U -invariant kernels, defect operators, and graded submodules

New York Journal of Mathematics

New York J. Math.22(2016) 677–709.

U -invariant kernels, defect operators, and graded submodules

Sameer Chavan and Rani Kumari

Abstract. Let κbe anU-invariant reproducing kernel and letH(κ) denote the reproducing kernel HilbertC[z1, . . . , zd]-module associated with the kernel κ. Let Mz denote the d-tuple of multiplication oper- ators Mz₁, . . . , Mz_d on H(κ). For a positive integer ν and d-tuple T = (T1, . . . , Td), consider the defect operator

DT^∗,ν:=

ν

X

l=0

(−1)^l ν l

! X

|p|=l

l!

p!T^pT^∗p.

The first main result of this paper describes all U-invariant kernels κ which admit finite rank defect operatorsDM_z^∗,ν. These areU-invariant polynomial perturbations of R-linear combinations of the kernels κν, whereκν(z, w) = _{(1−hz, wi)}¹ _ν for a positive integerν. We then formulate a notion of pure rowν-hypercontraction, and use it to show that certain rowν-hypercontractions correspond to anA-morphism. This result en- ables us to obtain an analog of Arveson’s Theorem F for graded submod- ules ofH(κν).It turns out that forµ < ν,there are no nonzero graded submodulesM ofH(κν) (ν≥2) with finite rank defectD(M_z|_M)^∗,µ.

Contents

1. U-invariant kernels and spherical tuples 677

2. Finite rank defect operators 684

3. Pure row ν-contractions 694

4. Finite rank graded submodules 700

References 708

1. U-invariant kernels and spherical tuples

The starting point of the investigations in this paper is the observation that the multiplication tupleMz,ν acting on the reproducing kernel Hilbert space H(κ_ν) associated with the kernel κ_ν(z, w) = _{(1−hz, wi)}¹ ν is a rank one

Received September 24, 2015.

2010 Mathematics Subject Classification. Primary 47A13, 47A15, 46E22; Secondary 47B10, 47B32.

Key words and phrases. U-invariant kernel, Graded submodule, A-morphism, Rowν- contraction.

ISSN 1076-9803/2016

677

row ν-hypercontraction, where ν is a positive integer. Needless to say, the multiplication tuple Mz,1 (commonly known as the Drury–Arveson shift) acting on H(κ₁) is the most outstanding example of a rank one row con- traction. A beautiful result in the theory of finite rank row contractions (to be referred to as Arveson’s Theorem F) says that any finite rank graded module ofH(κ₁) is necessarily of finite codimension [4]. In the same paper, W. Arveson asked whether this result remains true for general submodules.

This question was settled affirmatively by K. Guo in [14]. The paper [14] also contains a version of Arveson’s Theorem F for the submodules of H(κν), where the finite rank condition is replaced by finiteness of the ranks of cross- commutators of M_zi,1 and the orthogonal projection onto the submodules.

One of the main results in this paper provides an analog of Arveson’s The- orem F that takes into consideration the notion of row ν-hypercontraction.

Indeed, the present paper is an attempt to develop the theory of finite rank rowν-hypercontractions parallel to that of finite rank row contractions. It is worth noting that the class of rowν-hypercontractions is precisely the class of adjoints of jointν-hypercontractions. The later one is studied extensively in one and several variables (refer to [1], [5], [18], [3], etc.).

In this preliminary section, we discuss basics ofU-invariant reproducing kernel Hilbert spaces (RKHS) and weighted symmetric Fock spaces (the reader is referred to [15] and [17]). For future reference, we see below that the bounded multiplication tuple M_z on an U-invariant RKHS is unitarily equivalent to the creation tuple on certain weighted symmetric Fock space (Proposition 1.9). As far as we know, this fact appears to be unnoticed in the literature.

Throughout this paper, we use the following notations. For the set N of nonnegative integers, let N^d denote the Cartesian product N× · · · × N (d times). Let p ≡ (p1, . . . , pd) and n ≡ (n1, . . . , nd) be in N^d. We write |p|:= Pd

i=1p_i and p≤n if p_i ≤n_i for i= 1, . . . , d. For n∈ N^d, we let n! :=Qd

i=1n_i!.

