The Drury--Arveson space on the Siegel upper half-space and a von Neumann type inequality

In this work we study what we call Siegel--dissipative vector of commuting operators $(A_1,\ldots, A_{d+1})$ on a Hilbert space $\mathcal H$ and we obtain a von Neumann type inequality which involves the Drury--Arveson space $DA$ on the Siegel upper half-space $\mathcal U$. The operator $A_{d+1}$ is allowed to be unbounded and it is the infinitesimal generator of a contraction semigroup $\{e^{-i\tau A_{d+1}}\}_{\tau<0}$. We then study the operator $e^{-i\tau A_{d+1}}A^{\alpha}$ where $A^{\alpha}=A_1^{\alpha_1}\cdots A^{\alpha_d}_d$ for $\alpha\in\mathbb N^d_0$ and prove that can be studied by means of model operators on a weighted $L^2$ space. To prove our results we obtain a Paley--Wiener type theorem for $DA$ and we investigate some multiplier operators on $DA$ as well.


Introduction
The Drury-Arveson space DApB d`1 q on the unit ball of C d`1 is a renowned Hilbert space of holomorphic functions endowed with the inner product We refer the reader to [AMPS19, Theorem 6.1] for an exact integral representation of such inner product. The space DApB d`1 q is a reproducing kernel Hilbert space with kernel Kpz, wq " p1ź¨w q´1 and may be considered the natural multi-dimensional version of the Hardy space on the unit disc H 2 pDq; one of the main reasons being that DApB d`1 q plays the same role as H 2 pDq in a multi-dimensional version of the famous von Neumann Inequality.
space H and T : H Ñ H a contraction, i.e., }T } BpHq ď 1, the von Neumann Inequality states that, for any polynomial ppzq, }ppT q} BpHq ď }p} MpH 2 pDqq " }p} 8 . It is possible to prove an analogous result modeled on the upper-half plane, which is the unbounded biholomorphic realization of the unit disk D via the Cayley transform. Let A be any bounded dissipative operator on a Hilbert space H, that is, such that 1 i pA´A˚q ě 0, then }f pAq} BpHq ď sup Im zą0 |f pzq| for any rational function f which is bounded on the upper half-plane, see [Wak87].
We now describe the operator-theorical condition we shall work with.
Given two operators U and T on H, U densely defined and closed, T bounded, we say that U and T strongly commute if UT is an extension of T U. In other words T pHq Ď DompUq and T Uv " UT v for all v P DompUq.
We point out that from the conditions above and the Lumer-Philips Theorem (Theorem 2.5) it follows that iA d`1 is the infinitesimal generator of a (unique) semigroup of contractionste´i τ A d`1 u τ ď0 that commutes with all the A j 's, see Lemma 5.1. Our von Neumann type inequality reads as follows. Theorem 1.3. Let H be a Hilbert space and let pA 1 , . . . , A d`1 q be a Siegel-dissipative vector of commuting operators. For any τ j ă 0, j " 1, . . . , d, set M j " e´i τ j A d`1 A j and m j pζ, ζ d`1 q " e´i τ j ζ d`1 ζ j . Let p denote any polynomial in d variables. Then  A few remarks are now in order. The space DA is a reproducing kernel Hilbert space, and, with respect to the DA pnq -norm, its reproducing kernel is . Hence, notice that the condition (1.4) is modelled after the reproducing kernel of DA. In general, it is a well established phenomenon that von Neumann type inequalities hold as long as one can suitably interpret the condition K´1pA, Aq ě 0, where K is the reproducing kernel of a space of holomorphic functions in some domain in C d . A precise formulation of this result is stated and proved in [AEM02]. There the authors use the Dunford-Riesz functional calculus to define K´1pA, Aq. This is possible since they are working with bounded operators with some additional assumptions on their spectrum. Furthermore, they assume that the multiplication by any coordinate function is a continuous (bounded) operation. Our setting differs in these two latter aspects. The last operator A d`1 is allowed to be unbounded and also the multiplication by any coordinate is an unbounded operation in DA.
