In this note, it is proved that multiplier algebras of analytic reproducing kernel Hilbert spaces which are compatible with the action of the torus group possess Kraus’ completely contractive approximation property (CCAP) and, consequently, have the Property S_{&#92;sigma}. Our results apply in particular to the usual reproducing kernel Hilbert spaces on bounded symmetric domains

Barbian, Christoph

Barbian Christoph

University of Szeged

Approximation properties for multiplier algebras of reproducing kernel Hilbert spaces

Scientific publications of the Saarland University

Universita¨t des SaarlandesUN IV E R S IT A SSAR A V I E NS ISFachrichtung 6.1 – MathematikPreprint Nr. 209Approximation properties for multiplieralgebras of reproducing kernel Hilbert spacesChristoph BarbianSaarbru¨cken 2008Fachrichtung 6.1 – Mathematik Preprint No. 209Universita¨t des Saarlandes submitted: April 10, 2008Approximation properties for multiplieralgebras of reproducing kernel Hilbert spacesChristoph BarbianSaarland UniversityDepartment of MathematicsP.O. Box 15 11 5066041 Saarbru¨ckenGermanycb@math.uni-sb.deEdited byFR 6.1 – MathematikUniversita¨t des SaarlandesPostfach 15 11 5066041 Saarbru¨ckenGermanyFax: + 49 681 302 4443e-Mail: preprint@math.uni-sb.deWWW: http://www.math.uni-sb.de/AbstractIn this note, it is proved that multiplier algebras of analytic repro-ducing kernel Hilbert spaces which are compatible with the action ofthe torus group possess Kraus’ completely contractive approximationproperty (CCAP) and, consequently, have the Property Sσ. Our re-sults apply in particular to the usual reproducing kernel Hilbert spaceson bounded symmetric domains.1Approximation properties for multiplieralgebras of reproducing kernel Hilbert spacesChristoph BarbianApril 10, 20081 IntroductionSince the pioneering work of Grothendieck many different versions of approxi-mation properties for Banach and operator spaces have been considered. Thecommon ground of these properties is the question whether or not there ex-ists, over a given Banach or operator space X, a net of finite rank operatorsVi, possibly bounded in norm or cb norm, such that limi Vi(x) = x holds ina prescribed topology for all x ∈ X.In this note, we shall be concerned with the so-called σ-weak completelycontractive approximation property (CCAP) as introduced by J. Kraus in[10]: A σ-weakly closed subspace S of B(G,H) (G,H Hilbert spaces) is saidto have the CCAP if there exists a net (Vi)i of completely contractive σ-weakly continuous finite-rank operators on S such that limi Vi(S) = S holdsσ-weakly for all S ∈ S.In [10] it is shown that the dual algebra generated by an injective weightedshift has the CCAP. In particular this means that H∞(D), identified withthe multiplier algebra of the Hardy space H2(D), has the CCAP. It is themain result of this note to prove, more generally, that multiplier spaces ofholomorphic reproducing kernel Hilbert spaces have the CCAP whenever theunderlying reproducing kernel Hilbert spaces are circular. More precisely,let D ⊂ Cd be a circular domain (that is, eitD ⊂ D holds for all t ∈ R)containing the origin. Then a reproducing kernel Hilbert space H ⊂ O(D)is called circular if it contains the constant functions and if the reproducingkernel KH of H satisfies KH(eitz, eitw) = KH(z, w) for all z, w ∈ D andt ∈ R. This class of spaces includes the usual reproducing kernel spaces onbounded symmetric domains (see for instance [7] or [1]), which attracted alot of attention in recent time. In particular, the Hardy and Bergman spaceover the d-dimensional unit ball, and also the Arveson-Drury space (cf. [3]2for details) are circular in the above sense. The key step in the proof of ourmain result (Theorem 3.3) is the observation that every multiplier betweencircular reproducing kernel Hilbert spaces can be approximated pointwise bya sequence of polynomials which is bounded in multiplier norm.As an immediate consequence, multiplier algebras of circular reproducing ker-nel Hilbert spaces have the CCAP, and therefore have Kraus’ Property Sσ(cf. [9] and [10]). This can be used to prove a tensor product formula (Corol-lary 3.4) which generalizes the well-known equality H∞(D, T ) = H∞(D)⊗T ,where H∞(D, T ) is the space of all bounded holomorphic functions on D withvalues in a σ-weakly closed subspace T of B(D, E), and H∞(D)⊗T denotesthe normal spatial tensor product, that is, the σ-weak closure of H∞(D)⊗Tin B(H2(D)⊗D, H2(D)⊗E). A generalization of this formula in the settingof strictly pseudoconvex domains can be found in the recent paper [6].2 Reproducing kernel Hilbert spacesIn this preliminary section, we are going to recapitulate some basic factsabout reproducing kernel Hilbert spaces (see [2] for further reference). AHilbert space H of functions on a set X with values in a Hilbert space E iscalled a reproducing kernel Hilbert space if the point evaluations δz : H →E , f 7→ f(z), are continuous for all z ∈ X. Equivalently, there exists aunique function KH : X × X → B(E) (the reproducing kernel of H) withthe property that the functions KH(·, z)x : X → E belong to H and that〈f,KH(·, z)x〉 = 〈f(z), x〉 holds for all f ∈ H, z ∈ X and x ∈ E . Moreover,it is well known that KH is a positive definite B(E)-valued function andthat conversely every positive definite function is the reproducing kernel of auniquely determined reproducing kernel Hilbert space. Hence it makes senseto define, for a given scalar reproducing kernel Hilbert space H ⊂ CX and aHilbert space E , the so-called inflation HE as the reproducing kernel Hilbertspace associated with the positive definite function KH · 1E . It is not hardto see that HE coincides with the Hilbert space tensor product H ⊗ E viacanonical identification.Given two reproducing kernel Hilbert spaces G,H ⊂ CX and Hilbert spacesD, E , the multiplier space M(GD,HE) is defined as the collection of all func-tions φ : X → B(D, E) with the property that the pointwise product φ · fbelongs to HE for all f ∈ GD. Furthermore, we write MT (GD,HE) for thespace of all multipliers taking values in a prescribed subspace T of B(D, E).As a consequence of the closed graph theorem, the multiplication operatorMφ : GD →HE , f 7→ φ·f , associated with a multiplier φ, is continuous. Thisinduces a seminorm onM(GD,HE) by setting ‖φ‖M = ‖Mφ‖. This seminorm3is a norm if we assume that G and H contain the constant functions, sincein this case the assignment φ 7→ Mφ is one-to-one. In this situation, we shalloften regard M(GD,HE) or, more generally, MT (GD,HE) as an (operator)subspace of B(GD,HE).Lemma 2.1. Suppose that G,H ⊂ CX are reproducing kernel Hilbert spacescontaining the constant functions and that D, E are Hilbert spaces.(a) A function ψ : X → B(D, E) belongs to M(GD,HE) with ‖ψ‖M ≤ 1if and only if the function X × X → B(E) , (z, w) 7→ KH(z, w)1E −KG(z, w)ψ(z)ψ(w)∗, is positive definite.(b) Let T be a σ-weakly closed subspace of B(D, E). Then MT (GD,HE) isσ-weakly closed in B(GD,HE).(c) A bounded net (ψi)i inM(GD,HE) converges σ-weakly to ψ ∈M(GD,HE)if and only if (ψi(z))i converges weakly to ψ(z) for all z ∈ X.(d) The equality M(GD,HE) = M(G,H)⊗B(D, E) holds, up to canonicalidentification.Proof. Part (a) is Corollary 3.5 in [5]. Assertion (b) is proved in [4], and(c) is checked by a straightforward calculation. To prove (d), one firstverifies by use of (a) that for every multiplier ψ ∈ M(G,H) and everyT ∈ B(D, E), the function ψ ·T belongs toM(GD,HE) withMψ·T =Mψ⊗T .Then (b) yields the inclusion M(G,H)⊗B(D, E) ⊂M(GD,HE). Conversely,fix ψ ∈ M(GD,HE) and choose nets (Pi)i and (Qj)j of finite-rank projec-tions approximating σ-weakly the identities on D and E , respectively. Thenagain by (a), the functions ψi,j = Qj · ψ · Pi belong to M(GD,HE) with‖ψi,j‖M ≤ ‖ψ‖M. Moreover, each ψi,j belongs to the algebraic tensor prod-uct M(G,H) ⊗ B(D, E). In fact, for rank-one projections P = u ⊗ u andQ = v ⊗ v, we have Qψ(z)P = 〈ψ(z)u, v〉(v ⊗ u) for all z ∈ X and by(a), the function X → C , z 7→ 〈ψ(z)u, v〉, belongs to M(G,H). Sincelimi,j ψi,j(z) = ψ(z) weakly for all z ∈ X, an application of (c) proves theclaim.3 Main resultsThroughout this section, D ⊂ Cd denotes a circular domain containing theorigin. For the reader’s convenience, we start by recapitulating some factsabout the Feje´r kernelsFN : [−π, π]→ [0,∞) , FN (t) =1NN−1∑n=0Dn(t) =1N(sin Nt2)2(sin t2)2 (N ≥ 1),4whereDn : [−π, π]→ C , Dn(t) =n∑ν=−neiνt (n ≥ 0)is the nth-order