The open unit ball {z ∈ C^d :kzk₂ <1} will be denoted by B while the unit sphere {z∈C^d:kzk₂ = 1} will be denoted by ∂B,wherekzk₂ denotes the Euclidean norm ofz inC^d.

Let H be a complex, separable, infinite-dimensional Hilbert space. Let M be closed subspace of H.We reserve the notation P_M to denote an or- thogonal projection fromHonto M,and the symbolM^⊥ for the orthogonal complement ofM inH.By dimM, we understand the Hilbert space dimen- sion ofM.

For a Hilbert spaceH, letB(H) denote theC^∗-algebra of bounded linear operators on H. Let T ∈ B(H). If T is a positive operator then by T^1/2 (resp. traceT), we mean the positive square-root (resp. trace) of T. We reserve the notation ranT for the range ofT. If T is a finite rank operator then the rank rankT ofT is defined as dim ranT. Ifx, y∈ H then the rank

one operator x⊗y is defined through x⊗y(h) = hh, yix for h ∈ H. For S ∈B(H),set [S, T] :=ST −T S.

Letκ(z, w) be a reproducing kernel defined forz, w in the open unit ball BinC^d.We say thatκ(z, w) isU-invariantif

κ(U z, U w) =κ(z, w) for any unitary d×dmatrix U and z, w∈B. The following fact is certainly known, which we include for the sake of completeness (refer to [7, Section 1], [15, Section 4]).

Lemma 1.1. Assume thatκis holomorphic separately in the variableszand w on the unit ball B in C^d. If κ isU-invariant then there exists a sequence {a_n}_n≥0 of nonnegative numbers such that

(1.1) κ(z, w) =

∞

X

n=0

anhz, wiⁿ (z, w ∈B), where hz, wi=Pd

i=1ziwi for z= (z1, . . . , z_d) and w= (w1, . . . , w_d) in C^d. Proof. Sinceκ is holomorphic separately inzandw, by a result of Hartogs [20, Pg 6],κ(z, w) is holomorphic in (z, w).By general theory of Reinhardt domains [20, Theorem 1.5, Ch II], one can expandκ as a power series in z and won the open unit ball B:

κ(z, w) = X

p,q∈N^d

b_pqz^pw^q (z, w∈B).

Suppose now that κ is U-invariant. In particular, κ is invariant under di- agonal unitary d×d matrices. It is now easy to see by integrating term by term in polar coordinates that bpq = 0 if p 6= q. Thus κ takes the form κ(z, w) =P

p∈N^db_ppz^pw^p (z, w ∈B). Let U be a d×dunitary matrix that sends z= (z₁, . . . , z_d)∈Bto (kzk,0, . . . ,0)∈B.Note that

κ(z, w) =κ(U z, U w) = X

p∈N^d, p₂=· · ·=p_d= 0

bppkzk^p1(U w)^p₁¹

= X

p1∈N

ap1hU z, U wi^p1 = X

p1∈N

ap1hz, wi^p1

for some scalar sequence {a_n}n≥0. IfH(κ) denotes the reproducing kernel Hilbert space associated with κ then a_|α| = _kzα¹k²

α!

|α|! provided a_|α| 6= 0 [15, Proposition 4.1]. In particular, each a_n is nonnegative.

We always assume that anyU-invariant kernelκsatisfies the hypothesis of the preceding lemma. We reserve the notationκafor the kernelκassociated with{a_n}_n≥0 as given in (1.1).

Throughout this paper, we letEn stand for the orthogonal projection of the reproducing kernel Hilbert space H(κ_a) onto the space H_n generated by homogeneous polynomials of degree nin the variables z1, . . . , zd, where it is tacitly assumed thatHn is a subspace ofH(κa).

In what follows, we need frequently the following lemma, which is essen- tially included in [15, Propositions 4.1, 4.3 and Corollary 4.4].

Lemma 1.2. Let κa be anU-invariant kernel. Then the orthonormal basis of the reproducing kernel Hilbert spaceH(κ_a) associated withκ_a is given by

(r a_|α||α|!

α! z^α:a_|α|6= 0 )

.

Let Mz denote the d-tuple of the multiplications operatorsMz1, . . . , Mzd de- fined on H(κ_a). If s is the smallest nonnegative integer such that the se- quence {a_n}n≥s consists of positive numbers then we have the following:

(1) Fori= 1, . . . , d, M_zi is bounded if and only if sup_n≥sa^an

n+1 <∞.

