on real balls and von Neumann inequality Shift operators on harmonic Hilbert function spaces Journal of Functional Analysis

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Journal of Functional Analysis

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

Daniel Alpay^a, H. Turgay Kaptanoğlu^b,∗

aSchmidCollegeofScienceandTechnology,ChapmanUniversity,Orange,CA 92866,UnitedStatesofAmerica

bBilkentÜniversitesi,MatematikBölümü,06800Ankara,Turkey

a r t i c l e i n f o a b s t r a c t

Articlehistory:

Received5August2020 Accepted15April2021 Availableonline22April2021 CommunicatedbyStefaanVaes

MSC:

primary47A13,47B32,33C55 secondary31B05,33C45,42B35, 46E20,46E22,47B37

Keywords:

Harmonicshift Harmonictypeoperator VonNeumanninequality Drury-Arvesonspace

On harmonicfunction spaces, wedefine shift operators us- ingzonalharmonicsandpartialderivatives,anddeveloptheir basicproperties.Theseoperatorsturnout tobe multiplica- tionsby thecoordinatevariablesfollowedby projectionson harmonicsubspaces.Thisdualitygivesrisetoanewidentity forzonalharmonics.Weintroducelargefamiliesofreproduc- ingkernel Hilbertspacesof harmonicfunctionson theunit ballof Rⁿ andinvestigate theactionof theshift operators on them. We prove a dilation result for a commuting row contractionwhichisalso whatwecallharmonictype.Asa consequence,weshowthatthenormofoneofourspacesG is˘ maximalamongthosespaceswithcontractivenormsonhar- monicpolynomials.WethenobtainavonNeumanninequality forharmonicpolynomialsofacommutingharmonic-typerow contraction.Thisyieldsthemaximalityoftheoperatornorm ofaharmonicpolynomialoftheshiftonG making˘ thisspace anaturalharmoniccounterpartoftheDrury-Arvesonspace.

©2021ElsevierInc.Allrightsreserved.

* Correspondingauthor.

E-mailaddresses:alpay@chapman.edu(D. Alpay),kaptan@fen.bilkent.edu.tr(H.T. Kaptanoğlu).

URLs:http://www1.chapman.edu/~alpay/(D. Alpay),http://www.fen.bilkent.edu.tr/~kaptan/

(H.T. Kaptanoğlu).

https://doi.org/10.1016/j.jfa.2021.109058

0022-1236/©2021ElsevierInc. Allrightsreserved.

1. Introduction

AfterthepioneeringworkofDrury[10] andArveson[6] onextendingthevonNeumann inequality tocommuting operatortuples, there havebeen several other generalizations to Hilbertspace operatorsinother settings.Wecancite [20],[3],[16],[11],[13],and[8]

to name afew. There is alsothe earlier[19] on noncommutingoperator tuples. Butin allvon Neumanninequalitiesthatweknow of,thepolynomialsactingontheoperators are holomorphic functionsof theirvariables. It isthe aimof this workto obtainavon Neumanninequalityinwhichthepolynomials areharmonicintheusualsense inRⁿ.

MultivariableversionsofvonNeumanninequalityoftendependonshiftoperatorson a specific Hilbertfunction space. This immediately brings outthe first major obstacle in dealingwith harmonicity. Eventhe definitionof a shiftoperator on aspace of har- monic functionshasnotbeen madebefore, becauseharmonicity isnotpreservedunder multiplication, and amultiplication byacoordinatevariablemust be followedbysome form ofaprojectiononharmonicfunctions.Wemakeadefinitionandcheck itbyusing another approach.

Another obstacle is to decide which space among harmonic function spaces plays a role like thatofthe Drury-Arvesonspace among holomorphicspaces.We findoutthat consideringafamilyofreproducingkernelHilbertspacesGqofharmonicfunctionsonthe unitballofRⁿindexedbyq∈ R ismorefeasiblesinceitexposesthecompositionsofthe spaces better.Thenitiseasierto pickoneofthesespaces astheharmoniccounterpart of theDrury-Arvesonspaceusingitsextremalpropertiesinthefamily.

Onemoreobstacleisthatthemorecomplicatedstructure ofharmonicfunctionsper- sistsat theoperatorleveland weare obligedtorestrictourattentioninvon Neumann inequalitytoaclassofcontractionsthatwecall harmonictype.

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 thehomogeneouspolynomials ofdegree m and spherical harmonicsofdegreem on Rⁿ,respectively.LetHm:Pm → Hm be thestan- dard projection. The zonal harmonics Zm(x,y) arethe reproducing kernels of theHm

with respecttotheL² innerproductontheunitsphere.Forj = 1,. . . ,n,wedefine the shiftoperators Sj:Hm→ Hm+1actingonx by

SjZm(x, y) := 1 n + 2m

∂

∂y_jZm+1(x, y).

Our first main result shows thatS_j is closely relatedto the operatorof multiplication M_xj bythejthcoordinatevariable.

Theorem 1.1.Sj= Hm+1Mxj forallj = 1,. . . ,n andm= 0,1,2,. . ..

Inthecourseofprovingthistheorem,weobtainthefollowingidentitiesfortheGegen- bauer polynomials C_m^λ and theChebyshev polynomialsTm which seemnew.There, K

istheKelvintransformwhichtransformsaharmonicfunctionontheunitballtooneon itsexterior.

