SOME FINITE–DIMENSIONAL BACKWARD–SHIFT–INVARIANT SUBSPACES IN THE BALL AND A RELATED INTERPOLATION PROBLEM

DANIEL ALPAY^{∗} and H. TURGAY KAPTANOĞLU

We solve Gleason’s problem in the reproducing kernel Hilbert space with reproducing kernel 1/(1 − ∑_{1}^{N} z_{j}w_{j}^{∗}). We define and study some finite-dimensional resolvent-invariant subspaces that generalize the finite-dimensional de Branges-Rovnyak spaces to the setting of the ball.

1 Introduction

In this paper we study a family of finite-dimensional Hilbert spaces of rational functions analytic in the ball

𝔹_{N} = {(z_{1}, …, z_{N}) | ∑_{1}^{N} |z_{j}|^{2} < 1} ,  (1.1)

whose reproducing kernel is of the form

\frac{I_{n} − B(z)B(w)^{∗}}{1 − zw^{∗}} .  (1.2)

In this expression, z and w are in the unit ball, zw^{∗} = ∑_{1}^{N} z_{j}w_{j}^{∗}, and the function B is rational and matrix-valued. To provide motivation and set the framework, we first consider the case N = 1, that is, the case of the open unit disk 𝔻. First recall that the Hardy space of the open unit disk H_{2}(𝔻) consists of power series f(z) = ∑_{0}^{∞} f_{n}z^{n} with complex coefficients such that

∥f∥_{H_{2}(𝔻)}^{2} =^{def.} ∑_{0}^{∞} |f_{n}|^{2} < ∞  (1.3)

∗This research was supported by a grant from the United States-Israel Binational Science Foundation (BSF), Jerusalem, Israel, and by the Israeli Academy of Sciences.

and is the reproducing kernel Hilbert space with reproducing kernel \frac{1}{1−zw^{∗}}. We recall that this means the following: For every w ∈ 𝔻 the function z ↦ \frac{1}{1−zw^{∗}} belongs to H_{2}(𝔻) and

⟨f(z), \frac{1}{1 − zw^{∗}} ⟩_{H_{2}(𝔻)} = f(w)

for every f ∈ H_{2}(𝔻).

The space H_{2}(𝔻) is invariant under the backward shift (also called resolvent-like) operators R_{a} defined as

R_{a}f(z) = \frac{f(z) − f(a)}{z − a}, z ∈ 𝔻.  (1.4)

We recall that these operators satisfy the resolvent equation

R_{a} − R_{b} = (a − b)R_{a}R_{b}, a, b ∈ 𝔻,  (1.5)

hence their name. Furthermore, the adjoint of the operator R_{a} in H_{2}(𝔻) is given by

R_{a}^{∗}f(z) = \frac{zf(z)}{1 − za^{∗}} .  (1.6)

In particular, R_{0}^{∗} is the shift operator:

R_{0}^{∗}f(z) = zf(z).  (1.7)

The finite-dimensional R_{a}-invariant subspaces of H_{2}(𝔻) are exactly the spaces of the form H_{2}(𝔻) ⊖ bH_{2}(𝔻), where b(z) is a finite Blaschke product, that is,

b(z) = ∏_{1}^{m} \frac{z − a_{j}}{1 − za_{j}^{∗}} , a_{1}, …, a_{m} ∈ 𝔻  (1.8)

(see [20, Lemma 7.35, p. 189]). The space H_{2}(𝔻) ⊖ bH_{2}(𝔻) has reproducing kernel

\frac{1 − b(z)b(w)^{∗}}{1 − zw^{∗}} ,  (1.9)

and more generally, any R_{a}-invariant reproducing kernel Hilbert space M contractively included in H_{2}(𝔻) has a reproducing kernel of the form (1.9), where b(z) is now allowed to be a row vector. If M is finite-dimensional, then b is rational; see [8, Theorem 3.1.2, p. 18].

When one leaves the open unit disk and goes to the setting of several complex variables, at least two choices for the domain are possible, the polydisk and the ball. For the polydisk 𝔻^{N}, the Hardy space H_{2}(𝔻^{N}) is the reproducing kernel Hilbert space with reproducing kernel

∏_{1}^{N} \frac{1}{1 − z_{j}w_{j}^{∗}} ,  (1.10)

where (z_{1}, …, z_{N}) and (w_{1}, …, w_{N}) ∈ 𝔻^{N}. In this case there are N families of resolvent operators

R_{a}^{(j)}f(z) = \frac{f(z) − f(z_{1}, …, z_{j−1}, a, z_{j+1}, …, z_{N})}{z_{j} − a}, a ∈ 𝔻.  (1.11)

Resolvent-invariant subspaces of H_{2}(𝔻^{N}) were studied in [6] for N = 2. Reproducing kernel Hilbert spaces with reproducing kernel of the form

\frac{1 − S(z)S(w)^{∗}}{∏_{1}^{N}(1 − z_{j}w_{j}^{∗})} ,  (1.12)

where S is 1 × m-valued with m possibly ∞, are R_{a}^{(j)}-invariant and are contractively included in H_{2}(𝔻^{N}). One of the main conclusions of [6] is that when N > 1, there is no finite-dimensional nontrivial reproducing kernel Hilbert space with reproducing kernel of the form (1.12). It is also worth noting that the common eigenvectors in H_{2}(𝔻^{N}) are exactly the kernels (1.10).

In this paper we consider the case of the ball 𝔹_{N}. The function \frac{1}{1−zw^{∗}} (with zw^{∗} = ∑_{1}^{N} z_{j}w_{j}^{∗}) is positive in 𝔹_{N} and we will denote by H(𝔹_{N}) the corresponding reproducing kernel Hilbert space (definitions are reminded at the end of the Introduction). Note that this is not the classical Hardy space associated to 𝔹_{N}, which has reproducing kernel \frac{1}{(1−zw^{∗})^{N}} and which we will denote by H^{2}(𝔹_{N}). On the other hand, since we have the inequality

\frac{1}{(1 − zw^{∗})^{N}} − \frac{1}{1 − zw^{∗}} = (zw^{∗}) (\frac{∑_{0}^{N−2}(1 − zw^{∗})^{k}}{(1 − zw^{∗})^{N}} ) = (zw^{∗}) (∑_{0}^{N−2} \frac{1}{(1 − zw^{∗})^{N−k}} ) ≥ 0

in the sense of reproducing kernel spaces, the space H(𝔹_{N}) is contractively included in the Hardy space of the ball H^{2}(𝔹_{N}). As is well known, the inclusion is strict; see [1] and Lemma 2.2 below.