If (1)holds true then (2) Pd

i=1MziM_z^∗_i =P∞ n=s+1

an−1

an En. (3) Pd

i=1M_ziM_z^∗_i ≤I if and only if the sequence{a_n−1}_n≥s+1 is increas- ing.

(4) Let l be a positive integer such that l≥s.

X

|β|=l

l!

β!M_z^β(M_z^∗)^βz^α=

(a|α|−l

a|α| z^α if |α| ≥l, 0 if |α|< l.

Note. Throughout this paper, we will assume that the multiplication oper- atorsM_z1, . . . , M_zd are bounded linear operators onH(κ_a).

It is easy to describe allz-invariant spacesH(κ_a) (cf. [9, Theorem 2.12]).

Lemma 1.3. Let κ_a be an U-invariant kernel and let H(κ_a) be the re- producing kernel Hilbert space associated with κa. Then the following are equivalent:

(1) H(κ_a) isz_i-invariant for i= 1, . . . , d.

(2) There exist a nonnegative integersand a sequence {a_n}_n≥s of posi- tive numbers such that

κ_a(z, w) =

∞

X

n=s

a_nhz, wiⁿ (z, w∈B).

(3) There exists a nonnegative integerssuch that{z^α:α∈Z^d+,|α| ≥s}

forms an orthogonal basis for H(κa).

Proof. (1) implies (2): Let s be the least nonnegative integer such that a_s 6= 0. By Lemma 1.2, z^α ∈ H(κ_a) for all |α| = s. As H(κ_a) is z_i- invariant, for allα such that|α| ≥s,z^α ∈H(κa).By Lemma 1.2, we must have a_k 6= 0 for allk≥s.

(2) implies (3): This is immediate from Lemma 1.2.

(3) implies (1): This is obvious.

The multiplication tuples M_z on U-invariant reproducing kernel Hilbert space provide important examples of so-called spherical tuples. Before we recall the definition of spherical tuples, let us introduce some notations.

By acommuting d-tuple T on H,we mean the tuple (T₁, . . . , T_d) of com- muting bounded linear operatorsT1, . . . , T_don H.For a commutingd-tuple T on H, we interpret T^∗ to be (T₁^∗, . . . , T_d^∗), and T^p to be T₁^p1. . . T_d^pd for p= (p₁, . . . , p_d)∈N^d.

LetT be a d-tuple onH and letU(d) denote the group of complexd×d unitary matrices. For U = (u_jk)1≤j,k≤d ∈ U(d), the commuting operator d-tupleTU is given by

(T_U)j =

d

X

k=1

u_jkT_k (1≤j≤d).

Following [9], we say that T isspherical if for every U ∈ U(d),there exists a unitary operator Γ(U) ∈ B(H) such that Γ(U)T_j = (T_U)_jΓ(U) for all j= 1, . . . , d.If, further, Γ can be chosen to be a strongly continuous unitary representation of U(d) on Hthen we say thatT isstrongly spherical.

Remark 1.4. If (T1, . . . , T_d) is a spherical tuple then so is (π(T1), . . . , π(T_d)) for any unital∗-homomorphismπ :B(H)→B(K).

The reader is referred to [9] for the basics of spherical tuples. We remark that spherical tuples are nothing butU(d)-homogeneous tuples (cf. [6]).

It turns out that creation tuple on any weighted symmetric Fock space can be modelled as a spherical multiplication tuple on anU-invariant repro- ducing kernel Hilbert space (see Proposition 1.9 below). Before we see that, let us reproduce some notations and notions from [17].

Let E0 := C. Let E^⊗n and Eⁿ denote the full and symmetric tensor product ofncopies of E =C^d forn≥1 respectively. Recall that

{e_i1⊗ei2⊗ · · · ⊗ein : 1≤i1, . . . , in≤d}

is an orthogonal basis forE^⊗n and

{e_i1e_i2. . . e_in : 1≤i₁ ≤ · · · ≤i_n≤d}

is an orthogonal basis forEⁿ, whereξηdenotes the symmetric tensor product of ξ and η. Set

F^⊗(E) :=

∞

M

n=0

E^⊗n, F(E) :=

∞

M

n=0

Eⁿ.