Theorem1.2. Forξ,η∈ S, wehave K

(η· ∂)^mK[1]

(ξ) = (−1)^mm! C_m^n/2−1(ξ· η) (n≥ 3, m = 0, 1, 2, . . .), K

(η· ∂)^mK[ log| · | ]

(ξ) = (−1)^m(m− 1)! Tm(ξ· η) (n = 2, m = 1, 2, . . .).

Wedefine thespacethatweclaim tobe theharmonic versionoftheDrury-Arveson spaceasthereproducingkernelHilbertspaceG on˘ theunitballofRⁿ withreproducing kernel

G(x, y) :=˘

∞ m=0

1 Am

Z_m(x, y),

whereAm isthecoefficientof(x· y)^m intheexpansionofZm(x,y).Wecall commuting operators(T1,. . . ,Tn) arowcontractioniftheyareacontractionasatuple.Acontractive normisoneinwhichthetupleofshiftoperatorsisarowcontraction.Anothermainresult ofoursshowsthatthenormofG is˘ as largeas possible.

Theorem1.3. If · isacontractiveHilbertnormonharmonic polynomialsthatrespects theorthogonalityof L²,then· ≤ · G˘1.

Wecall anoperatortuple (T1,. . . ,Tn) harmonic type ifT1T1+· · · + TnTn = 0. The harmonicshiftS = ( ˘˘ S1,. . . ,S˘n) onG is˘ theprimeexampleofaharmonic-typeoperator.

Ourfinalmain resultisavonNeumanninequality.

Theorem1.4. Let(T₁,. . . ,T_n) beaharmonic-typerowcontraction onaHilbertspace. If u isaharmonic polynomial,thenu(T1.. . . ,Tn)≤ u( ˘S1,. . . ,S˘n).

Allterminology is explained indetail in an appropriate section inthe paper. After introducinginSection2thebasicnotation,wemakeareviewofharmonic polynomials inSection3andgiveformulasfortheZ_m,H_m,andK.InSection4,wedefinetheshift operators on harmonic spaces, explain the meaning of the coefficient 1/(n+ 2m), and thenproveTheorem1.1.InSection5,we introduceanewfamilyofreproducing kernel HilbertspacesofharmonicfunctionsandisolateoneofthemasG by˘ makingitclearwhy we need the coefficients1/A_m in G.˘ In Section 6, we find the basicproperties of shift operatorsandtheiradjointsactingontheHilbertspacesjustintroduced.InSection7,we investigatetherowcontractionsonharmonicHilbertspaces,explainthetermharmonic type,and proveanessentialdilation resultforharmonic-typeand self-adjoint operator tuples.InSection8,weproveTheorems1.3and1.4.

Acknowledgments

ThefirstauthorthankstheFosterG.andMaryMcGawProfessorshipinMathematical Sciences of Chapman University for its support. The second author thanks Aurelian GheondeaofBilkentUniversityandSerdarAyofAtılımUniversityforusefuldiscussions.

The authors also thank an anonymous referee for suggesting to consider self-adjoint operatorswhichencouragedustoobtaintheresultsonsuchoperatorsinthenexttolast section.

2. Notation

LetB andS betheopenunitballanditsboundarytheunitsphereinRⁿwithrespect to the usualinner product x· y = x1y1+· · · + xnyn and thenorm |x|=√

x· x, where alwaysn≥ 2.Wewritex= rξ, y = ρη withr =|x|,ρ=|y|,andξ,η∈ S,andusethese throughout withoutfurther comment.When n = 2,the ball is justthe unitdisc D in the complexplanebounded bytheunitcircleT , and x,y∈ D arecomplexnumbersof moduluslessthan1.

In afewplaces, we alsouse thecomplexspace C^N and its Hermitianinner product

z,w = z1w1+· · · + zNwN.Wecontinueto use|· |,B,andS in C^N too.

Weletσ andν bethesurfaceandvolumemeasuresonS andB normalizedasσ(S)= 1 and ν(B)= 1. Weabbreviatetheall-importantLebesgue classL²(σ) to simplyL². An overline(·) denotesclosureforsetsandcomplexconjugationforelements;forpolynomials inx,theconjugationaffectsonlythecoefficientsnaturally.Thegreatestintegerlessthan or equalto areal numberisshownby·.Therightside of:= definesitsleftside.

Harmonic functionsby definitionarethosesufficiently smoothfunctionsannihilated by the usual Laplacian Δ := ∂²/∂x²₁+· · · + ∂²/∂x²_n. We leth(B) denote thespace of complex-valued harmonic functionsonB with the topologyof uniform convergence on compact subsets.