The space H(𝔹_{N}) (even for N = ∞, that is, when one considers the unit ball of ℓ_{2}) has been considered by Agler and McCarthy in [1] in the study of Nevanlinna-Pick interpolation. Operator-valued functions for which the kernel (1.2) is positive have been studied by Ball, Trent, and Vinnikov in [17], and some of our formulas are special cases of the general representation formulas obtained in [17], but the starting point and emphasis in [17] and in the present paper are different. It is easy to check (see Lemma 2.1) that the operators (1.11) are unbounded in H(𝔹_{N}). On the other hand, the operators of multiplication by the variables z_{j} are bounded in H(𝔹_{N}), and we define the backward shift operators as their adjoints in H(𝔹_{N}). This provides the appropriate definition of backward-shift-invariant subspaces of H(𝔹_{N}), which is our starting point. It appears that these backward shift operators are exactly the operators considered in the solution of Gleason’s problem by Ahern and Schneider [2] and by Leibenson [23]. See [29] and Section 3 for the definitions. We refer the reader to [24] (in particular Theorem 5.14, p. 67) for more background material on division theorems.

We also note the following: The kernel k(z, w) = \frac{1}{1−zw^{∗}} is a complete Nevanlinna-Pick kernel, i.e., is such that 1/k(z, w) has one positive square. Let H(k) be the reproducing kernel Hilbert space associated to such a kernel. One knows (see [22], [25]) that subspaces of H(k) invariant under multiplication by the independent variables are of the form BH(k)^{n×1}, where n can be ∞, and B is an inner function in some sense. These results are due to Arveson [16] for the kernel \frac{1}{1−zw^{∗}}. Complete Nevanlinna-Pick kernels have been characterized to be those kernels for which Pick’s interpolation theorem holds; see [28]. This aspect plays an important role in the preprint [7] where we develop the counterpart of the Schur algorithm for Schur multipliers.

The outline of the paper is as follows: In Section 2 we study some properties of the space H(𝔹_{N}). In Section 3 we consider the resolvent operators, and in Section 4 we define Blaschke factors. Section 5 is devoted to homogeneous interpolation in the space H(𝔹_{N}) and in Section 6 we consider homogeneous interpolation in the Hardy space of the ball. Finally in the last section we study finite-dimensional resolvent-invariant spaces of H(𝔹_{N}). Some of the results presented in this paper have been announced in [10].

Some words on notation and definitions. ℂ denotes the complex numbers, and ℂ^{p×q} denotes the set of p × q matrices with complex entries. By a matrix-valued rational function we mean a matrix-valued function whose entries are rational functions, that is, quotients of polynomials in the variables z_{1}, …, z_{N}. We denote by H(𝔹_{N})^{p×q} the Hilbert space of matrix-valued functions F = (f_{ℓ,j}) with entries in H(𝔹_{N}) and with norm

(∑_{ℓ,j} ∥f_{ℓ,j}∥_{H(𝔹_{N})}^{2})^{1/2} .

Usually we have p or q equal to 1. We denote by the symbol T^{∗} the adjoint of a bounded operator T between Hilbert spaces, or the adjoint of a matrix, or complex conjugation. Finally, we note that the paper relies heavily on the theory of positive functions and of the associated reproducing kernel Hilbert spaces. We refer to [15], [31], [32] for the main definitions and theorems. We use the noation K(z, w) ≥ 0 for the fact that a function

K(z, w) : 𝔹_{N} × 𝔹_{N} ⟶ ℂ^{n×n}

is positive; that is, it satisfies K(z, w) = K(w, z)^{∗} (i.e., it is hermitian), and it is such that for all m ∈ ℕ and all choices of w_{1}, …, w_{m} ∈ 𝔹_{N} the m×m block hermitian matrix with ℓ, j block K(w_{j}, w_{ℓ}) is nonnegative (i.e., has all its eigenvalues positive or equal to 0). Associated to such a function is a unique reproducing kernel Hilbert space of functions 𝔹_{N} → ℂ^{m×1}, which we will denote by H(K), that is characterized by the following properties: For every vector ξ ∈ ℂ^{m×1} and every point w ∈ 𝔹_{N}, the function z ↦ K(z, w)ξ belongs to H(K) and

⟨f(z), K(z, w)ξ⟩_{H(K)} = ξ^{∗}f(w) for all f ∈ H(K).

2 The space H(𝔹_{N})

Since the kernel \frac{1}{1−zw^{∗}} is analytic in each z_{j} and each w_{j}^{∗}, it follows from a general theorem (see e.g. [14, p. 43]) that the elements of H(𝔹_{N}) are analytic in 𝔹_{N}. We now give an explicit description of the elements of H(𝔹_{N}). We first remark that

\frac{1}{1 − zw^{∗}} = ∑_{0}^{∞}(zw^{∗})^{n} = ∑_{0}^{∞} ∑_{\begin{matrix}(n_{1},... ,n_{N})∈ℕ^{N} \\ n_{1}+···+n_{N} =n \end{matrix}} \frac{n!}{n_{1}!n_{2}! ⋯ n_{N}!}(z_{1}w_{1}^{∗})^{n_{1}}(z_{2}w_{2}^{∗})^{n_{2}} ⋯ (z_{N}w_{N}^{∗})^{n_{N}} .

Thus

H(𝔹_{N}) = \{f(z_{1}, z_{2}, …, z_{N}) = ∑_{0}^{∞} ∑_{\begin{matrix}(n_{1},... ,n_{N})∈ℕ^{N} \\ n_{1}+···+n_{N} =n \end{matrix}} a_{n_{1},n_{2},... ,n_{N}} z_{1}^{n_{1}}z_{2}^{n_{2}} ⋯ z_{N}^{n_{N}} \}  (2.1)

with norm

\sqrt{∑_{0}^{∞} ∑_{\begin{matrix}(n_{1},... ,n_{N})∈ℕ^{N} \\ n_{1}+···+n_{N} =n \end{matrix}} \frac{|a_{n_{1},n_{2},... ,n_{N}} |^{2}}{\frac{n!}{n_{1}!n_{2}!···n_{N}!}}} .  (2.2)

and inner product

⟨z_{1}^{n_{1}} ⋯ z_{N}^{n_{N}} , z_{1}^{m_{1}} ⋯ z_{N}^{m_{N}} ⟩_{H_{2}(𝔹_{N})} = \begin{cases}\frac{n_{1}!n_{2}!···n_{N}!}{n!}, if n_{j} = m_{j}, j = 1, …, N, \\ 0, otherwise. \end{cases}  (2.3)

To prove these, we just verify that the Hilbert space defined by (2.1) and inner product associated to the norm (2.2) has indeed reproducing kernel \frac{1}{1−zw^{∗}}.

Specializing for simplicity to N = 2, this characterization allows us to show the following.

Lemma 2.1 The operator that associates to f the function

\frac{f(z_{1}, z_{2}) − f(z_{1}, 0)}{z_{2}}

is unbounded in H(𝔹_{2}).