Definition 1.5. A weighted full Fock space F_a^⊗(E) associated with a se- quence of nonnegative real numbers{a_n}_n≥0 is defined as the completion of finite sums of elements in E^⊗n, n≥0. Note that for⊕ξ_n,⊕η_n∈ F_a^⊗(E),

h⊕ξ_n,⊕η_ni:=

∞

X

n=0

anhξ_n, ηni_E⊗n.

Similarly, a weighted symmetric Fock space F_a(E) associated with a se- quence of nonnegative real numbers{a_n}n≥0 is defined as the completion of finite sums of elements in Eⁿ, n≥0. Note that for⊕ξ_n,⊕η_n∈ F_a(E),

h⊕ξ_n,⊕η_ni:=

∞

X

n=0

a_nhξ_n, η_ni_Eⁿ.

Remark 1.6. The weighted symmetric Fock (resp. full Fock) space F_a(E) (resp. F_a^⊗(E)) is a Hilbert space.

Since we are not aware of an appropriate reference, we include the follow- ing with details:

Lemma 1.7. For integers 1≤j1, . . . , jn≤d, we have

d

X

i1,...,in=1

an,νhe_i1ei2. . . ein, ej1 ⊗ej2⊗ · · · ⊗ejni_F⊗

ν(E)= 1.

where a_n,ν := ν(ν+1)...(ν+n−1)

n! (n≥0) and F_ν^⊗(E) denotes the weighted full Fock space endowed with the inner-product

h⊕ξ_n,⊕η_ni:=

∞

X

n=0

1

a_n,νhξ_n, ηni_E⊗n.

Proof. Without loss of generality, suppose that j1, . . . , jm are distinct inte- gers in the finite sequence{j₁, . . . , j_n}such thatj_p appearsk_p times, where p= 1, . . . , m. Clearly, k1+· · ·+km =n. Note that

d

X

i1,...,in=1

an,νhe_i1. . . ein, ej1 ⊗ · · · ⊗ejni_F⊗ ν(E)

=

d

X

i1,...,in=1

a_n,νhe_i1. . . e_in, e_j1. . . e_jni_Fν(E)

=an,ν

n!

k1!. . . km!ke_j1. . . ejnk²_F

ν(E)

=a_n,ν n!

k₁!...k_m!ke^k_j¹

1....e^k_j^m

mk²_F

ν(E). The desired conclusion now follows from the formula

keⁿ₁¹. . . eⁿ_d^dk²_F

ν(E)= 1 a_n,ν

n1!. . . n_d! (n₁+· · ·+n_d)!

(cf. [3, Lemma 3.8]).

Definition 1.8. The creation d-tuple S = (S₁, . . . , S_d) on the weighted symmetric Fock space F_a(E) is defined as follows : For 1 ≤ i ≤ d, Si : F_a(E)−→ F_a(E) is given by

Si(ξn) =eiξn forξn∈Eⁿ.

Proposition 1.9. LetF_a(E)denote a weighted symmetric Fock space asso- ciated with the sequence {a_n}n≥0. Then the following statements are equiv- alent:

(1) The weighted symmetric Fock space is invariant under the creation d-tuple S.

(2) There is a smallest nonnegative integer ssuch that {eⁿ₁¹eⁿ₂². . . eⁿ_d^d:|n| ≥s}

is an orthogonal basis forF_a(E).

(3) There exist a smallest integers≥0 such that the sequence {a_k}_k≥s consists of positive numbers and a unitary mapping

U :H(κ_b)−→ F_a(E)

such thatS_iU =U M_zi for i= 1, . . . , d, where b={1/a_k :k≥s}.

Proof. (1) implies (2): Note thatF_a(E) is the completion of linear span of {e_i1. . . ein ∈En :i1, . . . , in ∈ {1, . . . , d}, a_n 6= 0}. Since F_a(E) is invariant under the creationd-tupleS, (2) follows.

(2) implies (3): Since

keⁿ₁¹eⁿ₂². . . eⁿ_d^dk²_F

a(E)=a_|n|keⁿ₁¹eⁿ₂². . . eⁿ_d^dk²

E^|n|,

ak >0 fork≥s. Let b={1/a_k:k ≥s} and define U :H(κb)−→ F_a(E) by

U(z₁^α1. . . z_d^αd) =e^α₁¹. . . e^α_d^d for (α1, . . . , αd)∈N^d,

and extendU linearly to the subspace spanned by {zⁿ:|n| ≥s}.Since kU(z^α₁¹. . . z^α_d^d)k²_F

a(E)=ke^α₁¹. . . e^α_d^dk²_F

a(E)

= a_|α|α₁!. . . α_d!

|α|! =kz₁^α1. . . z_d^αdk²_H

(κb),

we may extendU continuously to H(κ_b).A routine verification shows that S_iU =U M_zi fori= 1, . . . , d.

