On a class of shift-invariant subspaces of the Drury-Arveson space

In the Drury-Arveson space, we consider the subspace of functions whose Taylor coefficients are supported in the complement of a set $Y\subset\mathbb{N}^d$ with the property that $Y+e_j\subset Y$ for all $j=1,\dots,d$. This is an easy example of shift-invariant subspace, which can be considered as a RKHS in is own right, with a kernel that can be explicitely calculated. Moreover, every such a space can be seen as an intersection of kernels of Hankel operators, whose symbols can be explicity calcuated as well. Finally, this is the right space on which Drury's inequality can be optimally adapted to a sub-family of the commuting and contractive operators originally considered by Drury.


Introduction
We begin by fixing some notation and delimiting the framework we work in. Let H be an abstract Hilbert space and for d ≥ 2 consider a d-tuple of operators A = (A 1 , . . . , A d ) : H → H d . It is not difficult to see that the formal adjoint operator A * : H d → H acts as follows Given a polynomial Q in d variables, say Q(z) = k c k z k , where z = (z 1 , . . . , z d ), k ∈ N d and the sum is finite, we write Q(A) for the operator from H to itself given by Following Drury, we will relate A to an operator acting on a Hilbert space of holomorphic functions of several variables on the unit ball. We write B d for the open unit ball {z = (z 1 , . . . , z d ) ∈ C d : |z| < 1}, where |z| 2 := Definition 1.1. The Drury-Arveson space is the space H d of functions f (z) = n∈N d a(n)z n holomorphic on the unit ball B d ⊂ C d , such that where the weight function β : N d → N is given by β(n) = |n|!/n!.
This space has a reproducing kernel. For f ∈ H d and z ∈ B d , we have f (z) = n a n z n = n a n z n β(n) β(n) = f, k z H d , with k z (w) = n β(n)z n w n for w ∈ D.
The series can be explicitly calculated and we get This function space was first introduced by Drury in [3], then further developed in [1]. See also [7]. It naturally arises as the right space to consider when trying to generalize to tuples of commuting operators a notable result by Von Neumann, saying that for any linear contraction A on a Hilbert space and any complex polinomial Q, it holds where M(H 2 ) = H ∞ denotes the multiplier space of the Hardy space of the unit disc H 2 .
In fact, Drury shows that for a d-tuples of operators A = (A 1 , . . . , The map T given by defines an isometric isomorphism from 2 (N d , β) to H d . This correspondence in particular tells us that the shift operator on 2 (N d , β), given by and the multiplication operator M j on H d are unitarily equivalent, i.e. it turns out that M j T = T S j for all j = 1, . . . , d.

A class of shift invariant subspaces of H d
We are interested in considering subspaces of H d of functions having Taylor coefficients with a prescribed support. Given some subset X of N d , we write 2 (X, β) for the closed subspace of 2 (N d , β) of functions supported in X. We say that a set X ⊆ N d is monotone, if its complement in N d is shift invariant, namely In all what follows we always consider X to be a monotone set. Given g ∈ 2 (N d \ X, β), for n ∈ X we have S j g(n) = 0 since n − e j ∈ X as well. Therefore 2 (N d \ X, β) is a shift-invariant subspace of 2 (N d , β). To any such a set X, we can associate the space H d (X) of functions of H d whose Taylor coefficients vanish on We can construct compressions of tuples of operators to the subspaces associated to the monotone set X.
In particular, let B j = S * j denote the backwards shift operator on 2 (N d , β), given by B j g(n) = g(n + e j ). We consider the d-tuple of operators where for each j = 1, . . . , d, Observe that the adjoint of B X is a row contraction from 2 (X, β) d to 2 (X, β), In the same way, we write M X z for the compressed d-tuple , P X being in this context the orthogonal projection from H d onto H d (X).