Proof: Take the function f(z_{1}, z_{2}) = z_{1}^{n}z_{2}. Then

\frac{f(z_{1}, z_{2}) − f(z_{1}, 0)}{z_{2}} = z_{1}^{n},

and using (2.3),

\frac{∥\frac{f(z_{1},z_{2})−f(z_{1},0)}{z_{2}}∥_{H(𝔹_{2})}^{2}}{∥f(z_{1}, z_{2})∥_{H(𝔹_{2})}^{2}} = \frac{1}{\frac{n!1!}{(n+1)!}} = n + 1 → ∞ as n → ∞.

□

Another consequence of (2.2) and (2.3) is:

Lemma 2.2 The space H(𝔹_{N}) is strictly included in the Hardy space of the ball.

Proof: The function

f(z_{1}, …, z_{N}) = ∑_{k} \frac{z_{1}^{k}}{\sqrt{k + 1}}

belongs to the Hardy space of the ball and not to H(𝔹_{N}). Indeed,

∑_{0}^{∞} \frac{1}{k + 1}∥z_{1}^{k}∥_{H(𝔹_{N})}^{2} = ∑_{0}^{∞} \frac{1}{k + 1} = ∞.

On the other hand, the functions z_{1}^{k} are also orthogonal in H^{2}(𝔹_{N}) and ∥z_{1}^{k}∥_{H^{2}(𝔹_{N})}^{2} = \frac{(N−1)!k!}{(N−1+k)!} using the definiton in [29, p. 84]. Thus

∑_{0}^{∞} \frac{1}{k + 1}∥z_{1}^{k}∥_{H^{2}(𝔹_{N})}^{2} = ∑_{0}^{∞} \frac{1}{k + 1} · \frac{(N − 1)!k!}{(N − 1 + k)!} < ∞

since

\frac{(N − 1)!k!}{(N − 1 + k)!} = \frac{2 · 3 ⋯ k}{N(N + 1) ⋯ (N + k − 2)} · \frac{1}{N + k − 1} ≤ \frac{1}{N + k − 1}.

□

Before proceeding on we need a definition and a result from the theory of reproducing kernel Hilbert spaces. Let K(z, w) be a positive function on the ball with the associated reproducing kernel Hilbert space H(K). The function φ : 𝔹_{N} → ℂ is called a multiplier if the operator M_{φ} of multiplication by φ is bounded in H(K). The function φ is a multiplier if and only if there is a positive constant C such that the function (C − φ(z)φ(w)^{∗})K(z, w) is positive in the ball; if C = 1, the operator M_{φ} is a contraction; see e.g. [18, Theorem 2.11]. When N = 1, the multipliers of H(𝔹_{1}) (that is, of the Hardy space of the open unit disk) are exactly the functions analytic and contractive in 𝔻 (i.e., Schur functions). When N > 1, the class of functions for which the operator of multiplication by the function is a contraction on H(𝔹_{N}) is smaller than the class of Schur functions, the unit ball of H^{∞}(𝔹_{N}). A whole family of Schur functions that are not Schur multipliers can be constructed using the following idea due to Rudin; see [29, p. 164]. Define numbers c_{j} via 1 − \sqrt{1 − t} = ∑_{j} c_{j}t^{j}, where |t| < 1. Then all c_{j} > 0, and

p_{m}(z_{1}, …, z_{N}) = z_{1} + c_{1}z_{2}^{2} + c_{2}z_{2}^{4} + ⋯ + c_{m}z_{2}^{2m}

is a Schur function. Now ∥1∥_{H(𝔹_{N})} = 1 and so the norm of the operator of multiplication by p_{m} on H(𝔹_{N}) is at least ∥p_{m}∥_{H(𝔹_{N})}. But in view of (2.3),

∥p_{m}∥_{H(𝔹_{N})}^{2} = ∥z_{1}∥_{H(𝔹_{N})}^{2} + c_{1}^{2}∥z_{2}^{2}∥_{H(𝔹_{N})}^{2} + ⋯ + c_{m}^{2}∥z_{2}^{2m}∥_{H(𝔹_{N})}^{2} = 1 + c_{1}^{2} + ⋯ + c_{m}^{2} > 1.

Thus the norm of the operator of multiplication by p_{m} on H(𝔹_{N}) is strictly bigger than 1. The case m = 1 of these examples reduces to the example of Misra [26, Example 4.4, p. 834] who first saw the difference between the two classes using different methods.

Proposition 2.3 Let a ∈ 𝔹_{N}. The multiplication operator M_{za^{∗}} in H(𝔹_{N}) is a strict contraction whose norm is equal to \sqrt{aa^{∗}}, and the operator of multiplication by \frac{1}{1−za^{∗}} is bounded in H(𝔹_{N}).

Proof: We first write

\frac{aa^{∗} − (za^{∗})(aw^{∗})}{1 − zw^{∗}} = aa^{∗} + \frac{z ((aa^{∗})I_{N} − a^{∗}a) w^{∗}}{1 − zw^{∗}} .

Note that the matrix (aa^{∗})I_{N} − a^{∗}a is positive, and so the function z ((aa^{∗})I_{N} − a^{∗}a) w^{∗} is positive in the ball. Since a product or a sum of positive functions is positive, we have

\frac{z ((aa^{∗})I_{N} − a^{∗}a) w^{∗}}{1 − zw^{∗}} ≥ 0

By a result on multipliers mentioned above, it follows that ∥M_{za^{∗}}∥ ≤ \sqrt{aa^{∗}}. To see that the norm is exactly \sqrt{aa^{∗}}, notice that the constants belong to H(𝔹_{N}) since they are just multiples of the function identically equal to 1, which is the kernel corresponding to w = 0. Thus by (2.2),

∥M_{za^{∗}}(1)∥_{H(𝔹_{N})}^{2} = ∥ ∑_{1}^{n} z_{j}a_{j}^{∗}∥_{H(𝔹_{N})}^{2} = ∑_{j} |a_{j}|^{2} = aa^{∗}

The second claim drops from the inequality ∥M_{\frac{1}{1−za^{∗}}} ∥ ≤ \frac{1}{1−∥M_{za^{∗}}∥}, which in turn follows from the power series expansion

\frac{1}{1 − za^{∗}} = ∑_{0}^{∞}(za^{∗})^{n}.

□

For related computations we refer to [11]. As a corollary, we have the following.

Corollary 2.4 Let a ∈ 𝔹_{N}. The operators T_{j}(a) defined by

(T_{j}(a)) f(z) = \frac{z_{j}f(z)}{1 − za^{∗}}  (2.4)

are bounded.

Using the theory of Hilbert spaces of analytic functions on bounded symmetric domains, one can find integral formulas for the inner product of H(𝔹_{N}). They are presented in [9].