Dirichlet kernel. It is well known (see Lemma 2.2 in [8])that∫ pi−piFN(t) dt = 2π, and that (FN )N converges to zero uniformly outside(−δ, δ), for every 0 < δ < π. The following lemma probably is well known.Since we are unable to find an exact reference, we include a proof.Lemma 3.1. Suppose that X is a Banach space and that u ∈ O(D,X) is aholomorphic function. Then the functionsuN : D → X , uN(z) =12π∫ pi−piu(eitz)FN (t) dt (1)are polynomials (of degree at most N−1) with coefficients in X. Furthermore,limNuN(z) = u(z) (in the norm of X)holds for all z ∈ D.Proof. First of all, uN obviously is analytic. By the definition of the Feje´rkernel, we see thatDαuN(z) =12πNN−1∑n=0n∑ν=−n∫ pi−piDαu(eitz)ei(|α|+ν)t dtholds for all z ∈ D and α ∈ Nd0. By Cauchy’s Theorem, the integrals on theright-hand side of the above expression are zero whenever |α| ≥ N . Since Dis connected, this shows that uN is a polynomial of degree at most N − 1. Itremains to prove that (uN)N converges pointwise to u. So, given z ∈ D andǫ > 0, we choose δ > 0 such that ‖u(eitz) − u(z)‖ < ǫ holds for all |t| < δ.Using the remarks preceding this lemma, we obtain that‖uN(z)− u(z)‖ ≤12π∫|t|≤δ‖u(eitz)− u(z)‖FN (t) dt+12π∫δ≤|t|≤pi‖u(eitz)− u(z)‖FN(t) dt≤ ǫ+M sup{FN(t) ; δ ≤ |t| ≤ π}for all N ≥ 1, where M = sup{‖u(eitz) − u(z)‖ ; |t| ≤ π}. As indicatedearlier, the sequence (FN)N converges to zero uniformly on compact subsetsof [−π, π]\{0}, which completes the proof.5Remark 3.2. We point out that the polynomials uN in the above lemma canbe computed explicitly: If u(z) =∑α cαzα is the Taylor expansion of u on aneighbourhood of 0, then a straightforward calculation reveals thatuN(z) =∑|α|<NcαN − |α|Nzαholds for all z ∈ C and all N ≥ 1. Since we shall not make further use ofthis observation, a proof is omitted.The main result now reads as follows.Theorem 3.3. Suppose that G,H ⊂ O(D) are circular reproducing kernelHilbert spaces, that D, E are Hilbert spaces, and that T is a norm closedsubspace of B(D, E). Then there exists a sequence of complete contractionsVN on MT (GD,HE) such that for every φ ∈ MT (GD,HE), VNφ is a polyno-mial of degree at most N − 1 with coefficients in T for all N , and such thatlimN(VNφ)(z) = φ(z) holds for all z ∈ D in the norm topology of B(D, E).If T is σ-weakly closed in B(D, E), then limN VNφ = φ holds σ-weakly for allφ ∈ MT (GD,HE). If in addition D and E are separable, then the mappingsVN are σ-weakly continuous.Proof. For φ ∈ MT (GD,HE), we write φt : D → B(D, E) , φt(z) = φ(eitz).By Lemma 3.1, the functionsφN : D → B(D, E) , φN(z) =12π∫ pi−piφt(z)FN (t) dt (N ≥ 1)are polynomials of degree at most N − 1 with coefficients in T , and thesequence (φN)N approximates φ pointwise in the norm topology of B(D, E).We claim that the functions φN are multipliers with ‖φN‖M ≤ ‖φ‖M. Infact, Lemma 2.1(a) and the circular symmetry of KG and KH imply that thefunctions φt belong to MT (GD,HE) with ‖φt‖M ≤ ‖φ‖M. So the mappingR → B(GD,HE) , t 7→Mφt , is well defined and moreover weakly continuous,since the functions of the form KH(·, z)x form a total subset of HE and sincethe family {Mφt , t ∈ R} is bounded. Thus it makes sense to define operatorsAN =12π∫ pi−piMφtFN(t) dt (N ≥ 1),as weak integrals. They obviously satisfy‖AN‖ ≤12π∫ pi−pi‖Mφt‖FN(t) dt ≤12π‖φ‖M∫ pi−piFN(t) dt = ‖φ‖M.6Furthermore,〈(ANf)(z), x〉 = 〈ANf,KH(·, z)x〉 =12π∫ pi−pi〈Mφtf,KH(·, z)x〉FN (t) dt=12π∫ pi−pi〈φt(z)f(z), x〉FN(t) dt = 〈φN(z)f(z), x〉holds for all f ∈ GD, z ∈ D and x ∈ E . This shows that φN ∈ MT (GD,HE)and that AN = MφN holds for all N ≥ 1. It remains to show that thecontractive operatorsVN :MT (GD,HE)→MT (GD,HE) , φ 7→ φN (N ≥ 1)actually are complete contractions. To see this, let us write VN = VN [D, E ],and observe that the kth amplification of VN [D, E ] coincides with the map-ping VN [Dk, Ek], which of course is contractive by what we have shown sofar.Now suppose that T is σ-weakly closed in B(D, E). Then by Lemma 2.1(b),the space MT (GD,HE) is σ-weakly closed in B(GD,HE). The assertion that(VNφ)N converges σ-weakly towards φ for every φ ∈ MT (GD,HE) followsimmediately from Lemma 2.1(c). If in addition D and E are separable Hilbertspaces, then also GD and HD are separable Hilbert spaces, and MT (GD,HE)is