Hankel operators and shift invariant subspaces
Shift-invariant subspaces for the Drury-Arveson space are characterized in [5], where it is shown that they can be represented as intersections of countably many kernels of Hankel operators, to be defined shortly. See also the PhD thesis [8].
Consider a Hilbert space H of holomorphic functions on the unit ball B d , such that functions holomorphic on B d are dense in it. The function b ∈ H is a symbol if there exists C > 0 such that Endowing the space H := {f : f ∈ H} with the inner product f ,ḡ H := g, f H , we say that On H d , consider the Hankel operator with symbol b(z) = z m , for some m ∈ N d . We have . This is the easiest example of shift-invariant subspace of the Drury-Arveson space with explicit symbol.
Actually, each set X satisfying (1) can be associated to a collection of Hankel symbols. Observe that X is bounded if and only if for all j there exists n ∈ N d \ X such that n ∈ Ne j . In such a case, X is a finite union of rectangles, X = k=1,...,K R m k and hence, If X is unbounded, then for every j such that N d \ X ∩ Ne j = ∅, we have an increasing sequence of rectangles covering the strip unbounded in the j − th direction. Summing up, it follows that

Drury type inequality
In the introduction we have defined polynomials valued on operators, Q(A). The concept of operators being variables of functions can be properly extended. Following Nagy and Foias [9], given a contraction A on a Hilbert space H one can define the holomorphic functional calculus Now, for any ϕ ∈ Hol(D), the function ϕ r (·) := ϕ(r·) is in the class A for r ∈ (0, 1). Moreover, if ϕ ∈ H ∞ , we have the uniform bound |ϕ r (z)| ≤ ϕ ∞ , for z ∈ D, 0 < r < 1. Hence, for every ϕ ∈ H ∞ it can be defined the functional calculus ϕ(A) = lim r→1 − ϕ r (A), whenever the above limit exists in the strong operator topology, which is always the case when A is a completely non-unitary contraction (see [9]).
In particular, for ϕ ∈ M(H d ) ⊂ H ∞ and A = M z , we can define the operator of multiplication by ϕ via the functional calculus . This defines a bounded operator from H d to itself, and its adjoint is clearly given by (M ϕ ) * = lim r→1 − (ϕ r (M z )) * .
We have the following version of Drury's inequality.
Let X be the complement in N d of the set N := {n ∈ N d : A n = 0}. Then for every complex polynomial Q of d variables, we have Proof. For N = ∅ we have X = N d and this is just Drury's theorem, while for N = N d \ {0}, A reduces to a d-tuple of zeros (we set 0 0 to be the identity). So, we suppose that N (and hence X) is a proper subspace of N d . It is enough to show that the theorem is true when (ii) is replaced by the stronger condition where r ∈ (0, 1). We write H(X) for the space 2 (X,Ȟ, β), whereȞ has the same underlying space as H but a different norm, h Ȟ = Dh H , where D is the defect operator of A, D = √ I − A * A, (see [3] for the details). Drury constructs an injective isometry θ : H → H(N d ), θh(n) := A n h, and shows that B m θ = θA m for all m ∈ N d (here B is the d-tuple of backshifts on H(N d )).
We rephrase this in our setting. Let π X be the orthogonal projection of H(N d ) onto H(X), B X j := π X B j | H(X) and ψ := π X • θ.
Since θ is an isometry, it is easy to see that that (4) ψ is an isometry ⇐⇒ θh = 0 on N d \ X ⇐⇒ A n = 0 for n ∈ N d \ X.
We have ψA j = π X B j θ, and B X j ψ = π X B j | H(X) π X θ = π X B j π X θ. For n ∈ X and h ∈ H, n + e j ∈ X θh(n + e j ) n + e j ∈ X which equals zero by (4). It follows that At this point, it is standard (for example follow [3]) that for every complex polynomial Q we have, The equality above follows from the intertwining relation M X j T = T (B X j ) * , where the operator T in our case is the isometric isomorphism from 2 (X, β) to H d (X) given by (T g)(z) := n∈X g(n)β(n)z n .
For f (z) = n a n z n ∈ H d (X), we have M X j f (z) = n∈X∩X+ej a n−ej z n , and so Then, all polynomials are multipliers for H d and Of course, there are in general many functions ϕ such that P X ϕ(M z ) = P X Q(M z ). In particular, let ϕ be a multiplier of H d such that ϕ(n) = Q(n) for n ∈ X. Then, for any g ∈ H d we have, The term on the right goes to zero as r → 1 − , so it follows P X Q(M z ) = P X ϕ(M z ). Then, (6) can be generalized as follows for any ϕ ∈ M(H d ) such that ϕ(n) = Q(n). We have then proved that,