In the multi-indexnotation, α = (α1,. . . ,αn) isan n-tuple of nonnegative integers,

|α|= α₁+· · · + αn, α!= α₁!· · · αn!,0⁰= 1,andx^α= x^α₁¹· · · x^αnⁿ. Letting∂_j := ∂/∂x_j and∂ := (∂1,. . . ,∂n),wealsohave∂^α= ∂₁^α1· · · ∂n^αn,∂^αx^α= α! andp(∂)=

α

aα∂^αfor apolynomialp(x)=

α

aαx^α.Soforp(x)=|x|²,

|x|²(∂) = Δ. (1)

ThePochhammersymbol (a)_b is definedby

(a)b=Γ(a + b) Γ(a)

when a and a+ b are off the pole set −N ofthe gamma function Γ.This is a shifted risingfactorialsince(a)k = a(a+ 1)· · · (a+ k− 1) forpositive integerk.Inparticular, (1)k = k! and(a)0= 1.Stirlingformulagives

Γ(c + a)

Γ(c + b) ∼ c^a−b, (a)c

(b)_c ∼ c^a−b, (c)a

(c)_b ∼ c^a−b (Re c→ ∞), (2) whereA∼ B meansthat|A/B| isboundedaboveandbelowbytwopositiveconstants, thatis,A=O(B) andB =O(A),forallA,B ofinterest.Soforexample,1−|x|∼ 1−|x|² forallx∈ B.Such constantsthatareindependent oftheparametersandthefunctions in the equation are all denoted by the generic unadorned upper case C. We also use A B to meanA=O(B).

WedenoteaninnerproductonafunctionspaceH by[·,·]_Handtheassociatednorm by· H.

Definition 2.1.A functionk(x,y) is called thereproducing kernel of aHilbertspace H offunctionsdefinedonB ifk(x,·)∈ H foreachx∈ B and

u(x) = [ u(·), k(x, ·) ]H (u∈ H, x ∈ B).

Thereisaone-to-onecorrespondencebetweenreproducingkernelHilbertspacesand positivedefinitekernels.Weusewordslike positiveandincreasingtomeannonnegative andnondecreasing.

Thealgebraofallboundedlinearoperators onacomplexHilbertspaceH isdenoted B(H).AnoperatorT onH iscalled positive andwewrite T ≥ 0 if[T v,v]_H ≥ 0 forall v∈ H.Fora,b∈ H,a⊗ b denotestherank-1 operatordefinedby(a⊗ b)(v)=

v,b

Ha forv∈ H.

3. Harmonicpolynomials

Wereview theessentialsof zonalharmonics andtheKelvin transformfor complete- ness,becausewerefertothese factsmanytimesinthepaper.These resultsaremostly well-knownandcanbe consultedin[7,Chapters4&5].

Form= 0,1,2,. . .,letPmdenotethecomplexvectorspaceofallpolynomialshomo- geneous(withrespecttoreal scalars)ofdegreem onRⁿ. Itisimmediatethat

(x· ∂)pm= m pm (pm∈ Pm). (3)

Let Hm be the subspace of Pm consisting of all harmonic homogeneous polynomials of degree m. By homogeneity, a p_m ∈ Pm is determined by its restriction to S, and we freely identifyp_m with its restriction. The restrictionsof those u_m ∈ Hm to S are calledsphericalharmonics.WealsoletP andH denoteallpolynomialsandallharmonic polynomialsonRⁿ.

WeregardPmandHmassubspacesofL²inissuesrequiringanormorinnerproduct.

Forexample,ifm= l,thenHmisorthogonal toHlwithrespect to[·,·]L². Proposition 3.1. [7,Exercise5.12] If um,vm∈ Hm,then

um(∂)(vm) = n(n + 2)· · · (n + 2m − 2)

S

umvmdσ

= 2^m(n/2)m

um, vm]_L².

Proof. Letum(x)= 

|α|=m

bαx^αandvm(x)= 

|α|=m

dαx^α.By[7,Theorem5.14],theright sidesare 

|α|=m

α!bαdα. Itiseasytocheck thatthisis equaltoum(∂)(vm). 

Ifpm∈ Pm,thenthereareuniqueul∈ Hl suchthat

p_m(x) = u_m(x) +|x|²u_m−2(x) +· · · + |x|^2ku_m−2λ(x) (x∈ Rⁿ),

where λ=m/2.This decompositionis simplyp_m = u_m+ u_m−2+· · · + um−2λ when restrictedto S.Let

H_m:Pm→ Hm, p_m→ um

bethemapthatprojectsp_mtou_m.TheexplicitformulaforH_mrequiressomeconstants c_m, m= 1,2,. . .,definedby

cm:=

(−1)^m2^m(n/2− 1)m, if n≥ 3, (−1)^m−12^m−1(m− 1)!, if n = 2.

It alsomakesuseoftheKelvin transform K definedonafunctionf by K[f ](x) :=|x|²⁻ⁿf (x^∗) (x= 0), where x^∗:= x

|x|²,

whichreducestoK[f ](x):= f (x^∗) whenn= 2.NotethatK[f ]= f onS foranyn≥ 2.

TheKelvintransform islinearandinvertiblewith K⁻¹= K.Afunctionis harmonicif andonlyifitsKelvintransformisharmonic.Sou(x)=|x|²⁻ⁿisharmonicespeciallyfor n≥ 3 whereveritisdefinedsinceitistheKelvintransformoftheconstant1.Forn= 2, itis replacedinformulasbytheharmonicfunctionu(x)= log|x|.

Theorem 3.2.[7,Theorem5.18] Letm≥ 1 and pm∈ Pm.

(a) H_m(p_m)(x)= 1 cm

K

p_m(∂)|x|²⁻ⁿ

, if n≥ 3, K

p_m(∂) log|x|

, if n = 2.

(b) When pm is restricted to S, then Hm (without any need for K) is an orthogonal projection with respectto[·,·]L²

ThespacesHm arefinitedimensional,henceclosed subspacesofL²,and

δm:= dimHm= (n + 2m− 2)(n− 1)m−1

m! (m≥ 1), (4)

which gives δm = 2 for m ≥ 1 when n = 2. When m = 0, H0 = C and δ0 = 1.