the dual of a separable Banach space. By the Krein-Smulian theorem, theσ-weak continuity of the operators VN can be proved by showing that everyVN is sequentially σ-weakly continuous on bounded sets. So suppose that(φk)k is a bounded sequence in MT (GD,HE) converging σ-weakly to someφ ∈ MT (GD,HE). By Lemma 2.1(c), this means exactly that the sequence(φk(z))k converges weakly to φ(z) for all z ∈ D. Fix z ∈ D and x ∈ D,y ∈ E . From‖φk(eitz)‖ ≤ ‖φk‖M‖1‖GKH(eitz, eitz)12 = ‖φk‖M‖1‖GKH(z, z)12(t ∈ R, k ∈ N) we deduce thatlimk〈(VNφk)(z)x, y〉 = limk∫ pi−pi〈φk(eitz)x, y〉FN(t) dt=∫ pi−pi〈φ(eitz)x, y〉FN(t) dt = 〈(VNφ)(z)x, y〉holds. By Lemma 2.1(c) we conclude that limk VNφk = VNφ σ-weakly, thusproving the claimed σ-weak continuity of VN .7Theorem 3.3 shows in particular that, for circular reproducing kernel Hilbertspaces G and H, the space M(G,H) ∩ C[z] is sequentially σ-weakly densein M(G,H). If M(G,H) contains the coordinate functions, then C[z] issequentially σ-weakly dense in M(G,H).As a second consequence, Theorem 3.3 implies that M(G,H) has Kraus’Property Sσ. Recall that a σ-weakly closed subspace S of B(G,H) is said tohave Property Sσ if, for all Hilbert spaces D,E and all σ-weakly closed sub-spaces T of B(D,E), the Fubini product F (S, T ) coincides with the normalspatial tensor product S⊗T . The Fubini product F (S, T ) can be defined asF (S, T ) = {X ∈ S⊗B(D,E) ; Rλ(X) ∈ T for all λ ∈ S∗}(cf. [10], p.119). Here, for λ ∈ S∗ (the space of all σ-weakly continuousfunctionals on S), the right slice mapping Rλ : S⊗B(D,E) → B(D,E) isthe unique σ-weakly continuous operator satisfying Rλ(S ⊗ T ) = λ(S)T forall S ∈ S and T ∈ B(D,E).Corollary 3.4. For circular reproducing kernel Hilbert spaces G,H ⊂ O(D),the dual spaceM(G,H) has the completely contractive σ-weak approximationproperty (CCAP). In particular, M(G,H) has Property Sσ. For every σ-weakly closed subspace T of B(D, E), the tensor product formulaM(G,H)⊗T =MT (GD,HE)holds.Proof. It is clear from Theorem 3.3 that M(GD,HE) has the CCAP. Thatthe CCAP implies Property Sσ is Theorem 2.10 in [10]. Furthermore, theinclusion M(G,H)⊗T ⊂ MT (GD,HE) is obvious, since the right-hand sideis σ-weakly closed. Suppose conversely that φ ∈ MT (GD,HE). Then, byLemma 2.1(d), φ ∈ M(G,H)⊗B(D, E), and Rλz(φ) = φ(z) ∈ T holds forall z ∈ D, where λz ∈ M(G,H)∗ is the point evaluation at z ∈ D. By asimple Hahn-Banach argument, the set of all λz is norm total in M(G,H)∗,which shows that Rλ(φ) ∈ T for all λ ∈ M(G,H)∗. This means (cf. [10],p.119) that φ ∈ F (M(G,H), T ). Since M(G,H) has Property Sσ, the claimis proved.Finally, we point out that the equality M(G,H)⊗T = MT (GD,HE) can bededuced directly from Theorem 3.3 if we suppose in addition that all poly-nomials belong to M(G,H). In fact, in this case, the polynomials VNφ asso-ciated with a multiplier φ ∈ MT (GD,HE) obviously belong to the algebraictensor product M(G,H)⊗ T .8References[1] J. Arazy, A survey of invariant Hilbert spaces of analytic functions onbounded symmetric domains, Multivariable operator theory, vol. 185,American Mathematical Society, 1993, pp. 7-65[2] N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc.68 (1950), 337-404[3] W. Arveson, Subalgebras of C∗-algebras. III. Multivariable operator the-ory, Acta Math. 181 (1998), 159-228[4] C. Barbian, A characterization of multiplication operators on repro-ducing kernel Hilbert spaces, Preprint (http://www.math.uni-sb.de/service/preprints/preprint208.pdf)[5] J. Burbea P. Masani, Banach and Hilbert spaces of vector-valued func-tions. Their general theroy and applications to holomorphy, ResearchNotes in Mathematics 90, Pitman Advanced Publishing Program,Boston, MA, 1984[6] M. Didas, On the slice map problem for H∞(Ω) and the reflexivity oftensor products, Preprint.[7] J. Faraut A. Koranyi, Function spaces and reproducing kernels onbounded symmetric domains, J. Funct. Anal. 88 (1990), 64-89[8] T.W. Ko¨rner, Fourier Analysis, Cambdridge University Press, Cam-bridge, 1988[9] J. Kraus, The slice map problem for σ-weakly closed subspaces of vonNeumann algebras, Trans. Amer. Math. Soc. 279 (1983), 357-376[10] J. Kraus, The slice map problem and approximation properties, J. Funct.Anal. 102 (1991), 116-1559