(3) implies (1): This follows from Lemma 1.3.

Remark 1.10. By Lemma 1.2(1),Si is bounded fori= 1, . . . , dif and only if

sup

n≥s

an

a_n+1 <∞.

With the notations of Proposition 1.9, we agree to say thatF_a(E) is the Fock space realization of H(κ_b).We further refer to the creation d-tuple S as theFock space realization of the multiplicationd-tupleMz. Forν≥1,we denote by F_ν the Fock space realization of the reproducing kernel Hilbert space H(κν) associated with theU-invariant kernel

κν(z, w) = 1

(1− hz, wi)^ν (z, w∈B).

Sometimes, we use the simpler notation Hν in place of H(κ_ν). The Fock space realization of the multiplication d-tuple Mz,ν on H(κν) will be de- noted byS^(ν). We use these notations interchangeably.

Here is the plan of the present paper. In Section 2, we describe all U- invariant kernels which admit finite rank defect operators DM_z^∗,ν (Theo- rem 2.10). Loosely speaking, the U-invariant kernels {κ_ν : ν ≥ 1} form basis for these kernels. In Section 3, we introduce a new notion of pure rowν-contraction. This notion combined with the theory of weighted sym- metric Fock spaces [17] enables us to show that certain rowν-contractions correspond to an A-morphism in the sense of Arveson (Theorem 3.6). As a consequence, we recover a ball analog of von Neumann inequality for row ν-hypercontractions (refer to [11], [3], [13], [17], [19] for variants of von Neumann-type inequalities). In the last section, we obtain an analog of Arveson’s Theorem F for finite rank graded submodules of H(κ_ν) (The- orem 4.9). We remark that the Arveson’s method of proof of Theorem F does not readily generalize to the kernels κ_ν. This is perhaps due to the fact that the kernelκν is not a complete NP kernel forν ≥2. Our method, build off of the ideas of K. Guo [14], gives an alternative proof of Arveson’s Theorem F. One rather striking consequence of Theorem 4.9 asserts that for ν ≥ 2 and 1 ≤ µ≤ ν −1, the defect operator D_(Mz|M)^∗_,µ can never be of finite rank for any nonzero graded submoduleM ofH(κ_ν) (Corollary 4.15).

2. Finite rank defect operators

Given a commutingd-tuple T = (T₁, . . . , T_d) onH,we set

(2.2) QT(X) :=

d

X

i=1

T_i^∗XTi (X ∈B(H)).

It is easy to see thatQⁿ_T(I) =P

|p|=nn!

p!T^∗pT^p.Consider thedefect operator DT ,k of orderk given by

(2.3) D_T,k:=

k

X

l=0

(−1)^l k

l

Q^l_T(I).

Unless it is specified, the sequence{a_n}_n≥0associated with theU-invariant kernel κa consists of positive numbers. The main result (Theorem 2.10) of this section extends naturally to the case in which first finitely many ele- ments of{a_n}_n≥0 are 0 (see Remark 2.11).

Let κa be an U-invariant reproducing kernel and let H(κa) denote the reproducing kernel Hilbert space associated withκ_a. LetM_z denote the d- tuple of bounded linear multiplication operators Mz1, . . . , Mzd on H(κa).

Recall thatE_n denotes the orthogonal projection of H(κ_a) onto the space Hn generated by homogeneous polynomials of degree n. We see in Lem- ma 2.3 below that there exists a sequence {α_n}_n≥0 of real numbers such

that

D_Mz^∗_,k =

∞

X

n=0

αnEn.

We are interested in the spaces H(κ_a) which admit finite rank defect op- erators DM_z^∗,k. Before we see concrete examples of such spaces, we find it convenient to introduce the following familyD_k,l ofU-invariant reproducing kernels κ_a for positive integersk, l:

{κ_a:DM_z^∗,k =α0E0+· · ·+αl−1El−1 for some scalarsα0, . . . , αl−1 ∈R}.