The next few are δ1 = n, δ2 = (n− 1)(n+ 2)/2, and δ3 = (n− 1)n(n+ 4)/6. Thus evaluation functionals at points η ∈ S arebounded on Hm, and Hm is a reproducing kernelHilbertspace.ItsreproducingkernelZ_m(ξ,η) withrespectto[·,·]_L² iscalledthe zonalharmonic ofdegreem;soZ_misapositivedefinitefunction.EachZ_misrealvalued andsymmetricinitsvariables,henceitisaharmonichomogeneouspolynomialineach ifitstwo variables.ThehomogeneityoftheZmgivesZm(x,y):= r^mρ^mZm(ξ,η);so

Zm(0, y) = Zm(x, 0) = 0 (m≥ 1). (5) Theirreproducingpropertywrittenexplicitlyis

u_m(x) =



S

u_m(η) Z_m(x, η) dσ(η) =

u_m(·), Zm(x,·)

L² (x∈ B, um∈ Hm). (6) ThePoissonkernelis

P (x, η) := 1− |x|²

|x − η|ⁿ =

∞ m=0

Z_m(x, η) (x∈ B, η ∈ S);

theseriesconvergesuniformly forx inacompactsubsetofB.

ThereisanexplicitformulafortheZ_mwhichisof majorinteresttous; itis

Zm(x, y) = (n+2m−2)

m/2

l=0

n(n+2)· · · (n+2m−2l−4)

(−1)^l2^ll! (m− 2l)! |x|^2l(x·y)^m−2l|y|^2l

= A_m0(x· y)^m+A_m1|x|²(x·y)^m−2|y|²+A_m2|x|⁴(x·y)^m−4|y|⁴+· · · , (7)

whereAm:= Am0 istheleadingcoefficientobtainedforl = 0.Then

A_m:= n(n + 2)· · · (n + 2m − 2)

m! = 2^m(n/2)_m

m! , (8)

where thenumeratoris thecoefficientinthe equationinProposition3.1. Thefirst few areA₀= 1,A₁= n,A₂= n(n+ 2)/2,andA₃= n(n+ 2)(n+ 4)/6.NotethatA_m= 2^m forallm= 0,1,2,. . . whenn= 2.

Interesting andusefulrelationsinclude

|Zm(ξ, η)| ≤ Zm(ξ, ξ) =

m/2

l=0

A_ml= δ_m (ξ, η∈ S). (9)

If{Ym1,. . . ,Ymδm} isanorthonormalbasisforHm⊂ L²,then

Zm(ξ, η) =

δm



k=1

Ymk(ξ) Ymk(η). (10)

In particular, Z0 ≡ 1 and we takeY01 ≡ 1;also Z1(x,y)= n(x· y) and we canchoose Y1k(x)=√

n xk fork = 1,. . . ,δ1= n. Itisalways possibleto choosetheYmk withreal coefficients.

Theorem 3.3. Every u ∈ h(B) has the homogeneous expansion u = ^∞

m=0

um in which um∈ HmandwhichconvergesabsolutelyanduniformlyoncompactsubsetsofB.Letting u_mk=

u_m,Y_mk

L²∈ C, wealso have

u(x) =

∞ m=0

δm



k=1

u_mkY_mk(x) (x∈ B)

with thesametype ofconvergence.

Whenn= 2,we canusecomplexanalysis andFourieranalysis toconnecttheabove theory to better knownobjects. Then for all m ≥ 1, an orthonormal basis for Hm is {Ym1(x) = x^m,Ym2(x) = x^m : x ∈ C }. The expansion of a u ∈ h(B) with the Ymk

inTheorem 3.3 withsuitableboundarybehavior is itsFourierseries ontheunitcircle.

Lettingξ = e^iφ andη = e^iψ,wecanwrite

Z_m(x, y) = x^my^m+ x^my^m= 2r^mρ^mcos(φ− ψ) (m≥ 1).

It isamazing that this simpleform is equal to thesum in(7) for n= 2, whichis still complicated.

A comparison with thecomplex case donein [15, Section3] when n iseven is very instructive.WethinkofRⁿ asC^N byequatingn= 2N .Sphericalharmonicscorrespond to thespace of holomorphic polynomials homogeneousof degree m whichsimply have the form 

|α|=m

bαz^α. Thedimensionofthis space is (N )m

m! , whichequals 1 foreverym when N = 1.Thecounterpartsofzonalharmonicsarethesesquiholomorphickernels

M_m(z, w) = (N )_m

m! z, w ^m.

Thus the complex version of the sum in (7) has only the leading term Mm, and its coefficientisexactlythedimensionofthespaceofwhichMm isthereproducingkernel.

Note thatAm= δminthe harmoniccaseeven when n= 2. However,writing(7) with x= y = ξ andusing(9),wesee that

m/2

l=0

Aml= δm.By(2),wehave

δ_m∼ mⁿ⁻² and A_m∼ 2^mm^n/2−1 (m→ ∞). (11) Yetthereproducingkernel oftheholomorphic Drury-Arvesonspace is ^∞

m=0z,w ^m and not ^∞

m=0

M_m(z,w).

