).

(2.10)

(5) For every choice ofF∈ HΩ(E_∗,E_∗) and ofF∈ HΩ(K,E_∗),

F·F _∧R

(T) = (F·F^∧R(T))^∧R(T).

(2.11)

Proof. Statement (1) is a consequence of the fact that the Martinelli–Vasilescu functional calculus reproduces constants. Statement (2) is an immediate conse- quence of equalities

Ω

H(z)·W dz=

Ω

H(z)dz·W,

Ω

W·H(z)dz=W·

Ω

H(z)dz for aL(E,K)-valued (2n−1)-formH(z) and aL(K,E_∗)-valued (2n−1)-formH(z).

Alternatively, the first equality in (2.8) follows from (2.10) for the special case ofF(z)≡W when combined with (2.7):

(F·W)^∧L(T) =

F^∧L(T)·W∧L

(T) =F^∧L(T)·W

and the second equality in (2.8) follows in much the same way from (2.11) for the special case ofF(z)≡W.

The first relation in (2.9) follows from (2.10) for the special case whenE =E andF(z) =z_jI_E. Indeed, in this case (2.10) gives

(z_jF(z))^∧L(T) = (F ·F)^∧L (2.12)

=

F^∧L(T)·F

)^∧L(T) =

F1·F^∧L(T)_∧L (T)

where F1(z) =z_jI_K. Applying the first relation in (2.8) (with F =F1 and W = F^∧L(T)) to the right-hand side in (2.12) we get

(z_jF(z))^∧L(T) =F₁^∧L(T)·F^∧L(T)

and since F₁^∧L(T) = T_j by (2.3), the first equality in (2.9) follows. To get the second equality, one can apply (2.11) to the special caseE_∗ =E_∗andF(z) =z_jI_E∗. Finally, statement (4) follows from (2.5) and statement (5) from (2.6) upon

settingT =T in (2.5) and (2.6).

3. The solvability criterion

In this section we prove the necessity part of Theorem1.7.

Proof of the necessity part in Theorem 1.7. Suppose that S ∈ SA_Q(E,E_∗) satisfies conditions (1.14) and (1.26)–(1.28) for some choice of associated functions H andGof the form (1.3) and (1.5) in the representation (1.4), (1.6), (1.7). LetP be defined as in (1.22)–(1.24). Interpolation conditions (1.26)–(1.28) mean thatP can be represented as

P = T^∗_L

T^∗_R T_L T_R (3.1)

where the operatorsTL: C^p⊗ K → HandTR: C^q⊗ K→ Hare given by TL=

(X_LH₁)^∧L(T)_∗ . . .

(X_LH_p)^∧L(T)_∗ (3.2)

and

TR=

(G1Y_R)^∧R(T). . .(G_qY_R)^∧R(T) . (3.3)

Comparing (3.1) with (1.22) we conclude that

P_L=T^∗_LT_L, P_R=T^∗_RT_R, P_LR=T^∗_LT_R. (3.4)

It follows from (3.1) thatP ≥0 and thus, it remains to show that P satisfies the Stein identity (1.20). To this end, note that by the first property in (2.9),

(pF)^∧L(T) =p(T)·F^∧L(T)

for every polynomialpinnvariables and everyF ∈ H_Q(K,E). In particular, taking into account the block structure (1.3) ofH and (1.1) ofQ, we get

(X_LHQ)^∧L(T) = _p

i=1

qi1X_LH_i. . . p

i=1

qiqX_LH_{i ∧L}

(T)

= _p

i=1

qi1(T) (X_LH_i)^∧L(T). . . p i=1

qiq(T) (X_LH_i)^∧L(T)

which can be written in terms of (1.17) and (3.2) as (X_LHQ)^∧L(T) =

Q·1(T)^∗T^∗_L. . .Q·p(T)^∗T^∗_L . (3.5)

Note also that according to decomposition (1.3), (X_LH)^∧L(T) =

(X_LH1)^∧L(T). . .(X_LH_p)^∧L(T)

,