Caution. In the definition of D_k,l, k is the order of the defect operator D_Mz^∗_,k, butlis notthe rank of D_Mz^∗_,k.

Remark 2.1. For any integer m≥l,D_k,l⊆ D_k,m.

The results in this section are motivated mainly by the following basic question.

Question 2.2. What is the structure of the cone D_k,l ofU-invariant repro- ducing kernels?

Before we answer this question, we gather some preliminary results. The first of which provides a handy formula for the defect operatorD_Mz^∗_,k. Lemma 2.3. Let Mz be the multiplicationd-tuple onH(κa). Then

D_Mz^∗_,k=

∞

X

n=0 n

X

i=0

(−1)ⁱ k

i an−i

a_n

! En, where we used the standard convention that ^k_i

= 0 for any positive integer i > k.

Proof. By Lemma 1.2,Q^l_M∗

z(I) =P∞ n=l

an−l

an E_n.It follows that D_Mz^∗_,k =

k

X

l=0

(−1)^l k

l ∞

X

n=l

an−l

a_n En

=

∞

X

n=0

E_n− k

1 ^∞

X

n=1

an−1

a_n E_n+· · ·+ (−1)^k

∞

X

n=k

an−k

a_n E_n. We can see that forn≤k, the coefficient ofE_n is

1− k

1 an−1

an

+ k

2 an−2

an

+· · ·+ (−1)ⁿ k

n a0

an

. Otherwise, the coefficient ofEn is

1− k

1 an−1

a_n + k

2 an−2

a_n +· · ·+ (−1)^kan−k

a_n .

This completes the proof of the lemma.

Remark 2.4. Note that the coefficient ofE0 equals 1.

Lemma 2.5. Let κ_a be an U-invariant kernel. Suppose the defect operator DM_z^∗,k=P∞

n=0αnEn for a scalar sequence{α_n}n≥0. Then we have:

(1) If 0 < n < k, then α_n = 0 if and only if there exists a nonnegative polynomialp in i of degree at most n−1 such that

an−i=p(i)(n+ 1−i). . .(k−i) (0≤i≤n),

(2) If n ≥ k, then α_n = 0 if and only if an−i is a polynomial in i of degree at most k−1.

Proof. The proof relies on the following well-known fact, which may be derived from Newton’s Interpolation Formula: For a sequence {b_k}ⁿ_k=0 of positive real numbers,P_n

k=0(−1)^{k n}_k

b_k = 0 if and only ifb_kis a polynomial ink of degree less than or equal ton−1.

Suppose that 0< n < k. By Lemma 2.3,αn= 0 if and only if

n

X

i=0

(−1)ⁱ k

i

an−i = 0.

The later one is equivalent to

n

X

i=0

(−1)ⁱ n

i k

i

n i

an−i= 0,

and hence by the observation stated in the last paragraph, α_n = 0 if and only if there exists a polynomial piniof degree less than or equal to n−1 such that ^n!_k!(^k_i)

(ⁿ_i)an−i=p(i).The desired conclusion in (1) is now immediate.

The same argument yields the conclusion in (2).

Remark 2.6. The Dirichlet kernel κa with an = _n+1¹ does not belong to D_k,l for any k, l≥1.

We are now ready to give examples ofU-invariant kernels with finite rank defect operators, which in some sense play the role of building blocks for the familyD_k,l.

Example 2.7. For an integer ν≥1,consider the U-invariant kernel κ_ν(z, w) = 1

(1− hz, wi)^ν (z, w ∈B)

and let Mz,ν be the multiplication d-tuple on H(κ_ν). We contend that D_Mz,ν^∗ _,ν =E₀,that is, κ_ν ∈ D_ν,1.

We may rewriteκν(z, w) asP∞

n=0an,νhz, wiⁿfor a sequence{a_n,ν}n≥0 of nonnegative real numbers (see (1.1)). It is easy to see using Multinomial Theorem that

an,ν = ν(ν+ 1). . .(ν+n−1)

n! (n≥0).

Note that an−i,ν = (n−i+1)...(n−i+ν−1)

(ν−1)! is a polynomial in i of degree ν−1.

By Lemma 2.5, the coefficient ofEn is 0 for n≥ν.Also, for 1≤n≤ν,we have

an−i,ν

(n+ 1−i). . .(ν−i) = 1 (ν−1)!

