Shift operators on harmonic Hilbert function spaces on real balls and von Neumann inequality

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Abstract

On harmonic function spaces, we define shift operators using zonal harmonics and partial derivatives, and develop their basic properties. These operators turn out to be multiplications by the coordinate variables followed by projections on harmonic subspaces. This duality gives rise to a new identity for zonal harmonics. We introduce large families of reproducing kernel Hilbert spaces of harmonic functions on the unit ball of Rn and investigate the action of the shift operators on them. We prove a dilation result for a commuting row contraction which is also what we call harmonic type. As a consequence, we show that the norm of one of our spaces G˘ is maximal among those spaces with contractive norms on harmonic polynomials. We then obtain a von Neumann inequality for harmonic polynomials of a commuting harmonic-type row contraction. This yields the maximality of the operator norm of a harmonic polynomial of the shift on G˘ making this space a natural harmonic counterpart of the Drury-Arveson space.

Introduction

After the pioneering work of Drury [10] and Arveson [6] on extending the von Neumann inequality to commuting operator tuples, there have been several other generalizations to Hilbert space operators in other settings. We can cite [20], [3], [16], [11], [13], and [8] to name a few. There is also the earlier [19] on noncommuting operator tuples. But in all von Neumann inequalities that we know of, the polynomials acting on the operators are holomorphic functions of their variables. It is the aim of this work to obtain a von Neumann inequality in which the polynomials are harmonic in the usual sense in Rn.

Multivariable versions of von Neumann inequality often depend on shift operators on a specific Hilbert function space. This immediately brings out the first major obstacle in dealing with harmonicity. Even the definition of a shift operator on a space of harmonic functions has not been made before, because harmonicity is not preserved under multiplication, and a multiplication by a coordinate variable must be followed by some form of a projection on harmonic functions. We make a definition and check it by using another approach.

Another obstacle is to decide which space among harmonic function spaces plays a role like that of the Drury-Arveson space among holomorphic spaces. We find out that considering a family of reproducing kernel Hilbert spaces Gq of harmonic functions on the unit ball of Rn indexed by qR is more feasible since it exposes the compositions of the spaces better. Then it is easier to pick one of these spaces as the harmonic counterpart of the Drury-Arveson space using its extremal properties in the family.

One more obstacle is that the more complicated structure of harmonic functions persists at the operator level and we are obliged to restrict our attention in von Neumann inequality to a class of contractions that we call harmonic type.

We now present our major results; for them it helps to have some familiarity with the classical knowledge on harmonic polynomials summarized in Section 3. For m=0,1,2,, let Pm and Hm denote the homogeneous polynomials of degree m and spherical harmonics of degree m on Rn, respectively. Let Hm:PmHm be the standard projection. The zonal harmonics Zm(x,y) are the reproducing kernels of the Hm with respect to the L2 inner product on the unit sphere. For j=1,,n, we define the shift operators Sj:HmHm+1 acting on x bySjZm(x,y):=1n+2myjZm+1(x,y). Our first main result shows that Sj is closely related to the operator of multiplication Mxj by the jth coordinate variable.

Theorem 1.1

Sj=Hm+1Mxj for all j=1,,n and m=0,1,2,.

In the course of proving this theorem, we obtain the following identities for the Gegenbauer polynomials Cmλ and the Chebyshev polynomials Tm which seem new. There, K is the Kelvin transform which transforms a harmonic function on the unit ball to one on its exterior.

Theorem 1.2

For ξ,ηS, we haveK[(η)mK[1]](ξ)=(1)mm!Cmn/21(ξη)(n3,m=0,1,2,),K[(η)mK[log||]](ξ)=(1)m(m1)!Tm(ξη)(n=2,m=1,2,).

We define the space that we claim to be the harmonic version of the Drury-Arveson space as the reproducing kernel Hilbert space G˘ on the unit ball of Rn with reproducing kernelG˘(x,y):=m=01AmZm(x,y), where Am is the coefficient of (xy)m in the expansion of Zm(x,y). We call commuting operators (T1,,Tn) a row contraction if they are a contraction as a tuple. A contractive norm is one in which the tuple of shift operators is a row contraction. Another main result of ours shows that the norm of G˘ is as large as possible.

Theorem 1.3

If is a contractive Hilbert norm on harmonic polynomials that respects the orthogonality of L2, then G˘1.

We call an operator tuple (T1,,Tn) harmonic type if T1T1++TnTn=0. The harmonic shift S˘=(S˘1,,S˘n) on G˘ is the prime example of a harmonic-type operator. Our final main result is a von Neumann inequality.

Theorem 1.4

Let (T1,,Tn) be a harmonic-type row contraction on a Hilbert space. If u is a harmonic polynomial, then u(T1.,Tn)u(S˘1,,S˘n).