We also mention that, in the case when all the operators are bounded, a proof of the von Neumann type inequality (1.5) could probably be obtained by means of the Cayley transform, the classical inequality of Drury on the unit ball and some classical results in operator theory as it was pointed out to us by M. Hartz in a private communication [Har]. Nevertheless, in the footsteps of Drury's proof, we prefer to use a direct approach, without relying on the known results on the unit ball, for two main reasons. First, we have greater generality by allowing one operator to be unbounded. Second, and more importantly, we develop some machinery we believe it is interesting in its own right and has the potential to be used in the context of the theory of shift-invariant subspaces of DA, as in the spirit of Buerling-Lax's Theorem [Lax59], in the Siegel half-space setting.
In order to follow such a plan, we further investigate some operator-theorical properties of DA. On the space N d 0ˆR´w e consider the measure with dλ denoting the Lebesgue measure on R´and dα the counting measure on N d 0 . Theorem 1.4. The map S : is a conjugate linear surjective isometry. The inverse map S´1 is explicitly given in (3.1).
With our assumptions it follows that L 2 p∆q is a pre-Hilbert space. Before stating our result on d`1-tuples of operators we also need to study some weighted shift operators on L 2 pN d 0ˆR´, dµq, which correspond to some multiplier operators on DA (see Section 4). If α, γ P N d 0 we write α ě γ meaning that α i ě γ i for all i " 1, . . . , d. Then, we prove the following.
Theorem 1.5. Let γ P N d 0 and τ ă 0. Then, the operator extends to a bounded linear operator on L 2 pN d 0ˆR´, dµq unitarily equivalent to the multiplier operator on DA with multiplier ζ γ e´i τ ζ d`1 . Furthermore, its adjoint is given by the formula We remark that, if we call the multipliers m j , j " 1, . . . , d, that appear in Theorem 1.3 the shift operators on DA, then the operators S γ,τ correspond to the operators S´1m j S . Finally, we have the following result.
Theorem 1.6. Let pA 1 , . . . , A d`1 q be a strongly Siegel-dissipative vector of commuting operators and set commutes.
We conclude this introduction pointing out that the Drury-Arveson space DApB d`1 q on the unit ball of C d`1 has drawn considerable interest since its first appearance in the works of Drury and of Arveson [Dru78], [Arv98]. We mention [ARS08, ARS10, AL18], [FX15], [RS16] and references therein. The Drury-Arveson space on U was studied in [ARS10] and [AMPS19]; see also [CP21] for the case of more general Siegel type domains.
The paper is organized as follows. In Section 2 we recall some preliminary facts, in Section 3 we study the space DA and we prove Theorem 1.4. In Section 4 we study some multiplier operators on DA and we prove Theorem 1.5. We conclude proving Theorem 1.6 and Theorem 1.3 in Section 5.

Preliminary facts
In this section we first recall some basic facts on the Siegel half-space, the Heisenberg group and the group Fourier transform. Then we also recall some simple properties of semigroups of operators on a Hilbert space.
Hence, it is possible to introduce a group structure on BU itself.
Definition 2.1. The Heisenberg group H d is the set C dˆR endowed with product rw, ssrz, ts " " w`z, s`t´1 2 Impw¨zq ‰ .
The right and left Haar measures on the Heisenberg group coincide with the Lebesgue measure on C dˆR . In particular, the Lebesgue measure is both left and right translation invariant.