Thereforewe must find theharmonic counterparts of thez,w ^m and we are ledto the

Xm(x, y) := 1

A_mZm(x, y) = (x· y)^m− · · · , (12) whichwecallthexonalharmonics.ThefirstfewareX₀= 1,X₁(x,y)= x· y,

X2(x, y) = (x· y)²−|x|²|y|²

n , X3(x, y) = (x· y)³− 3

n + 2|x|²(x· y)|y|². Byhomogeneity, (9),(4),and (8),weseethatforallx,y∈ B,

|Xm(x, y)| = r^mρ^m|Xm(ξ, η)| = (rρ)^m|Zm(ξ, η)|

Am ≤ (rρ)^mδ_m

Am ≤ (rρ)^m< 1 (13) incomplete analogywith|z,w ^m|≤ |z|^m|w|^m< 1 for z,w intheunitballofC^N. 4. Shiftoperators

Wedefinetheshiftoperatorsonharmonicfunctionsfirstinanunusualway,butlater showthattheyareequivalent essentiallyto multiplicationsbythecoordinatevariables.

Themotivationforourdefinitionliesintheobservation 1

m + 1

∂

∂w_j

z, w ^m+1

= z_jz, w ^m

forz,w∈ C^N andtherealizationthatXm(x,y) replacesz,w ^m.

Definition4.1. For1≤ j ≤ n,wedefine thejth shiftoperator S_j:Hm→ Hm+1 acting onthevariablex byfirstletting

S_jX_m(x, y) := 1 m + 1

∂

∂y_jX_m+1(x, y)

and thenextendingtoallofHmbylinearityandthedensityoftheXm(·,y) inHm. Note that all this make sense; a partial derivative of a harmonic function is again harmonic,andfinitelinearcombinationsofthereproducingkernelsZ_mandhenceofthe XmaredenseinHmin· L².Alsonote thattheshiftactsonthefirstvariablex,but the partial derivativeis with respect to the second variable y. So occasionally we also use notation like S^x or ∂y to indicatethe variables onwhich theyact. In termsof the morefamiliar zonalharmonics,

S_j^xZm(x, y) = Am

A_m+1 1 m + 1

∂

∂y_jZm+1(x, y) = 1 n + 2m

∂

∂y_jZm+1(x, y).

Let’s denoteby S_j^∗:Hm→ Hm−1 the adjointofSj with respectto [·,·]L² inwhich the reproducingkernel ofHmis Zm= AmXm. Thisis ofcoursethe jthbackward shift operator.FirstwesetS_j^∗(X0)= S_j^∗(1)= 0.Nextform≥ 1,using(6),symmetryofXm

initsvariables,anditsreal-valuedness, weobtain (S_j^x)^∗Xm(x, y) =

(S_j^t)^∗Xm(t, y), A_m−1X_m−1(x, t)

L²

= A_m−1

X_m(t, y), S_j^tX_m−1(x, t)

L²

= A_m−1

X_m(t, y), 1 m

∂

∂x_jX_m(x, t) 

L²

= 1 m

Am−1

A_m

AmXm(t, y), ∂

∂x_jXm(x, t) 

L²

= 1 m

A_m−1 Am

∂

∂xj

X_m(x, y),

wherey actsjustlikeaparameter.ThislastformulaforS_j^∗isindependentofthepartic- ular form ofthefunction onwhichit acts,so worksequallywell foru_m∈ Hm inplace of Xmbylinearityand densityagain. Thus

S_j^∗u_m= 1 m

A_m−1 Am

∂_ju_m= 1

n + 2m− 2∂_ju_m (u_m∈ Hm, m≥ 1). (14) It is clearthat theS_j^∗ commute with eachother. Sothe shift S = (S1,. . . ,Sn) is a commutingtuple.

Shiftoperatorsonholomorphicfunctionspacesareoperatorsofmultiplicationbythe coordinate variables. Here we make a distinction between the two, becausethe latter doesnotingeneralcarryharmonicfunctionstoharmonicfunctionsunliketheformer.If f,g arefunctionsonthesamedomain,welet

Mgf = gf be theoperatorofmultiplication byg.

OneofourresultsconcernsalimitedversionoftheobviousfactthatifS isthetuple of shift operators ona space of holomorphic functionson a domain in C^N and p is a holomorphicpolynomialinN complexvariables,then p(S)= Mp.It isafirstresulton theconnectionbetweenshiftsand multiplicationoperators.

Proposition 4.2.If u isa harmonic polynomial,then u(S)(1)= u.In other words,1 is acyclic vectorforh(B).

Proof. Itsufficesto consideru= u_m∈ Hm. By(6),repeated useof(14),and Proposi- tion3.1,

u_m(S^x)(1)(x) =

u_m(S^η)(Z₀)(η, y), Z_m(x, η)

L²

=

Z₀(η, y), u_m((S^η)^∗)(Z_m)(x, η)

L²

=

Z₀(η, y), 1

n(n + 2)· · · (n + 2m − 2)u_m(∂_η)(Z_m)(x, η) 

L²

= 1

2^m(n/2)m

u_m(∂_η)(Z_m)(x, η), Z₀(y, η)

L²

= 1

2^m(n/2)_mu_m(∂_y)(Z_m)(x, y)

=



S

um(·) Zm(x,·) dσ(·) = um(x).