Universaar

Scientific documents from the Saarland University,

Universita¨t des Saarlandes
U
N I
V E R S IT A S
S
A
R A V I E N
S I
S
Fachrichtung 6.1 – Mathematik
Preprint Nr. 209
Approximation properties for multiplier
algebras of reproducing kernel Hilbert spaces
Christoph Barbian
Saarbru¨cken 2008
Fachrichtung 6.1 – Mathematik Preprint No. 209
Universita¨t des Saarlandes submitted: April 10, 2008
Approximation properties for multiplier
algebras of reproducing kernel Hilbert spaces
Christoph Barbian
Saarland University
Department of Mathematics
P.O. Box 15 11 50
66041 Saarbru¨cken
Germany
cb@math.uni-sb.de
Edited by
FR 6.1 – Mathematik
Universita¨t des Saarlandes
Postfach 15 11 50
66041 Saarbru¨cken
Germany
Fax: + 49 681 302 4443
e-Mail: preprint@math.uni-sb.de
WWW: http://www.math.uni-sb.de/
Abstract
In this note, it is proved that multiplier algebras of analytic repro-
ducing kernel Hilbert spaces which are compatible with the action of
the torus group possess Kraus’ completely contractive approximation
property (CCAP) and, consequently, have the Property Sσ. Our re-
sults apply in particular to the usual reproducing kernel Hilbert spaces
on bounded symmetric domains.
1
Approximation properties for multiplier
algebras of reproducing kernel Hilbert spaces
Christoph Barbian
April 10, 2008
1 Introduction
Since the pioneering work of Grothendieck many different versions of approxi-
mation properties for Banach and operator spaces have been considered. The
common ground of these properties is the question whether or not there ex-
ists, over a given Banach or operator space X, a net of finite rank operators
Vi, possibly bounded in norm or cb norm, such that limi Vi(x) = x holds in
a prescribed topology for all x ∈ X.
In this note, we shall be concerned with the so-called σ-weak completely
contractive approximation property (CCAP) as introduced by J. Kraus in
[10]: A σ-weakly closed subspace S of B(G,H) (G,H Hilbert spaces) is said
to have the CCAP if there exists a net (Vi)i of completely contractive σ-
weakly continuous finite-rank operators on S such that limi Vi(S) = S holds
σ-weakly for all S ∈ S.
In [10] it is shown that the dual algebra generated by an injective weighted
shift has the CCAP. In particular this means that H∞(D), identified with
the multiplier algebra of the Hardy space H2(D), has the CCAP. It is the
main result of this note to prove, more generally, that multiplier spaces of
holomorphic reproducing kernel Hilbert spaces have the CCAP whenever the
underlying reproducing kernel Hilbert spaces are circular. More precisely,
let D ⊂ Cd be a circular domain (that is, eitD ⊂ D holds for all t ∈ R)
containing the origin. Then a reproducing kernel Hilbert space H ⊂ O(D)
is called circular if it contains the constant functions and if the reproducing
kernel KH of H satisfies KH(e
itz, eitw) = KH(z, w) for all z, w ∈ D and
t ∈ R. This class of spaces includes the usual reproducing kernel spaces on
bounded symmetric domains (see for instance [7] or [1]), which attracted a
lot of attention in recent time. In particular, the Hardy and Bergman space
over the d-dimensional unit ball, and also the Arveson-Drury space (cf. [3]
2
for details) are circular in the above sense. The key step in the proof of our
main result (Theorem 3.3) is the observation that every multiplier between
circular reproducing kernel Hilbert spaces can be approximated pointwise by
a sequence of polynomials which is bounded in multiplier norm.
As an immediate consequence, multiplier algebras of circular reproducing ker-
nel Hilbert spaces have the CCAP, and therefore have Kraus’ Property Sσ
(cf. [9] and [10]). This can be used to prove a tensor product formula (Corol-
lary 3.4) which generalizes the well-known equality H∞(D, T ) = H∞(D)⊗T ,
where H∞(D, T ) is the space of all bounded holomorphic functions on D with
values in a σ-weakly closed subspace T of B(D, E), and H∞(D)⊗T denotes
the normal spatial tensor product, that is, the σ-weak closure of H∞(D)⊗T
in B(H2(D)⊗D, H2(D)⊗E). A generalization of this formula in the setting
of strictly pseudoconvex domains can be found in the recent paper [6].
2 Reproducing kernel Hilbert spaces
In this preliminary section, we are going to recapitulate some basic facts