We now recall the basic facts for the Fourier transform on the Heisenberg group. For λ P Rzt0u define the Fock space |F pzq| 2 e´λ 2 |z| 2 dz ă`8 * when λ ą 0, and F λ " F |λ| when λ ă 0. The Fock space is a reproducing kernel Hilbert space with reproducing kernel e |λ| 2 z¨w . A complete orthonormal basis of F λ is given by the normalized For rz, ts P H d , the Bargmann representation σ λ rz, ts is the operator acting on F λ given by, if λ ą 0, and, if λ ă 0, as σ λ rz, ts " σ´λrz,´ts, that is, If f P L 1 pH d q, for λ P Rzt0u, σ λ pf q is the operator acting on F λ as where }σ λ pf q} 2 HS " ř α }σ λ pf qe α } 2 F λ is the Hilbert-Schmidt norm of σ λ pf q and e α " z α {}z α } F λ . If f P L 1 X L 2 pH d q the following inversion formula holds: 2.2. The Drury-Arveson space on U. In [AMPS19] a family of holomorphic function spaces on U depending on a real parameter ν was studied and characterized by means of the group Fourier transform on H d . This family of spaces includes weighted Bergman spaces, the Hardy space, weighted Dirichlet spaces and the Dirichlet space. In particular, the Drury-Arveson space DA was identified as a particular weighted Dirichlet space. Here we recall the results in [AMPS19] we need in the rest of the paper.
Definition 2.2. We define the space L 2 DA as the space of functions τ on Rzt0u such that: piq τ pλq P HSpF λ q for every λ, i.e., τ pλq : F λ Ñ F λ is a Hilbert-Schmidt operator; piiq τ pλq " 0 for λ ą 0; piiiq ranpτ pλqq Ď spant1u; The following result holds true. (i) T 0 " Id; (ii) T t`s " T t T s , @s, t ą 0. If in addition T t converges to the identity operator Id in the strong operator topology as t OE 0, the semigroup is called strongly continuous or C 0 . From now on we will focus exclusively on C 0 semigroups. The infinitesimal generator G of a C 0 semigroup tT t u tě0 is a linear operator defined on the subspace DompGq of v P H such that the limit lim tOE0 T t v´v t ": Gv exists in the norm topology. It can be shown that G on its domain DompGq is a linear densely defined closed operator [Lax02, Section 34, Theorem 4]. In particular, we will be interested in the characterization of infinitesimal generators for contraction semigroups, i.e., semigroups tT t u tě0 such that each T t is a contraction, which is provided by the Lumer-Philips Theorem.
Definition 2.4. A densely define operator G on a Hilbert space H is called dissipative if Re xGv, vy H ď 0, @v P DompGq.
It is called maximal dissipative, if it is dissipative and its resolvent set rpGq includes R`" p0,`8q.
For the following renowned theorem we refer the reader, for instance, to [Lax02, p. 432].

Theorem 2.5 (Lumer-Philips). A densely defined operator G is the infinitesimal generator of a (unique) semigroup of contractions if and only if it is maximal dissipative.
The following is a very well-known lemma that we shall need in what follows.
Lemma 2.6. Let tT t u tě0 be a semigroup of bounded operators on a Hilbert space H. Suppose that: (i) there exists δ ą 0 such that sup 0ătăδ }T t } BpHq ă`8; (ii) for some dense subset D Ď H, T t f Ñ f for all f P D. Then, tT t u tě0 is a strongly continuous semigroup.
We remark that we will work with semigroups that appear to have a negative parameter τ ; this is to stay consistent with [AMPS19]. When we use this notation we simply mean that the semigroup has the positive parameter t "´τ .

Proof of Theorem 1.4
The proof of Theorem 1.4 follows at once from the next two lemmas. Recall that the measure µ on N d 0ˆR´i s defined in (1.6). Now define ϕpα, λq " }z α }´1 F λ xe 0 , σ λ pf 0 qpe α qy F λ .
Lemma 3.1. The map Φ : DA X H 2 Ñ L 2 pN d 0ˆR´, dµq defined as Φpf qpα, λq " ϕpα, λq, Proof. First we assume that f P DA X H 2 where H 2 denotes the Hardy space on U. This intersection is dense in DA ( [AMPS19, Lemma 4.2]) and every f P H 2 admits a boundary value function f 0 P L 2 pBUq. Moreover, for such function f the function τ P L 2 DA in formula (2.2) actually coincides with the Fourier transform of its boundary value function, that is, τ pλq " σ λ pf 0 q (see [AMPS19,OV79]).