3 The resolvent operators

When N = 1, that is, in the case of the disk, the adjoint of the operator T_{1}(0) (defined by (2.4)) is just the backward shift operator (1.4). When N > 1, the natural analogues of these operators would seem to be the operators (1.11), but they are unbounded, as we showed above. On the other hand, the adjoints of the operators T_{j}(a) are of course bounded. These will be our backward shift operators here.

Definition 3.1 The operator T_{j}(a)^{∗} will be called the j-th backward shift operator in H(𝔹_{N}) at the point a.

These operators in general do not satisfy the resolvent equation (1.5). They do when one restricts a to have only one fixed nonzero component. We cannot give a closedform expression for the backward shift operators acting on arbitrary f ∈ H(𝔹_{N}). On the other hand, as is usual in reproducing kernel spaces, we can compute the adjoint on the kernels. This is a special case of a general theorem on multipliers in reproducing kernel Hilbert spaces (see e.g. [4, formula (2.3.5), p. 34]), but we give a proof for completeness.

Lemma 3.2 We have

(T_{j}(a)^{∗}) (\frac{1}{1 − zw^{∗}} ) = \frac{w_{j}^{∗}}{1 − aw^{∗}} \frac{1}{1 − zw^{∗}}  (3.1)

Proof: Let v be in the ball. Then

(T_{j}(a)^{∗} \frac{1}{1 − zw^{∗}} ) (v) = ⟨T_{j}(a)^{∗} \frac{1}{1 − zw^{∗}} , \frac{1}{1 − zv^{∗}} ⟩_{H(𝔹_{N})}
= ⟨\frac{1}{1 − zw^{∗}} , T_{j}(a)\frac{1}{1 − zv^{∗}} ⟩_{H(𝔹_{N})}
= ⟨\frac{1}{1 − zw^{∗}} , \frac{z_{j}}{1 − za^{∗}} \frac{1}{1 − zv^{∗}} ⟩_{H(𝔹_{N})}
= (\frac{w_{j}}{1 − wv^{∗}} \frac{1}{1 − wa^{∗}} )^{∗}
= \frac{w_{j}^{∗}}{(1 − vw^{∗})(1 − aw^{∗})}.
□

The main result of this section is the following.

Theorem 3.3 Let f ∈ H(𝔹_{N}). Then there exist functions g_{1}(z), …, g_{N}(z) ∈ H(𝔹_{N}) such that

f(z) − f(a) = ∑_{1}^{N}(z_{j} − a_{j})g_{j}(z).  (3.2)

We define the functions g_{j}(z) in a unique way by the formula g_{j}(z) = T_{j}(a)^{∗}f. The decomposition is not unique in general if one does not impose further restrictions on the g_{j}(z), as is illustrated by the example

1 = 1 + z_{1}(0) + z_{2}(0)
= 1 + z_{1}(z_{2}) + z_{2}(−z_{1}),

which holds since the functions z_{1} and z_{2} and the constant functions belong to H(𝔹_{N}).

Proof of Theorem 3.3: We first consider f(z) to be a kernel, i.e., f(z) = \frac{1}{1−zw^{∗}}. Let g_{j} = T_{j}(a)^{∗}f. Then

f(z) − f(a) = \frac{1}{1 − zw^{∗}} − \frac{1}{1 − aw^{∗}}
= ∑_{j} \frac{1}{(1 − zw^{∗})(1 − aw^{∗})}(z_{j} − a_{j})w_{j}^{∗}
= ∑_{j}(z_{j} − a_{j})g_{j}(z).

Thus the equality

f(z) = f(a) + ∑_{j}(z_{j} − a_{j})(T_{j}(a)^{∗}f)(z)  (3.3)

holds on the linear span of the kernel functions, and hence on all of H(𝔹_{N}) by continuity.

□

We can rewrite (3.3) as

∑_{j} M_{z_{j}−a_{j}}T_{j}(a)^{∗} = I − C_{a}  (3.4)

where C_{a} is the evaluation at the point a.

The problem of existence of such functions g_{j} in a given domain and in a given class of functions is called Gleason’s problem. See [29, p. 116] for a list of spaces where this problem has a solution. Rudin gives two formulas for the g_{j}(z) in various spaces, and in particular in the Hardy space of the ball. The first formula is due to Ahern and Schneider (see [2]) and is valid in the space H^{1}(𝔹_{N}) associated to the ball, and is equal to

g_{j}(z) = ∫_{S} \frac{C(z, u) − C(a, u)}{(z − a)u^{∗}}u_{j}^{∗}f(u)dσ(u).  (3.5)

In this expression, C(z, u) = (1 − zu^{∗})^{−N} is the Cauchy kernel, S is the boundary of the ball and σ is the Lebesgue measure on S such that σ(S) = 1; see [29, pp. 12–13]. The second formula holds for a = 0, is valid for functions analytic in the ball and continuous up to the boundary (i.e., in the ball algebra), and is due to Leibenson (see [23]). The g_{j}(z) are now given (when a = 0) by

g_{j}(z) = ∫_{0}^{1} \frac{∂}{∂z_{j}} f(tz)dt.  (3.6)

A long computation using Propositions 1.4.8 and 1.4.9 of [29] shows that the Ahern-Schneider solution coincides with our solution for any a ∈ 𝔹_{N} for N = 2. The computations for N > 2 and a ≠ 0 seem similar but longer. On the other hand, for a = 0 the situation is simpler and we have the following.

Lemma 3.4 Let N > 1 and a = 0. The operator T_{j}(0)^{∗} coincides with the formulas of Ahern and Schneider and of Leibenson on H(𝔹_{N}).

Proof: We first consider an element of the form f(z) = \frac{1}{1−zw^{∗}}. Since it is in the ball algebra, it follows that the formulas of Ahern and Schneider and of Leibenson coincide; see [29, §6.6.3, p. 117]. We now check that for such f

∫_{0}^{1} \frac{∂}{∂z_{j}} f(tz)dt = \frac{w_{j}^{∗}}{1 − zw^{∗}} .

But

\frac{∂}{∂z_{j}} (\frac{1}{1 − zw^{∗}} ) (tz) = \frac{w_{j}^{∗}}{(1 − tzw^{∗})^{2}} ,

and so

∫_{0}^{1} \frac{∂}{∂z_{j}} f(tz)dt = ∫_{0}^{1} \frac{w_{j}^{∗}}{(1 − tzw^{∗})^{2}} dt = \frac{w_{j}^{∗}}{1 − zw^{∗}} .

The result follows by continuity. □

We conclude with some further properties of the operators T_{j}(a)^{∗}.

Proposition 3.5 It holds that T_{j}(a)^{∗}(1) = 0 and

T_{j}(a)^{∗}z_{i} = δ_{ij},

where δ_{ij} denotes the Kronecker symbol.