about reproducing kernel Hilbert spaces (see [2] for further reference). A
Hilbert space H of functions on a set X with values in a Hilbert space E is
called a reproducing kernel Hilbert space if the point evaluations δz : H →
E , f 7→ f(z), are continuous for all z ∈ X. Equivalently, there exists a
unique function KH : X × X → B(E) (the reproducing kernel of H) with
the property that the functions KH(·, z)x : X → E belong to H and that
〈f,KH(·, z)x〉 = 〈f(z), x〉 holds for all f ∈ H, z ∈ X and x ∈ E . Moreover,
it is well known that KH is a positive definite B(E)-valued function and
that conversely every positive definite function is the reproducing kernel of a
uniquely determined reproducing kernel Hilbert space. Hence it makes sense
to define, for a given scalar reproducing kernel Hilbert space H ⊂ CX and a
Hilbert space E , the so-called inflation HE as the reproducing kernel Hilbert
space associated with the positive definite function KH · 1E . It is not hard
to see that HE coincides with the Hilbert space tensor product H ⊗ E via
canonical identification.
Given two reproducing kernel Hilbert spaces G,H ⊂ CX and Hilbert spaces
D, E , the multiplier space M(GD,HE) is defined as the collection of all func-
tions φ : X → B(D, E) with the property that the pointwise product φ · f
belongs to HE for all f ∈ GD. Furthermore, we write MT (GD,HE) for the
space of all multipliers taking values in a prescribed subspace T of B(D, E).
As a consequence of the closed graph theorem, the multiplication operator
Mφ : GD →HE , f 7→ φ·f , associated with a multiplier φ, is continuous. This
induces a seminorm onM(GD,HE) by setting ‖φ‖M = ‖Mφ‖. This seminorm
3
is a norm if we assume that G and H contain the constant functions, since
in this case the assignment φ 7→ Mφ is one-to-one. In this situation, we shall
often regard M(GD,HE) or, more generally, MT (GD,HE) as an (operator)
subspace of B(GD,HE).
Lemma 2.1. Suppose that G,H ⊂ CX are reproducing kernel Hilbert spaces
containing the constant functions and that D, E are Hilbert spaces.
(a) A function ψ : X → B(D, E) belongs to M(GD,HE) with ‖ψ‖M ≤ 1
if and only if the function X × X → B(E) , (z, w) 7→ KH(z, w)1E −
KG(z, w)ψ(z)ψ(w)
∗, is positive definite.
(b) Let T be a σ-weakly closed subspace of B(D, E). Then MT (GD,HE) is
σ-weakly closed in B(GD,HE).
(c) A bounded net (ψi)i inM(GD,HE) converges σ-weakly to ψ ∈M(GD,HE)
if and only if (ψi(z))i converges weakly to ψ(z) for all z ∈ X.
(d) The equality M(GD,HE) = M(G,H)⊗B(D, E) holds, up to canonical
identification.
Proof. Part (a) is Corollary 3.5 in [5]. Assertion (b) is proved in [4], and
(c) is checked by a straightforward calculation. To prove (d), one first
verifies by use of (a) that for every multiplier ψ ∈ M(G,H) and every
T ∈ B(D, E), the function ψ ·T belongs toM(GD,HE) withMψ·T =Mψ⊗T .
Then (b) yields the inclusion M(G,H)⊗B(D, E) ⊂M(GD,HE). Conversely,
fix ψ ∈ M(GD,HE) and choose nets (Pi)i and (Qj)j of finite-rank projec-
tions approximating σ-weakly the identities on D and E , respectively. Then
again by (a), the functions ψi,j = Qj · ψ · Pi belong to M(GD,HE) with
‖ψi,j‖M ≤ ‖ψ‖M. Moreover, each ψi,j belongs to the algebraic tensor prod-
uct M(G,H) ⊗ B(D, E). In fact, for rank-one projections P = u ⊗ u and
Q = v ⊗ v, we have Qψ(z)P = 〈ψ(z)u, v〉(v ⊗ u) for all z ∈ X and by
(a), the function X → C , z 7→ 〈ψ(z)u, v〉, belongs to M(G,H). Since
limi,j ψi,j(z) = ψ(z) weakly for all z ∈ X, an application of (c) proves the
claim.
3 Main results
Throughout this section, D ⊂ Cd denotes a circular domain containing the
origin. For the reader’s convenience, we start by recapitulating some facts
about the Feje´r kernels
FN : [−π, π]→ [0,∞) , FN (t) =
1
N
N−1∑
n=0
Dn(t) =
1
N
(
sin Nt
2
)2
(
sin t
2
)2 (N ≥ 1),
4
where
Dn : [−π, π]→ C , Dn(t) =
n∑
ν=−n
eiνt (n ≥ 0)
is the nth-order Dirichlet kernel. It is well known (see Lemma 2.2 in [8])
that
∫ pi
−pi
FN(t) dt = 2π, and that (FN )N converges to zero uniformly outside
(−δ, δ), for every 0 < δ < π. The following lemma probably is well known.
Since we are unable to find an exact reference, we include a proof.
Lemma 3.1. Suppose that X is a Banach space and that u ∈ O(D,X) is a
holomorphic function. Then the functions
uN : D → X , uN(z) =
1
2π
∫ pi
−pi
u(eitz)FN (t) dt (1)