Let te α u α be the orthonormal basis of normalized monomials of the Fock space F λ and let F " ř α F α e α be a function in F λ . Then, for every function f P DA X H 2 , we have σ λ pf 0 qpF q " ÿ thanks to property (iii) in Definition 2.2. Thus, In particular, we deduce that }z α } F λ ϕpα, λq " xΦ λ , e α y F λ . Now, since te α u α is an orthonormal basis in F λ . However, In conclusion, ϕpα, λq| 2 dµpα, λq.

(3.2)
We now see that the isometry Φ is surjective as well. We first specialize the inversion formula (2.2).
Using the fact that τ pλq is a rank one operator such that ranktτ pλqu Ď spante 0 u for every λ we get trpτ pλqσ λ rz, ts˚q " trpσ λ rz, ts˚τ pλqq where P 0 denotes the orthogonal projection onto the subspace generated by e 0 . Moreover, it holds (see also p26q in [AMPS19]) Thus, Hence,

Pointwise multipliers on the Drury-Arveson space
In this section we explicitly study some multiplier operators on DA and we prove Theorem 1.5. Recall that given a function m the associated multiplier operator is the operator f Þ Ñ mf . The problem of characterizing the multiplier algebra of a given reproducing kernel Hilbert function space is a classical problem. In the case of the Drury-Arveson space on the unit ball DApB d`1 q this problem turned out to be very challenging; a first important result is due to J. Ortega and J. Fàbrega [OF00]. Their result reads as follows. Let n P N 0 be such that 2n ą d and for f in HolpB d`1 q define the measure where R is the radial derivative, dν is the normalized Lebesgue measure on B d`1 and w P B d`1 . Then, f is a DApB d`1 q-multiplier if and only if f P H 8 pB d`1 q and dν f is a Carleson measure for DApB d`1 q. A few years later N. Arcozzi, R. Rochberg and E. Sawyer in [ARS08] completely characterized the Carleson measures of DApB d`1 q; see also [Tch08b,Tch08a], [VW12]. Hence, the multiplier algebra of the Drury-Arveson space on the unit ball is completely characterized. Nonetheless, there is still interest in finding an easier characterization and we refer the reader, for instance, to [FX15].
From [OF00,ARS08] we can also deduce an indirect characterization of the multiplier algebra for DA on the Siegel half-space. Indeed, let C be the multi-dimensional Cayley transform defined in (1.2). Then, up to an irrelevant multiplicative constant, DA denote the reproducing kernel of DA and DApB d`1 q respectively, and pz, z d`1 q, pw, w d`1 q P B d`1 . From the abstract theory of reproducing kernel Hilbert spaces (see, for instance, [AMK02, Chapter 2.6]) we deduce that f Þ Ñ p1´z d`1 q´1pf˝Cq is a surjective isometry from DA onto DApB d`1 q and that m is a multiplier for DA if and only if pm˝Cq is a multiplier for DApB d`1 q.
For our goal we need to study the multiplier operators associated to the functions ζ γ " ζ γ 1 1¨¨¨ζ γ d d , e´i τ ζ d`1 for τ ă 0 and ζ γ e´i τ ζ d`1 . However, we do not rely on the multiplier characterization on the unit ball since it is easier to study them directly. The proof of the following lemma is standard and we omit it.
Lemma 4.1. The set is dense in L 2 pN d 0ˆR´, dµq. We now study the multiplier operator associated to the monomial ζ γ , γ P N d 0 . Although this operator is unbounded on DA, as it is easily seen, it is closed and densely defined. Proof. Let us consider D γ " tf P DA : ζ γ f P DAu as domain of our multiplier operator. Let tf n u n Ď D γ be a sequence such that f n Ñ f in DA and ζ γ f n Ñ g P DA. Since DA is a reproducing kernel Hilbert space we also have that f n Ñ f and ζ γ f n Ñ g uniformly on compact sets. Hence, ζ γ f n Ñ ζ γ f uniformly on compact sets as well and ζ γ f " g. In particular both f and ζ γ f are in DA so that f belongs to the domain D γ . Thus, our operator is closed.