Above,S^η and∂η mustbeinterpretedasS^y

y=η and∂y

y=η,respectively.Notethatwe usetheharmonicityofu_monlyinpassingtothelast line. _

Ifthefunctionactedonismorecomplicatedthan1,thenweofferthefollowingpartial result.

Proposition4.3. If pm∈ Pm,then

pm(S^x)(X)(x, y) = 1 (1 + )m

pm(∂y)(Xm+)(x, y).

Proof. FollowingthesameideaandnotationintheproofofProposition4.2, pm(S^x)(Z)(x, y) =

pm(S^η)(Z)(η, y), Zm+(x, η)

L²

=

Z(η, y), pm((S^η)^∗)(Zm+)(x, η)

L²

=

Z(η, y), pm(∂η)(Zm+)(x, η)

(n + 2 )(n + 2 + 2)· · · (n + 2 + 2m − 2) 

L²

= 1

2^m(n/2 + )m

pm(∂η)(Zm+(x, η), Z(y, η)

L²

= 1 2^m(n/2 + )m

pm(∂y)(Zm+)(x, y).

Lastly, we pass to the xonal harmonics X_ = Z_/A_ and simplify the resulting coeffi- cient. 

Our goal now is to express the shift operators S_j : Hm → Hm+1 in terms of the operators of multiplication by the coordinate variables M_xj : Hm → Pm+1, where for both j = 1,. . . ,n.ItisTheorem 1.1andwerestateit.

Theorem 4.4. For all j = 1,. . . ,n and m = 0,1,2. . ., if S_j : Hm → Hm+1, then S_j= H_m+1M_xj.

So the harmonic shiftsare really Toeplitz operators. Wefacilitate the longproof of this theorem withsomecomputationallemmas inwhichj = 1,. . . ,n,y is aparameter, and allpartial derivativesandKelvintransformsarewithrespecttox.

Lemma 4.5.Fora,b∈ R,easy computations give

∂_j|x|^a= a|x|^a−2x_j and ∂_j(x· y)^b= b(x· y)^b−1y_j, (y· ∂)|x|^a= a|x|^a−2(x· y) and (y· ∂)(x · y)^b= b(x· y)^b−1|y|².

Lemma 4.6. If a polynomial p has |x|² as a factor, then p(∂)|x|²⁻ⁿ = 0 and also p(∂)log|x|= 0 whenn= 2.Consequently

Xm(∂, y)|x|²⁻ⁿ= (y· ∂)^m|x|²⁻ⁿ and Xm(∂, y) log|x| = (y · ∂)^mlog|x| (n = 2).

Proof. The firststatementfollows immediately from (1) and theharmonicity of|x|²⁻ⁿ andoflog|x| whenn= 2.Thesecondstatementfollowsfromtheexplicitformsofzonal harmonics in (7), because all the terms in X_m(x,y) except the first have a factor of

|x|². _

This lemma isveryuseful, becauseitletsustreatXm asthesingle term(x· y)^m in thepresenceof K orHm likeitsholomorphiccounterpartz,w ^m.

Lemma 4.7.Form≥ 1,

(y· ∂)^m|x|²⁻ⁿ= c_m Xm(x, y)

|x|^n+2m−2 (n≥ 3), (y· ∂)^mlog|x| = cm

Xm(x, y)

|x|^2m (n = 2);

and hence

K

(y· ∂)^m|x|²⁻ⁿ

= cmXm(x, y) (n≥ 3), K

(y· ∂)^mlog|x|

= c_mX_m(x, y) (n = 2).

The identitiesform≥ 3 aretrue also form= 0 ifweset c0= 1.

FromLemma4.7,Theorem1.2 followseasilywhichwerestate.

Theorem4.8. Forξ,η∈ S, wehave K

(η· ∂)^mK[1]

(ξ) = (−1)^mm! C_m^n/2−1(ξ· η) (n≥ 3, m = 0, 1, 2, . . .), K

(η· ∂)^mK[ log| · | ]

(ξ) = (−1)^m(m− 1)! Tm(ξ· η) (n = 2, m = 1, 2, . . .), where C_m^λ is the Gegenbauer (ultraspherical) polynomial of degree m and indexλ, and T_mistheChebyshev polynomialof thefirstkindof degree m.

Proof. Let’s first note that K[1] = |x|²⁻ⁿ for n = 3 and K[ log|x|] = −log|x| for n= 2. Thefirstidentity isaconsequenceof thewell-knownrelationbetweenthe zonal harmonicsandGegenbauerpolynomials;see[12,(14.8)] forexample.Thesecondidentity holdsbecauseofthecloseconnectionbetweenGegenbauerpolynomials withparameter 0 andChebyshevpolynomials;see[18,18.1.1] forexample. 

The identities in Lemma 4.7 and Corollary 4.8 seem new; we are unable to locate theminstandardreferencessuchas[18].Theyalsogivefurtherindicationthatthexonal harmonicsXmareimportantintheirownrightsincethecoefficientsintheidentitiesin Lemma4.7 withtheZmarenotsimpleknownones.

Proof of Lemma4.7. Thesecondset ofidentitiesfollowsimmediately fromthefirstset, andfortheseweproceedbyinductiononm.Wegivetheproofonlyforn≥ 3;theproof for n = 2 is obtained by replacing |x|²⁻ⁿ by log|x| and setting n = 2 in appropriate places.Form= 1,byLemma4.5,

(y· ∂)|x|²⁻ⁿ= (2− n)|x|⁻ⁿ(x· y) = c1

X1(x, y)

|x|ⁿ .