Remark 4.2.
Observe that the first inequality in the theorem is optimal if the backshift d-tuple B X satisfies (i) and (ii) and if {n ∈ N d : (B X ) n = 0} equals N . It is clear that condition (ii) holds for B X , for every choice of X. Also, n ∈ N ⇐⇒ n + m ∈ N for all m ∈ N d and since N = N d \ X this is equivalent as asking f (n+m) = 0 for all m ∈ N d , f ∈ 2 (X, β). But f (n+m) = (B X ) n f (m) and so {n ∈ N d : (B X ) n = 0} = N . On the other hand, the commuting property (i) is not fulfilled on most sets X. Of course, if X is chosen such that 2 (X, β) is backshift-invariant, then B X = B 2 (X,β) and (i) and (ii) hold, see [3]. More in general, doing standard calculations it is not hard to see that B X satisfies (i) if and only if (7) n, n + e i + e j , n + e i ∈ X =⇒ n + e j ∈ X, for i, j = 1, . . . , d.
This is a shape-condition on the set X, saying that it cannot have any subset with one of the following configurations It is clear that X = N C satisfies (7), since n + e j ∈ N d \ X for some j would imply n + e j + e i ∈ N d \ X for all i = 1, . . . , d. It follows that the inequality in the theorem is optimal.

Further considerations
We want to look closer at the inequality in (3). In particular, we are interested in understanding if it is an equality indeed. The reason to be optimistic in this sense comes from a theorem proved by Sarason in [6] (see also [4,Theorem 3.1]) in the one-dimensional case, i.e. for the Hardy space. Let K be a closed backshift-invariant subspace of the Hardy space H 2 , and write S K for the compression of the shift operator to this subspace. Sarason proved the following. Theorem 5.2. Let k(z, w) be a nondegenerate positive kernel on a domain Ω such that 1/k has 1 positive square. Let H(k) be the associated RKHS. Suppose that W ⊂ H(k) is a -invariant subspace and that T is a bounded linear contraction from W to itself such that for all ϕ ∈ M(H(k)). Then, there exists a a multiplier ψ ∈ M(k) such that (M ψ ) ≤ 1 and Asking that 1/k has 1 positive square means that the self adjoint matrix {1/k(z i , z j )} N i,j=1 has exactly one positive eigenvalue, counted with multiplicity, for every finite set of disjoint points {z 1 , · · · , z N } ⊂ B d . It is well known that the Drury-Arveson kernel has this property.
So, in order to apply the theorem, take H d as the RKHS and let W = H d (X). We have to Suppose that the multiplier function ϕ has the power series expansion ϕ(z) = n a n z n . Then ϕ r (z) = n a n (r)z n , where a n (r) = a n r |n| . Using the fact that (M nj j ) * = (M * j ) nj and the uniform absolute convergence of the series, we get we get k X (w, z) = n∈X β(n)z n w n = N1 n1=0 ∞ n2=0 n 1 + n 2 n 2 t n1 1 t n2 2 = N1 n1=0 t n1 . Now, suppose that (12) holds on N d−1 . Again, suppose to re-order the basis e 1 , . . . , e d so that j = 1. On N d we have