are polynomials (of degree at most N−1) with coefficients in X. Furthermore,
lim
N
uN(z) = u(z) (in the norm of X)
holds for all z ∈ D.
Proof. First of all, uN obviously is analytic. By the definition of the Feje´r
kernel, we see that
DαuN(z) =
1
2πN
N−1∑
n=0
n∑
ν=−n
∫ pi
−pi
Dαu(eitz)ei(|α|+ν)t dt
holds for all z ∈ D and α ∈ Nd0. By Cauchy’s Theorem, the integrals on the
right-hand side of the above expression are zero whenever |α| ≥ N . Since D
is connected, this shows that uN is a polynomial of degree at most N − 1. It
remains to prove that (uN)N converges pointwise to u. So, given z ∈ D and
ǫ > 0, we choose δ > 0 such that ‖u(eitz) − u(z)‖ < ǫ holds for all |t| < δ.
Using the remarks preceding this lemma, we obtain that
‖uN(z)− u(z)‖ ≤
1
2π
∫
|t|≤δ
‖u(eitz)− u(z)‖FN (t) dt
+
1
2π
∫
δ≤|t|≤pi
‖u(eitz)− u(z)‖FN(t) dt
≤ ǫ+M sup{FN(t) ; δ ≤ |t| ≤ π}
for all N ≥ 1, where M = sup{‖u(eitz) − u(z)‖ ; |t| ≤ π}. As indicated
earlier, the sequence (FN)N converges to zero uniformly on compact subsets
of [−π, π]\{0}, which completes the proof.
5
Remark 3.2. We point out that the polynomials uN in the above lemma can
be computed explicitly: If u(z) =
∑
α cαz
α is the Taylor expansion of u on a
neighbourhood of 0, then a straightforward calculation reveals that
uN(z) =
∑
|α|<N
cα
N − |α|
N
zα
holds for all z ∈ C and all N ≥ 1. Since we shall not make further use of
this observation, a proof is omitted.
The main result now reads as follows.
Theorem 3.3. Suppose that G,H ⊂ O(D) are circular reproducing kernel
Hilbert spaces, that D, E are Hilbert spaces, and that T is a norm closed
subspace of B(D, E). Then there exists a sequence of complete contractions
VN on MT (GD,HE) such that for every φ ∈ MT (GD,HE), VNφ is a polyno-
mial of degree at most N − 1 with coefficients in T for all N , and such that
limN(VNφ)(z) = φ(z) holds for all z ∈ D in the norm topology of B(D, E).
If T is σ-weakly closed in B(D, E), then limN VNφ = φ holds σ-weakly for all
φ ∈ MT (GD,HE). If in addition D and E are separable, then the mappings
VN are σ-weakly continuous.
Proof. For φ ∈ MT (GD,HE), we write φt : D → B(D, E) , φt(z) = φ(e
itz).
By Lemma 3.1, the functions
φN : D → B(D, E) , φN(z) =
1
2π
∫ pi
−pi
φt(z)FN (t) dt (N ≥ 1)
are polynomials of degree at most N − 1 with coefficients in T , and the
sequence (φN)N approximates φ pointwise in the norm topology of B(D, E).
We claim that the functions φN are multipliers with ‖φN‖M ≤ ‖φ‖M. In
fact, Lemma 2.1(a) and the circular symmetry of KG and KH imply that the
functions φt belong to MT (GD,HE) with ‖φt‖M ≤ ‖φ‖M. So the mapping
R → B(GD,HE) , t 7→Mφt , is well defined and moreover weakly continuous,
since the functions of the form KH(·, z)x form a total subset of HE and since
the family {Mφt , t ∈ R} is bounded. Thus it makes sense to define operators
AN =
1
2π
∫ pi
−pi
MφtFN(t) dt (N ≥ 1),
as weak integrals. They obviously satisfy
‖AN‖ ≤
1
2π
∫ pi
−pi
‖Mφt‖FN(t) dt ≤
1
2π
‖φ‖M
∫ pi
−pi
FN(t) dt = ‖φ‖M.
6
Furthermore,
〈(ANf)(z), x〉 = 〈ANf,KH(·, z)x〉 =
1
2π
∫ pi
−pi
〈Mφtf,KH(·, z)x〉FN (t) dt
=
1
2π
∫ pi
−pi
〈φt(z)f(z), x〉FN(t) dt = 〈φN(z)f(z), x〉
holds for all f ∈ GD, z ∈ D and x ∈ E . This shows that φN ∈ MT (GD,HE)
and that AN = MφN holds for all N ≥ 1. It remains to show that the
contractive operators
VN :MT (GD,HE)→MT (GD,HE) , φ 7→ φN (N ≥ 1)
actually are complete contractions. To see this, let us write VN = VN [D, E ],
and observe that the kth amplification of VN [D, E ] coincides with the map-
ping VN [D
k, Ek], which of course is contractive by what we have shown so
far.
Now suppose that T is σ-weakly closed in B(D, E). Then by Lemma 2.1(b),
the space MT (GD,HE) is σ-weakly closed in B(GD,HE). The assertion that
(VNφ)N converges σ-weakly towards φ for every φ ∈ MT (GD,HE) follows
immediately from Lemma 2.1(c). If in addition D and E are separable Hilbert
spaces, then also GD and HD are separable Hilbert spaces, and MT (GD,HE)
is the dual of a separable Banach space. By the Krein-Smulian theorem, the
σ-weak continuity of the operators VN can be proved by showing that every
VN is sequentially σ-weakly continuous on bounded sets. So suppose that
(φk)k is a bounded sequence in MT (GD,HE) converging σ-weakly to some
φ ∈ MT (GD,HE). By Lemma 2.1(c), this means exactly that the sequence
(φk(z))k converges weakly to φ(z) for all z ∈ D. Fix z ∈ D and x ∈ D,