All terminology is explained in detail in an appropriate section in the paper. After introducing in Section 2 the basic notation, we make a review of harmonic polynomials in Section 3 and give formulas for the Zm, Hm, and K. In Section 4, we define the shift operators on harmonic spaces, explain the meaning of the coefficient 1/(n+2m), and then prove Theorem 1.1. In Section 5, we introduce a new family of reproducing kernel Hilbert spaces of harmonic functions and isolate one of them as G˘ by making it clear why we need the coefficients 1/Am in G˘. In Section 6, we find the basic properties of shift operators and their adjoints acting on the Hilbert spaces just introduced. In Section 7, we investigate the row contractions on harmonic Hilbert spaces, explain the term harmonic type, and prove an essential dilation result for harmonic-type and self-adjoint operator tuples. In Section 8, we prove Theorem 1.3, Theorem 1.4.

The first author thanks the Foster G. and Mary McGaw Professorship in Mathematical Sciences of Chapman University for its support. The second author thanks Aurelian Gheondea of Bilkent University and Serdar Ay of Atılım University for useful discussions. The authors also thank an anonymous referee for suggesting to consider self-adjoint operators which encouraged us to obtain the results on such operators in the next to last section.

Section snippets

Notation

Let B and S be the open unit ball and its boundary the unit sphere in Rn with respect to the usual inner product xy=x1y1++xnyn and the norm |x|=xx, where always n2. We write x=rξ, y=ρη with r=|x|, ρ=|y|, and ξ,ηS, and use these throughout without further comment. When n=2, the ball is just the unit disc D in the complex plane bounded by the unit circle T, and x,yD are complex numbers of modulus less than 1.

In a few places, we also use the complex space CN and its Hermitian inner product z

Harmonic polynomials

We review the essentials of zonal harmonics and the Kelvin transform for completeness, because we refer to these facts many times in the paper. These results are mostly well-known and can be consulted in [7, Chapters 4 & 5].

For m=0,1,2,, let Pm denote the complex vector space of all polynomials homogeneous (with respect to real scalars) of degree m on Rn. It is immediate that(x)pm=mpm(pmPm). Let Hm be the subspace of Pm consisting of all harmonic homogeneous polynomials of degree m. By

Shift operators

We define the shift operators on harmonic functions first in an unusual way, but later show that they are equivalent essentially to multiplications by the coordinate variables.

The motivation for our definition lies in the observation1m+1wj(z,wm+1)=zjz,wm for z,wCN and the realization that Xm(x,y) replaces z,wm.

Definition 4.1

For 1jn, we define the jth shift operator Sj:HmHm+1 acting on the variable x by first lettingSjXm(x,y):=1m+1yjXm+1(x,y) and then extending to all of Hm by linearity and the

Harmonic Hilbert function spaces

We are inspired by a few earlier works in defining new reproducing kernels with desired properties. In [16], families of weighted symmetric Fock spaces of holomorphic functions that include the Drury-Arveson space are studied following [6]. In [12], Bergman-Besov kernels are defined as weighted infinite sums of zonal harmonics much like the Poisson kernel. And we have already noted that the right tool is the xonal harmonics rather than the zonal harmonics.

Definition 5.1

Let β:={βm>0:β0=1,m=0,1,2,} be a

Adjoints

Having families of harmonic Hilbert function spaces at hand, we now investigate the action of shift operators on them and of their adjoints. When we extend the Sj to all of Gβ and Gq by linearity and density, we name them Sβj and Sqj.

We start by obtaining the adjoints, the backward shifts, Sβj:HmHm1 now with respect to [,]Gβ using (21) this time. Repeating the computation that leads to (14), we similarly obtain Sβj(1)=0 andSβjum=1mβm1βmjum(umHm,m1).

We then extend both the Sj and the S

Row contractions

Our aim in this section is to show that the norm of G˘ is maximal among all contractive Hilbert norms. We start by recalling the necessary terms.

Definition 7.1

A commuting operator tuple T=(T1,,Tn):HHH on a Hilbert space H is called a row contraction if T1, that is, ifT1u1++TnunH2u1H2++unH2(u1,,unH).

We note that being commuting is part of the definition of a row contraction.

Proposition 7.2

The shift Sβ is a row contraction if and only if {βm} is an increasing sequence, and Sq is a row contraction if and

Von Neumann inequality

Definition 8.1

A norm on H derived from an inner product that respects the orthogonality in L2 is called contractive if the shift operator is a row contraction in this norm.

We are now ready to prove Theorem 1.3 which we restate.

Theorem 8.2

Let be a contractive norm on H. ThenuuG˘1(uH).

Proof

Let H be the Hilbert space that is the completion of H in the norm . Let also S=(S1,,Sn) be the shift on H and S its adjoint with respect to the inner product [,] of H. We have [Sj1,v]=[1,Sjv]=0 for all vH by

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