Proof: Let v and w in the ball. We have

(T_{j}(a)^{∗}1)(v) = ⟨T_{j}(a)^{∗}1, \frac{1}{1 − zv^{∗}} ⟩_{H(𝔹_{N})}
= ⟨1, \frac{z_{j}}{(1 − za^{∗})(1 − zv^{∗})}⟩_{H(𝔹_{N})}
= (\frac{z_{j}}{(1 − za^{∗})(1 − zv^{∗})}|_{z=0})^{∗} = 0.

Similarly,

(T_{j}(a)^{∗}z_{ℓ})(v) = ⟨T_{j}(a)^{∗}z_{ℓ}, \frac{1}{1 − zv^{∗}} ⟩_{H(𝔹_{N})}
= ⟨z_{ℓ}, \frac{z_{j}}{(1 − za^{∗})(1 − zv^{∗})}⟩_{H(𝔹_{N})}
= ⟨\frac{∂}{∂w_{ℓ}^{∗}} (\frac{1}{1 − zw^{∗}} ) |_{w=0}, \frac{z_{j}}{(1 − za^{∗})(1 − zv^{∗})}⟩_{H(𝔹_{N})}
= (\frac{∂}{∂z_{ℓ}} \frac{z_{j}}{(1 − za^{∗})(1 − zv^{∗})}) |_{z=0}

and the result follows. □

Proposition 3.6 The operators T_{j}(a)^{∗} commute.

Proof: It is enough to prove that their adjoints commute on a dense set, but this is clear from

T_{j_{1}}(a)T_{j_{2}}(a)f = \frac{z_{j_{1}}z_{j_{2}}}{(1 − za^{∗})^{2}} f(z)

for f(z) = \frac{1}{1−zw^{∗}}. □

Proposition 3.7 The common eigenvectors of the resolvent operators T_{j}(a)^{∗} in H(𝔹_{N}) are exactly the kernels \frac{1}{1−zw^{∗}}.

Proof: Let f ∈ H(𝔹_{N}) such that T_{j}(a)^{∗}f = λ_{j}f for possibly different complex numbers λ_{j}. Then it follows from (3.3) that f(z)(1 − ∑_{j} λ_{j}(z_{j} − a_{j})) = f(a), that is,

f(z) = \frac{f(a)}{1 + ∑_{j} λ_{j}a_{j} − ∑_{j} λ_{j}z_{j}} .

As already remarked, f is analytic in the ball and in particular in a neighborhood of the origin, and so 1 + ∑_{j} λ_{j}a_{j} ≠ 0. Thus f(z) = \frac{c}{1−zw^{∗}} with c = \frac{f(a)}{1+∑_{j} λ_{j}a_{j}} and

w^{∗} = \frac{1}{1 + ∑_{j} λ_{j}a_{j}} ( λ_{1} λ_{2} ⋯ λ_{N} ).

If |w| ≥ 1, the function f would not be in H(𝔹_{N}). So |w| < 1 and f is a kernel. □

We have already remarked that a similar result holds in the setting of the polydisk. It is of interest to note that it also holds in the setting of de Branges-Rovnyak spaces defined on compact real Riemann surfaces (see [13], [12]).

4 Blaschke factors

It is well known that one can extract zeros of functions in H_{2}(𝔻) via Blaschke factors. The counterpart here is the following.

Proposition 4.1 Let a ∈ 𝔹_{N}. Then the ℂ^{1×N}-valued function

b_{a}(z) = \frac{(1 − aa^{∗})^{1/2}}{1 − za^{∗}} (z − a)(I_{N} − a^{∗}a)^{−1/2}  (4.1)

satisfies

\frac{1 − b_{a}(z)b_{a}(w)^{∗}}{1 − zw^{∗}} = \frac{1 − aa^{∗}}{(1 − za^{∗})(1 − w^{∗}a)}, z, w ∈ 𝔹_{N}.  (4.2)

In particular,

b_{a}(z)b_{a}(z)^{∗} \begin{cases}< 1, if zz^{∗} < 1, \\ = 1, if zz^{∗} = 1. \end{cases}  (4.3)

Lastly, b_{a} belongs to H(𝔹_{N})^{1×N}.

Formula (4.2) appears in Rudin’s book on the ball [29, Theorem 2.2.2, p. 26] (with an apparently different choice of b_{a}, but in fact, up to sign, the same; see Lemma 4.2). It expresses the fact that the one-dimensional vector space spanned by the function z ↦ \frac{1}{1−za^{∗}} endowed with the metric of H(𝔹_{N}) has reproducing kernel of the form \frac{1−b_{a}(z)b_{a}(w)^{∗}}{1−zw^{∗}}. That formula in Rudin’s book was the trigger that made us understand that part of the analysis of H_{2}(𝔻) could be extended to the space H(𝔹_{N}).

We also note that (4.1) can be rewritten as

b_{a}(z) = b_{a}(0) + b_{a}(z) − b_{a}(0)
= −(1 − aa^{∗})^{1/2}a(I_{n} − a^{∗}a)^{−1/2} + \frac{z}{1 − za∗} ((1 − aa^{∗})^{1/2}(I_{N} − a^{∗}a)^{−1/2} − a^{∗}a)
= −a + (1 − aa^{∗})^{1/2}z(1 − za^{∗})^{−1}(I_{N} − a^{∗}a)^{1/2}.  (4.4)

To get to this last formula we first use that

(1 − aa^{∗})^{1/2}a = a(I_{N} − a^{∗}a)^{1/2},

and so (1 − aa^{∗})^{1/2}a(I_{N} − a^{∗}a)^{−1/2} = a. Next we have to check that

(1 − aa^{∗})^{1/2}(I_{N} − a^{∗}a)^{−1/2} − a^{∗}a = (1 − aa^{∗})^{1/2}(I_{N} − a^{∗}a)^{1/2}.

This is readily done by multiplying both sides of this equality by (I_{N} − a^{∗}a)^{1/2} on the right and making use of

(1 − aa^{∗})^{1/2}a^{∗}a = a^{∗}(1 − aa^{∗})^{1/2}a = a^{∗}a(I_{N} − a^{∗}a)^{1/2}.  (4.5)

The matrix

( \begin{matrix}a^{∗} & (I_{N} − a^{∗}a)^{1/2} \\ (1 − aa^{∗})^{1/2} & −a \end{matrix} )

is unitary and formula (4.4) is the dimension-one case of the general representation formula (2.3) in [17, §2].