y ∈ E . From
‖φk(e
itz)‖ ≤ ‖φk‖M‖1‖GKH(e
itz, eitz)
1
2 = ‖φk‖M‖1‖GKH(z, z)
1
2
(t ∈ R, k ∈ N) we deduce that
lim
k
〈(VNφk)(z)x, y〉 = lim
k
∫ pi
−pi
〈φk(e
itz)x, y〉FN(t) dt
=
∫ pi
−pi
〈φ(eitz)x, y〉FN(t) dt = 〈(VNφ)(z)x, y〉
holds. By Lemma 2.1(c) we conclude that limk VNφk = VNφ σ-weakly, thus
proving the claimed σ-weak continuity of VN .
7
Theorem 3.3 shows in particular that, for circular reproducing kernel Hilbert
spaces G and H, the space M(G,H) ∩ C[z] is sequentially σ-weakly dense
in M(G,H). If M(G,H) contains the coordinate functions, then C[z] is
sequentially σ-weakly dense in M(G,H).
As a second consequence, Theorem 3.3 implies that M(G,H) has Kraus’
Property Sσ. Recall that a σ-weakly closed subspace S of B(G,H) is said to
have Property Sσ if, for all Hilbert spaces D,E and all σ-weakly closed sub-
spaces T of B(D,E), the Fubini product F (S, T ) coincides with the normal
spatial tensor product S⊗T . The Fubini product F (S, T ) can be defined as
F (S, T ) = {X ∈ S⊗B(D,E) ; Rλ(X) ∈ T for all λ ∈ S∗}
(cf. [10], p.119). Here, for λ ∈ S∗ (the space of all σ-weakly continuous
functionals on S), the right slice mapping Rλ : S⊗B(D,E) → B(D,E) is
the unique σ-weakly continuous operator satisfying Rλ(S ⊗ T ) = λ(S)T for
all S ∈ S and T ∈ B(D,E).
Corollary 3.4. For circular reproducing kernel Hilbert spaces G,H ⊂ O(D),
the dual spaceM(G,H) has the completely contractive σ-weak approximation
property (CCAP). In particular, M(G,H) has Property Sσ. For every σ-
weakly closed subspace T of B(D, E), the tensor product formula
M(G,H)⊗T =MT (GD,HE)
holds.
Proof. It is clear from Theorem 3.3 that M(GD,HE) has the CCAP. That
the CCAP implies Property Sσ is Theorem 2.10 in [10]. Furthermore, the
inclusion M(G,H)⊗T ⊂ MT (GD,HE) is obvious, since the right-hand side
is σ-weakly closed. Suppose conversely that φ ∈ MT (GD,HE). Then, by
Lemma 2.1(d), φ ∈ M(G,H)⊗B(D, E), and Rλz(φ) = φ(z) ∈ T holds for
all z ∈ D, where λz ∈ M(G,H)∗ is the point evaluation at z ∈ D. By a
simple Hahn-Banach argument, the set of all λz is norm total in M(G,H)∗,
which shows that Rλ(φ) ∈ T for all λ ∈ M(G,H)∗. This means (cf. [10],
p.119) that φ ∈ F (M(G,H), T ). Since M(G,H) has Property Sσ, the claim
is proved.
Finally, we point out that the equality M(G,H)⊗T = MT (GD,HE) can be
deduced directly from Theorem 3.3 if we suppose in addition that all poly-
nomials belong to M(G,H). In fact, in this case, the polynomials VNφ asso-
ciated with a multiplier φ ∈ MT (GD,HE) obviously belong to the algebraic
tensor product M(G,H)⊗ T .
8
References
[1] J. Arazy, A survey of invariant Hilbert spaces of analytic functions on
bounded symmetric domains, Multivariable operator theory, vol. 185,
American Mathematical Society, 1993, pp. 7-65
[2] N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc.
68 (1950), 337-404
[3] W. Arveson, Subalgebras of C∗-algebras. III. Multivariable operator the-
ory, Acta Math. 181 (1998), 159-228
[4] C. Barbian, A characterization of multiplication operators on repro-
ducing kernel Hilbert spaces, Preprint (http://www.math.uni-sb.de/
service/preprints/preprint208.pdf)
[5] J. Burbea P. Masani, Banach and Hilbert spaces of vector-valued func-
tions. Their general theroy and applications to holomorphy, Research
Notes in Mathematics 90, Pitman Advanced Publishing Program,
Boston, MA, 1984
[6] M. Didas, On the slice map problem for H∞(Ω) and the reflexivity of
tensor products, Preprint.
[7] J. Faraut A. Koranyi, Function spaces and reproducing kernels on
bounded symmetric domains, J. Funct. Anal. 88 (1990), 64-89
[8] T.W. Ko¨rner, Fourier Analysis, Cambdridge University Press, Cam-
bridge, 1988
[9] J. Kraus, The slice map problem for σ-weakly closed subspaces of von
Neumann algebras, Trans. Amer. Math. Soc. 279 (1983), 357-376
[10] J. Kraus, The slice map problem and approximation properties, J. Funct.
Anal. 102 (1991), 116-155
9


https://publikationen.sulb.uni-saarland.de/bitstream/20.500.11880/26427/1/preprint_209_08.pdf

Approximation properties for multiplier algebras of reproducing kernel Hilbert spaces

Abstract

Similar works

Full text

Available Versions

University of Szeged

Scientific publications of the Saarland University

Universaar

Scientific documents from the Saarland University,