NextweassumethefirstidentityinLemma4.7holdsform,andshowthatitalsoholds form+ 1.Byapplying theinductionhypothesis,differentiatingwiththequotient rule, andusingLemma4.5,weobtain

(y· ∂)^m+1|x|²⁻ⁿ= (y· ∂)

c_m Xm(x, y)

|x|^n+2m−2



= cm|x|^n+2m−2(y· ∂)Xm− Xm(n + 2m− 2)|x|^n+2m−4(x· y)

|x|^2n+4m−4

= cm|x|²(y· ∂)Xm− (n + 2m − 2)(x · y)Xm

|x|^n+2m .

Writing outthecoefficients,wemustshow

(x· y)Xm− |x|²

n + 2m− 2(y· ∂)Xm= X_m+1. By(7) andLemma4.5,theleftsideequals

n + 2m− 2 A_m

m/2

l=0

n(n + 2)· · · (n + 2m − 2l − 4)

(−1)^l2^ll! (m− 2l)! |x|^2l(x· y)^m−2l+1|y|^2l

− 1 Am

m/2

l=0

n(n + 2)· · · (n + 2m − 2l − 4)

(−1)^l2^ll! (m− 2l)! 2l|x|^2l(x· y)^m−2l+1|y|^2l

− 1 Am

m/2

l=0

n(n + 2)· · · (n + 2m − 2l − 4)

(−1)^l2^ll! (m− 2l)! (m− 2l)|x|^2l+2(x· y)^m−2l−1|y|^2l+2

= 1 Am

m/2

l=0

n(n + 2)· · · (n + 2m − 2l − 2)

(−1)^l2^ll! (m− 2l)! |x|^2l(x· y)^m−2l+1|y|^2l

+ 1 A_m

m/2+1

l=1

n(n + 2)· · · (n + 2m − 2l − 4)

(−1)^l2^l−1(l− 1)! (m − 2l + 1)!|x|^2l(x· y)^m−2l+1|y|^2l

= m + 1 A_m

m/2

l=1

n(n + 2)· · · (n + 2m − 2l − 2)

(−1)^l2^ll! (m + 1− 2l)! |x|^2l(x· y)^m+1−2l|y|^2l+ extra term + (x· y)^m+1,

where theextra termis due to l =m/2+ 1 in the second sumon theprevious line.

Theright sideequals

n + 2m Am+1

(m+1)/2

l=1

n(n + 2)· · · (n + 2m − 2l − 2)

(−1)^l2^ll! (m + 1− 2l)! |x|^2l(x· y)^m+1−2l|y|^2l+ (x· y)^m+1. Since (m+ 1)/A_m = (n+ 2m)/A_m+1, the two sides are equal except for the extra termandthatthesumontherightsideendsperhapsatalargervalue.Whenm iseven, thel givingrisetotheextratermequalsm/2+ 1,andthenm− 2l + 1=−1< 0 inthat term, sothere isreallynoextraterm.Also(m+ 1)/2= m/2=m/2,andthesums onbothsidesendatthesamevalue.Whenm isodd,thel givingrisetotheextraterm equals (m+ 1)/2.Also(m+ 1)/2= (m+ 1)/2.Sotheextratermandthelasttermin thesumontherightarethesame. 

Proof of Theorem4.4. Againwewritetheproofonlyforn≥ 3.Itsufficestodotheproof onlyforum(·)= Xm(·,y) asinDefinition4.1.InviewofLemma4.6andbyTheorem3.2, allweneedtoshowis

1 m + 1

∂

∂yj

Xm+1(x, y) = 1 cm+1

K ∂

∂xj

(y· ∂)^m|x|²⁻ⁿ 

forallm= 0,1,2,. . .,whereK and∂ arewithrespecttox.ApplyingLemma4.7onthe rightandcombiningtheconstants,thisequationtakes theform

∂

∂yj

X_m+1(x, y) =− m + 1 n + 2m− 2K

∂

∂xj

X_m(x, y)

|x|^n+2m−2



(15)

By(7) andLemma4.5, theleft sideequals n + 2m

A_m+1

_(m+1)/2



l=0

n(n + 2)· · · (n + 2m − 2l − 2)

(−1)^l2^ll! (m− 2l)! x_j|x|^2l(x· y)^m−2l|y|^2l

+

(m+1)/2

l=1

n(n + 2)· · · (n + 2m − 2l − 2)

(−1)^l2^l−1(l− 1)! (m + 1 − 2l)!yj|x|^2l(x· y)^m+1−2l|y|^2l−2

 .

ByLemma4.5,

∂

∂xj

X_m(x, y)

|x|^n+2m−2



= |x|^n+2m−2∂_jX_m− Xm(n + 2m− 2)|x|^n+2m−4x_j

|x|^2n+4m−4

= |x|²∂jXm− (n + 2m − 2)xjXm

|x|^n+2m .