Proof of Proposition 4.1: For a ∈ 𝔹_{N}, let H(a) denote the Halmos extension of a, that is, let H(a) be the matrix

H(a) = ( \begin{matrix}\frac{1}{(1−aa^{∗})^{1/2}} & −a(I_{N} − a^{∗}a)^{−1/2} \\ −\frac{a^{∗}}{(1−aa^{∗})^{1/2}} & (I_{N} − a^{∗}a)^{−1/2} \end{matrix} ) .  (4.6)

Then

\begin{aligned}( 1 z ) H(a) & = ( \frac{1−za^{∗}}{(1−aa^{∗})^{1/2}} (z − a)(I_{N} − aa^{∗})^{−1/2} ) \\ & = \frac{1 − za^{∗}}{(1 − aa^{∗})^{1/2}} ( 1 b_{a}(z) ) . \end{aligned}  (4.7)

Let

J = ( \begin{matrix}1 & 0 \\ 0 & −I_{N} \end{matrix} ) .  (4.8)

One has

H(a)JH(a)^{∗} = J

(see [21, Theorem 1.2, p. 17]) and therefore, for any choice of z, w ∈ 𝔹_{N}, we have

1 − zw^{∗} = ( 1 z ) J ( \begin{matrix}1 \\ w^{∗} \end{matrix} )
= ( 1 z ) H(a)JH(a)^{∗} ( \begin{matrix}1 \\ w^{∗} \end{matrix} )
= \frac{(1 − za^{∗})(1 − aw^{∗})}{1 − aa^{∗}}(1 − b_{a}(z)b_{a}(w)^{∗})

and hence the result.

The last claim follows from the fact that \frac{1}{1−za^{∗}} belongs to H(𝔹_{N}) and that multiplication by the independent variables are bounded operators in H(𝔹_{N}). □

The function b_{a}(z) is the counterpart in the setting of 𝔹_{N} of the Blaschke factor \frac{z−a}{1−za^{∗}}. We now give another expression for b_{a}(z) which appears in [29].

Lemma 4.2 The function b_{a}(z) can be written as

b_{a}(z) = −\frac{a − \frac{za^{∗}}{aa^{∗}} a − (1 − aa^{∗})^{1/2}(z − \frac{za^{∗}}{aa^{∗}} a)}{1 − za^{∗}}.  (4.9)

Proof: We rewrite the right side of (4.9) as

−a + z \frac{\frac{a^{∗}a}{aa^{∗}} + (1 − aa^{∗})^{1/2}(I_{N} − \frac{a^{∗}a}{aa^{∗}}) − aa^{∗}}{1 − za^{∗}} .

So, in view of formula (4.4), we have to check that

(1 − aa^{∗})^{1/2}(I_{N} − a^{∗}a)^{1/2} = \frac{a^{∗}a}{aa^{∗}} + (1 − aa^{∗})^{1/2}(I_{N} − \frac{a^{∗}a}{aa^{∗}}) − a^{∗}a.  (4.10)

Taking into account (4.5), we see that (4.10) is equivalent to

(1 − aa^{∗})^{1/2} ((I_{N} − a^{∗}a)^{1/2} − I_{N}) = \frac{a^{∗}a}{aa^{∗}} − \frac{a^{∗}a}{aa^{∗}}(I_{N} − a^{∗}a)^{1/2} − a^{∗}a.  (4.11)

By looking at the power series expansion in powers of a^{∗}a of (I_{N} − a^{∗}a)^{1/2}, we see that both sides of (4.11) are multiples of a^{∗} and so it is legitimate to multiply both sides of (4.11) by a on the left and get the equivalent expression

a(1 − aa^{∗})^{1/2} ((I_{N} − a^{∗}a)^{1/2} − I_{N}) = a (\frac{a^{∗}a}{aa^{∗}} − \frac{a^{∗}a}{aa^{∗}}(I_{N} − a^{∗}a)^{1/2} − a^{∗}a)
= a − a(I_{N} − a^{∗}a)^{1/2} − (aa^{∗})a.  (4.12)

This in turn is readily seen after remarking that

a(1−aa^{∗})^{1/2}(I_{N}−a^{∗}a)^{1/2} = (1−aa^{∗})^{1/2}a(I_{N}−a^{∗}a)^{1/2} = a(I_{N}−a^{∗}a)^{1/2}(I_{N}−a^{∗}a)^{1/2} = a(I_{N}−a^{∗}a)

and that

a(1 − aa^{∗})^{1/2} = (1 − aa^{∗})^{1/2}a = a(I_{N} − a^{∗}a)^{1/2}.
□

The right-side of (4.9) is an automorphism of the ball, and is the expression used in [29] to prove (4.2). In fact, Lemma 4.2 implies equation (4.2) by Theorem 2.2.5 of [29]. This result is also the case M = 1 of the structure Theorem 7.2.

The analogy of b_{a}(z) with Blaschke factors is also stressed by the following result.

Proposition 4.3 Let a ∈ 𝔹_{N} and let f ∈ H(𝔹_{N}). Then

f(a) = 0 ⟺ f(z) = b_{a}(z)u(z)

for some u ∈ H(𝔹_{N})^{N×1}.

Proof: The ⟸ direction is trivial. Conversely, assume that f(a) = 0. By Theorem 3.3 we can write

f(z) − f(a) = f(z) = ∑_{1}^{N}(z_{j} − a_{j})g_{j}(z) = (z − a)g(z),

where

g(z) = ( g_{1}(z) ; ⋮ ; g_{N}(z) ) ∈ H(𝔹_{N})^{N×1}.

So

f(z) = \frac{(1 − aa^{∗})^{1/2}}{1 − za^{∗}} (z − a)(I_{N} − a^{∗}a)^{−1/2}(I_{N} − a^{∗}a)^{1/2} \frac{1 − za^{∗}}{(1 − aa^{∗})^{1/2}} g(z) = b_{a}(z)u(z)

with

u(z) = \frac{1 − za^{∗}}{(1 − aa^{∗})^{1/2}} (I_{N} − a^{∗}a)^{1/2}g(z) ∈ H(𝔹_{N})^{N×1},

since by Proposition 2.3 the function (za^{∗})g(z) belongs to H(𝔹_{N})^{N×1}. □

Proposition 4.4 Let n ∈ ℕ, let d ∈ ℂ^{n×1} and let a ∈ 𝔹_{N}. The one-dimensional subspace of H(𝔹_{N})^{n×1} spanned by the function

z ↦ \frac{d}{1 − za^{∗}}  (4.13)

has reproducing kernel of the form

\frac{I_{n} − B(z)B(w)^{∗}}{1 − zw^{∗}}  (4.14)

where B(z) is a rational ℂ^{n×(n+N−1)}-valued function taking coisometric values on the boundary of the ball. Here coisometric means B(z)B(z)^{∗} = I_{n} for zz^{∗} = 1.

Proof: We first consider the special case

d_{0} = ( 1 0 ⋮ 0 ) .