To prove that D γ is dense we exploit the previous Lemma 4.1. Let ϕ P D and define f P DA with the inversion formula (3.3). Then, where χ tαěγu pαq is the characteristic function of the set tα P N d 0 : α i ě γ i , i " 1, . . . , du. Recall that supp ϕ is a compact subset of N d 0ˆR´a nd assume that pα, λq P supp ϕ implies |α| ă N for some positive integer N and λ P I where I is a compact subset of R´. From (3.2) we have where c d pϕq is a constant depending on the compact support of ϕ (and on the dimension d). The density of D γ in DA now follows from the density of D in L 2 pN d 0ˆR´, dµq and Theorem 1.4. We now investigate the other multiplier operators we are interested in and see that they are actually bounded on the Drury-Arveson space.
where χ p´8,τ q is the characteristic function of the interval p´8, τ q. Thus, by (3.2) we get ϕpα, λq| 2 dµpα, λq " }f } 2 DA . Hence, we conclude that e´i τ ζ d`1 is a contractive multiplier. The semigroup property is automatically satisfied. Since e´i τ ζ d`1 f converges in norm to f for all f P D and the semigroup has uniformly bounded seminorm, it is strongly continuous by Lemma 2.6. Then, by definition, the infinitesimal generator is given by and the proof is concluded.
Substituting the value for λ in the expression above we find 2 |γ| |α| |α| τ |α| |γ| |α| 1 τ |α`γ|`| α| Note that the first factor is constant, the second is decreasing in |α| and tends to e´| γ| as |α| Ñ`8. For the last term notice that This proves that the multiplier operator associated to m γ,α is bounded with norm less than where we used Stirling's asymptotic.

The von Neumann type inequality
In this section we prove our main results Theorem 1.6 and Theorem 1.3. We first prove a couple of lemmas.
Lemma 5.1. Let T be a bounded operator on a Hilbert space H and let U be a densely defined closed operator. Assume that U is the infinitesimal generator of a (unique) C 0 semigroup te τ U u τ ą0 and assume that T and U strongly commute. Then, T and e τ U commute for all τ ą 0.
By the uniqueness of the infinitesimal generator we conclude that Z τ " e τ U . In other words, e τ U T " T e τ U for all τ ą 0, as we wished to show.
Proof of Theorem 1.6. Set A " pA 1¨¨¨Ad q, let v P H and notice that, since for 1 ď i ď d the operators A i strongly commute with A d`1 , by [Lax02, Section 34, Theorem 4(i)] we can infer that e´i τ A d`1 A α v P Dom A d`1 . Therefore, the map Θ is well-defined. Furthermore, xA α`e i e´i λA d`1 v, A α`e i e´i λA d`1 vy H |λ| |α| 2 |α|`1 α! dαdλ.
Letting m j be the multiplier on DA given by m j pζ, ζ d`1 q " ζ j e´i τ j ζ d`1 , j " 1, . In the general case let pA 1 , . . . A d`1 q be Siegel-dissipative. Then, if we replace A d`1 by A d`1`i ε Id we get a strongly Siegel-dissipative tuple of operators. Applying the von Neumann type inequality we have }ppe ετ 1 M 1 , . . . , e ετ k M d q} BpHq ď }ppm 1 , . . . , m d q} MpDAq , However, the right hand side of the inequality does not depend on ε and the left hand side converges in the operator norm as ε Ñ 0`. In fact, it suffices to prove the convergence for each term of the polynomial separately; we have }e´i τ A d`1 e τ ε´e´iτ A d`1 } " }e´i τ A d`1 }p1´e τ ε q ď p1´e τ ε q τ Õ0 ÝÑ 0. Therefore, we can pass to the limit and obtain the desired inequality.