AftertakingtheKelvintransform,therightsideequals

(m + 1)

xjXm− 1

n + 2m− 2|x|²∂Xm

∂x_j



= m + 1 A_m



(n + 2m− 2)

m/2

l=0

n(n + 2)· · · (n + 2m − 2l − 4)

(−1)^l2^ll! (m− 2l)! x_j|x|^2l(x· y)^m−2l|y|^2l

−

m/2

l=0

n(n + 2)· · · (n + 2m − 2l − 4)

(−1)^l2^ll! (m− 2l)! 2l xj|x|^2l(x· y)^m−2l|y|^2l

−

m/2

l=0

n(n + 2)· · · (n + 2m − 2l − 4)

(−1)^l2^ll! (m− 2l − 1)! yj|x|^2l+2(x· y)^m−2l−1|y|^2l



= m + 1 Am

_m/2



l=0

n(n + 2)· · · (n + 2m − 2l − 2)

(−1)^l2^ll! (m− 2l)! x_j|x|^2l(x· y)^m−2l|y|^2l

+

m/2+1

l=1

n(n + 2)· · · (n + 2m − 2l − 2)

(−1)^l2^l−1(l− 1)! (m + 1 − 2l)!yj|x|^2l(x· y)^m+1−2l|y|^2l−2

 .

Since thecoefficients multiplying thesums areequal,the twosides areequalexcept perhaps inthe upperlimitsof thesums.Let m beeven. Theupper limitonthe leftis l =(m+ 1)/2= m/2=m/2.Inthesumwithx_j,thisisalsotheupperlimitonthe right.Thesumwithy_j ontherightendswithl = m/2+ 1,butthenm+ 1−2l = −1< 0, so this term is not really there. Next let m be odd. The upper limit on the left is l = (m+ 1)/2= (m+ 1)/2 =m/2+ 1. Inthe sumwith y_j,this is also theupper limit onthe right.Thesum withx_j onthe right endswith l = (m− 1)/2,so it seems as ifthetermontheleftwithl = (m+ 1)/2 isextra,butthenm− 2l = −1< 0,sothis termisnotreallythere,either. 

Example 4.9. Let’s compute the action of shifts on a very simple harmonic function.

Let’s find S1u andS2u for u(x)= x1. We apply Theorem 4.4 and follow therecipe in Theorem 3.2separately for n≥ 3 and n= 2.Straightforward computations yield that S1x1 = x²₁− |x|²/n for any n ≥ 2. On theother hand,simply S2x1 = x2x1 since this product isalreadyharmonic.

5. HarmonicHilbertfunctionspaces

Weareinspiredbyafewearlierworksindefiningnewreproducingkernelswithdesired properties.In[16],familiesofweightedsymmetricFockspacesofholomorphicfunctions thatincludetheDrury-Arvesonspacearestudiedfollowing [6]. In[12],Bergman-Besov kernels are definedas weightedinfinite sumsofzonal harmonics muchlike the Poisson kernel.Andwehavealreadynotedthattherighttoolisthexonalharmonicsratherthan thezonal harmonics.

Definition 5.1.Letβ :={βm> 0: β0= 1, m= 0,1,2,. . .} beasequencesatisfying

lim sup

m→∞

δ_m Am

β_m

1/m

≤ 1. (16)

Wedefine positivedefinite kernelsby

Gβ(x, y) :=

∞ m=0

βmXm(x, y) (x, y∈ B)

and spacesGβ asthereproducingkernelHilbertspacesgeneratedbythese kernels.

Wecanalsowrite

G_β(x, y) =

∞ m=0

δm



k=1

βm

Am

Y_mk(x) Y_mk(y) (x, y∈ B)

by(10).Nothingabouttheboundedness,summability,ormonotonicityofβ isassumed atthispoint.Butthecondition(16),via(13),ensuresthattheseriesdefiningthekernels Gβ converge absolutely and uniformly on compact subsets of B× B and hencedefine harmonicfunctionsof x,y∈ B.Forany β,

G_β(0, y) = G_β(x, 0) = 1 (17)

by(5) since β0 = 1, Gβ(x,y)= Gβ(y,x), and Gβ is real-valued. The Gβ depend onx andy viax· y sincetheZm areconstantmultiplesof Gegenbauerpolynomials ofx· y;

see[12,(14.8)]

Theorem5.2. Theelements of Gβ are harmonicfunctions onB.

Proof. This is by [5, p. 43]; the result there is stated for sesquiholomorphic kernels, butworks equallywell for harmonic kernels since both function classeshave thesame topology,thetopologyofuniformconvergenceoncompactsubsets.Thehypothesesthere aresatisfied,becauseeachGβ(x,y) islocallybounded by(13) andaharmonicfunction ineachvariableonB. 

AlsoeveryGβ− βmXm= ^∞

l=mβlXl ispositive definite.Then by[5,Theorem II.1.2], everyHm iscontinuouslyimbeddedineachGβ.

Theorem5.3. Thespace Gβ coincideswith thespaceofharmonic functionsu on B with expansionsasin Theorem3.3forwhich

u²_Gβ :=

∞ m=0

um²_Gβ:=

∞ m=0

Am

β_m um²L² <∞ (18) equippedwith theinnerproduct

u, v

Gβ :=

∞ m=0

um, vm



Gβ :=

∞ m=0

Am

β_m

um, vm



L². (19)

Moreover,



W_mk^β :=

βm

A_mYmk: k = 1, . . . , δm, m = 0, 1, 2, . . .



isanorthonormal basisforGβ.

Proof. Weadapttheproofof[5,Theorem III.3.1] tooursituationmainlytoshowhow theL² normcomesin.Considerthespaceofu∈ h(B) satisfyingthefinitenesscondition