From Proposition 4.1 we have

\frac{1 − aa^{∗}}{(1 − za^{∗})(1 − w^{∗}a)}d_{0}d_{0}^{∗} = \frac{I_{n} − ( \begin{matrix}b_{a}(z)b_{a}(w)^{∗} & 0 \\ 0 & I_{n−1} \end{matrix} )}{1 − zw^{∗}} = \frac{I_{n} − B_{0}(z)B_{0}(w)^{∗}}{1 − zw^{∗}} ,

where

B_{0}(z) = ( \begin{matrix}b_{a}(z) & 0_{1×(n−1)} \\ 0_{(n−1)×N} & I_{n−1} \end{matrix} ) ,  (4.15)

in which, e.g., 0_{1×(n−1)} denotes the 1 × (n − 1) matrix with all entries equal to 0.

For the general case, without loss of generality, we can assume that d^{∗}d = 1. Then consider d as the first column of a n × n unitary matrix U, that is,

d = U ( 1 0 ⋮ 0 ) .

We then obtain the result by multiplying the equality

\frac{d_{0}d_{0}^{∗}(1 − |a|^{2})}{(1 − za^{∗})(1 − w^{∗}a)} = \frac{I_{n} − B_{0}(z)B_{0}(w)^{∗}}{1 − zw^{∗}}

by U on the left and by U^{∗} on the right, and get

B(z) = U ( \begin{matrix}b_{a}(z) & 0_{1×(n−1)} \\ 0_{(n−1)×N} & I_{n−1} \end{matrix} ) .  (4.16)
□

The tangential version of Proposition 4.3 is the following.

Proposition 4.5 Let 0 ≠ c ∈ ℂ^{1×n}, a ∈ 𝔹_{N}, and let f ∈ H(𝔹_{N})^{n×1}. Then

cf(a) = 0 ⟺ f(z) = B(z)u(z),

where B is as in (4.16), and u is an arbitrary element of H(𝔹_{N})^{(N+n−1)×1}.

Before turning to the proof, we recall some facts on the sum of positive functions and on the associated reproducing kernel Hilbert spaces. If K_{1}(z, w) and K_{2}(z, w) are two ℂ^{n×n}-valued functions positive in the ball, their sum is of course still positive. The associated reproducing kernel Hilbert spaces of ℂ^{n×1}-valued functions satisfy

H(K) = H(K_{1}) + H(K_{2}),

but this decomposition is not in general orthogonal. The spaces H(K_{1}) and H(K_{2}) are contractively included in H(K); see [19]. When H(K_{1}) is isometrically included in H(K), so is H(K_{2}), and the sum is an orthogonal sum. One can see this, if need be, from e.g. [4, (2.3.8), p. 36].

Proof of Proposition 4.5: We first check the ⟸ direction. We construct U from c as in the proof of Proposition 4.4. By the choice of U we have

c U = ( 1 0 ⋯ 0 ) ,

and so

cf(a) = cB(a)u(a)
= ( 1 0 ⋯ 0 ) ( \begin{matrix}b_{a}(a) & 0_{1×(n−1)} \\ 0_{(n−1)×N} & I_{n−1} \end{matrix} ) u(a)
= ( 1 0 ⋯ 0 ) ( \begin{matrix}0 & 0_{1×(n−1)} \\ 0_{(n−1)×N} & I_{n−1} \end{matrix} ) u(a)
= 0u(a) = 0.

Conversely, assume that cf(a) = 0. Thus the function f is orthogonal in H(𝔹_{N})^{n×1} to the function (4.13). Let M denote the subspace of H(𝔹_{N})^{n×1} spanned by the function (4.13). The reproducing kernel of M is, as already seen, of the form \frac{I_{n}−B(z)B(w)^{∗}}{1−zw^{∗}}, where B is defined by (4.16). Since

\frac{I_{n}}{1 − zw^{∗}} = \frac{I_{n} − B(z)B(w)^{∗}}{1 − zw^{∗}} + \frac{B(z)B(w)^{∗}}{1 − zw^{∗}}

it follows from the discussion before the beginning of the proof that the orthogonal complement of M has reproducing kernel \frac{B(z)B(w)^{∗}}{1−zw^{∗}}. But this means that the elements of M^{⊥} are of the form B(z)u(z), where u belongs to the closed linear span of the functions \frac{B(w)^{∗}ξ}{1−zw^{∗}} in H(𝔹_{N})^{n×1} as w runs through the ball and ξ runs through ℂ^{n+N−1}. □

When N = 1, the function B(z) is square and B(z)U^{∗} reduces to the matrix-valued Blaschke factor

(I_{n} − \frac{c^{∗}c}{cc^{∗}}) + \frac{c^{∗}c}{cc^{∗}} \frac{z − a}{1 − za^{∗}} , c ∈ ℂ^{1×n} a ∈ 𝔻

introduced by Potapov; see [27], [21].

5 Homogeneous interpolation in H(𝔹_{N})^{n×1}

We wish to solve the following interpolation problem.

Problem 5.1 Given a_{1}, …, a_{M} ∈ 𝔹_{N} and vectors c_{1}, …, c_{M} ∈ ℂ^{1×n} different from 0_{1×n}, find all f ∈ H(𝔹_{N})^{n×1} such that

c_{j}f(a_{j}) = 0, j = 1, …, M.

Theorem 5.2 The set of solutions of Problem 5.1 is of the form

f(z) = B(z)u(z)

where B(z) is a rational ℂ^{n×(n+k(N−1))}-valued function for some integer k ≤ M taking coisometric values on the boundary of 𝔹_{N}, and u is an arbitrary element in H(𝔹_{N})^{(n+k(N−1))×1}.

Proof: We solve the problem by induction. First let M = 1. We use Proposition 4.5 and find that f(z) = B_{1}(z)f_{1}(z) where B_{1}(z) is given by (4.16) and f_{1} is an arbitrary function in H(𝔹_{N})^{(N+n−1)×1}. Assume now the result proved up to M_{0}. So all the solutions are of the form f(z) = B_{M_{0}}(z)u(z), where B_{M_{0}}(z) is rational with coisometric values on the boundary of the ball and is ℂ^{n×n_{M_{0}}}-valued for some n_{M_{0}}. If c_{M_{0}+1}B_{M_{0}}(a_{M_{0}+1}) = 0, the next interpolation condition is empty. Otherwise, the next interpolation condition becomes

(c_{M_{0}+1}B_{M_{0}}(a_{M_{0}+1})) u(a_{M_{0}+1}) = 0,

and the set of all solutions u is given by

u(z) = B_{M_{0}+1}(z)v(z),

where B_{M_{0}+1}(z) is ℂ^{n_{M_{0}}×(n_{M_{0}}+N−1)}-valued and given by (4.16). This concludes the induction with

B_{M_{0}+1}(z) = B_{M_{0}}(z)B_{M_{0}+1}(z).
□

One could in the same way handle tangential interpolation problems with derivatives, i.e., problems of the type

c_{0}f(a_{0}) = 0
c_{0} \frac{∂f}{∂z_{1}} (a_{0}) + c_{1}f(a_{0}) = 0
⋮
c_{0} \frac{∂^{n}f}{∂z_{1}^{n}} (a_{0}) + ⋯ + c_{n}f(a_{0}) = 0,

where the c_{j} are in ℂ^{1×m} and f ∈ H(ℍ_{N})^{1×m}. When N = 1, i.e., in the case of the disk, the function B(z) is square and one recovers part of the results of [5].

6 Homogeneous interpolation in the Hardy space of the ball

In this section we only remark that the analysis in the previous section holds in fact in the Hardy space (H^{2}(𝔹_{N}))^{n×1} of the ball; the g_{j}(z) are now given by the formulas of Ahern and Schneider or of Leibenson. The Blaschke factor b_{a}(z) is in the ball algebra A(𝔹_{N}), and in particular is a multiplier of the Hardy space. There is an important difference between the case of the space H(𝔹_{N}) and of the Hardy space. In (H(𝔹_{N}))^{n×1} we saw in the previous section that the decomposition

\frac{I_{n}}{1 − zw^{∗}} = \frac{I_{n} − B(z)B(w)^{∗}}{1 − zw^{∗}} + \frac{B(z)B(w)^{∗}}{1 − zw^{∗}}

leads to an orthogonal decomposition of the space (H(𝔹_{N}))^{n×1}. Now in the Hardy space, the corresponding decomposition

\frac{I_{n}}{(1 − zw^{∗})^{N}} = \frac{I_{n} − B(z)B(w)^{∗}}{(1 − zw^{∗})^{N}} + \frac{B(z)B(w)^{∗}}{(1 − zw^{∗})^{N}}  (6.1)

does not lead in general to an orthogonal decomposition in the Hardy space. Indeed, take n = 1 and B(z) = b_{a}(z). The orthogonal complement of the set of functions in the Hardy space that vanish at the point a is spanned by the function 1/(1−za^{∗})^{N} and has reproducing kernel \frac{(1−aa^{∗})^{N}}{(1−za^{∗})^{N} (1−w^{∗}a)^{N}}, which is equal to

\frac{(1 − b_{a}(z)b_{a}(w)^{∗})^{N}}{(1 − zw^{∗})^{N}} .

Thus a decomposition of the kernel 1/(1 − zw^{∗})^{N} that leads to an orthogonal decomposition of the Hardy space is

\frac{I_{n}}{(1 − zw^{∗})^{N}} = \frac{(I_{n} − B(z)B(w)^{∗})^{N}}{(1 − zw^{∗})^{N}} + \frac{I_{n} − (I_{n} − B(z)B(w)^{∗})^{N}}{(1 − zw^{∗})^{N}}  (6.2)

and not (6.1).

Finally we remark that for an inner function B of the ball (which exists; see [30], [3]), multiplication by B is an isometry from the Hardy space of the ball into itself and we have the orthogonal decomposition

H^{2}(𝔹_{N}) = (H^{2}(𝔹_{N}) ⊖ BH^{2}(𝔹_{N})) ⊕ BH^{2}(𝔹_{N}),

corresponding to the decomposition (6.2). On the other hand, we do not know if inner functions of the ball are multipliers of the space H(𝔹_{N}).

7 Some finite-dimensional resolvent-invariant subspaces

The structure of general T_{j}(a)^{∗}-invariant subspaces of H(𝔹_{N}) seems at present elusive. In this section we present some partial results. We begin with the following.

Proposition 7.1 Assume M is a finite-dimensional resolvent-invariant subspace of H(𝔹_{N}). Then there exist matrices C, A_{1}, …, A_{N} such that the columns of the matrix-valued rational function

C(I − ∑_{j} z_{j}A_{j})^{−1}  (7.1)

form a basis of M.

Proof: Let f_{1}, …, f_{M} be a basis of M and set

F (z) = ( f_{1}(z) f_{2}(z) ⋯ f_{M}(z) ) .

Then there exist matrices m_{1}, …, m_{M} ∈ ℂ^{M×M} such that

T_{j}(a)^{∗}(F (z)ξ) = F (z)m_{j}ξ, ξ ∈ ℂ^{N×1}, j = 1, …, N.

Hence using (3.3), we get F (z) − F (a) = ∑_{1}^{N}(z_{j} − a_{j})F (z)m_{j}; so

F (z)(I_{M} − ∑_{1}^{N}(z_{j} − a_{j})m_{j}) = F (a),

and the result follows. □

The condition in the lemma is only necessary and is not sufficient as illustrated by the example

C = ( 1 0 ) , A_{1} = ( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ) , A_{2} = ( \begin{matrix}0 & 0 \\ 0 & \frac{1}{2} \end{matrix} )

when N = 2. The corresponding space M is spanned by the functions 1 and \frac{z_{1}}{1− \frac{z_{2}}{2}} . One checks readily that

T_{1}(0)^{∗} (\frac{z_{1}}{1 − \frac{z_{2}}{2}} ) = \frac{1}{1 − \frac{z_{2}}{2}}

and thus the space is not T_{1}(0)^{∗}-invariant.

We do not know how to characterize general T_{j}(a)^{∗}-invariant subspaces of H(𝔹_{N}). Here is a partial result.

Theorem 7.2 Let M be a finite-dimensional resolvent-invariant subspace of H(𝔹_{N}) spanned by M distinct kernel functions \frac{1}{1−zw_{j}^{∗}}, j = 1, …, M. Then there exists a row-valued function B such that the reproducing kernel of M is

\frac{1 − B(z)B(w)^{∗}}{1 − zw^{∗}} .  (7.2)

Proof: It suffices to note that the elements of M^{⊥} satisfy the interpolation conditions

f(w_{j}) = 0, j = 1, …, M

and apply Theorem 5.2. □

Acknowledgments: The first author thanks TÜBİTAK of Turkey for supporting a visit of Daniel Alpay to Middle East Technical University, Ankara, and to Professor Hayri Körezlioğlu for initiating the visit. The second author wishes to thank Middle East Technical University and Ben-Gurion University for supporting a visit to Ben-Gurion University, Beer-Sheva. It is also a pleasure to thank Professor Szafraniec for pointing out to us G. Misra’s paper [26].

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Daniel Alpay Department of Mathematics Ben-Gurion University of the Negev Beer-Sheva 84105, Israel

H. Turgay Kaptanoğlu Mathematics Department Middle East Technical University Ankara 06531, Turkey

2000 Mathematics Subject Classification. Primary: 47A57; Secondary: 32A70.