(n−i+ 1). . .(n−i+ν−1) (n−i+ 1). . .(ν−i)

= (ν−i+ 1). . .(ν−i+n−1)

(ν−1)! ,

which is a polynomial in iof degree n−1. By another application of Lem- ma 2.5, the coefficient ofE_n is 0 for 1≤n≤ν.

Remark 2.8. Let ν be a positive integer bigger than 1 and let µ be a positive integer less thanν.Sincean−i,ν is a polynomial iniof degreeν−1, by Lemma 2.5(2), the defect operator DM_z,ν^∗ ,µ is of infinite rank.

The conclusion of Example 2.7 is well-known in the case of Drury–Arveson kernel. In the special case of Bergman kernel on the unit disc, it is observed in [16, Pg 618] with the help of Berezin transform.

Here is an example of κ_a ∈ D_k,l,which does not belong to D_k,m for any m= 1, . . . , l−1.

Example 2.9. Fork≥2,consider theU-invariant kernelκ_awitha₀= 1 and a_i=i^k−1 fori≥1.It may be concluded from Lemma 2.5 thatκ_a∈ D_k,k+1, butκa does not belong to D_k,m for any 1≤m≤k.

We remark that up to a scalar multiple,κ_ν is the onlyU-invariant kernel inD_ν,1.Although this follows from the main result of this section, we regard this observation as the starting point of our investigations here, and hence we wish to outline a direct proof of it.

Suppose κ_b ∈ D_ν,1. By Lemma 2.3, forn≥1,the coefficient of E_n is 0, that is,

n

X

i=0

(−1)ⁱ ν

i bn−i

bn

= 0.

Clearly, b1 =νb0 =a1,νb0.We will prove by induction that bn =an,νb0 for n≥1.Supposeb_j =a_j,νb₀ for 1≤j≤n−1.Now

b_n=

n

X

i=1

(−1)ⁱ⁻¹ ν

i

an−i,ν

! b₀.

In view of the calculations of Example 2.7, we have bn=an,νb0.This com- pletes the induction. We thus obtain κ_b(z, w) =b₀κ_ν(z, w).

We now state the main result of this section.

Theorem 2.10. Let κ_a be an U-invariant kernel. Recall that κ_ν(z, w) =

∞

X

n=0

a_n,νhz, wiⁿ, where a_n,ν = ν(ν+ 1). . .(ν+n−1)

n! .

Then any one of the following cases occurs:

(1) k < l: κ_a ∈ D_k,l if and only if there exist real numbers α₁, . . . , α_k and a complex polynomial pm of degree at most m in one variable such that

κ_a(z, w) =

k

X

ν=1

α_νκ_ν(z, w) +pl−k−1(hz, wi).

(2) k = l: κ_a ∈ D_k,l if and only if there exist real numbers α₁, . . . , α_k such that

κa(z, w) =

k

X

ν=1

ανκν(z, w).

(3) k > l: κa ∈ D_k,l if and only if there exist real numbers α1, . . . , αk

such that

κ_a(z, w) =

k

X

ν=1

α_νκ_ν(z, w),

k−1

X

ν=1

α_νXⁿ

i=0

(−1)ⁱ k

i

an−i,ν

= 0 for n=l, l+ 1, . . . , k−1.

Remark 2.11. Suppose κ_b is an U-invariant kernel in D_k,l. Let s be the smallest positive integer such that {b_n}n≥s consists of positive numbers.

Consider the kernel κa with positive coefficients {a_n} such that an :=a_n,k for n= 0, . . . , s−1 and a_n=b_n forn ≥s. Then κ_a belongs toD_k,l. Since κ_b(z, w) =κ_a(z, w)−Ps−1

n=0a_n,khz, wiⁿ, κ_bis also a polynomial perturbation of a linear combination of κν (ν= 1, . . . , k).

Before we present a proof of Theorem 2.10, we would like to discuss some of its consequences.

Example 2.12. Let us see two instructive examples:

(1) TheU-invariant kernel

κa(z, w) = 2− hz, wi

(1− hz, wi)² +hz, wi belongs toD_2,4 with α₁ = 1 =α₂ and p₁(x) =x:

κa(z, w) =κ1(z, w) +κ2(z, w) +hz, wi.

(2) TheU-invariant kernel _(1−hz,wi)^{hz, wi} 2 belongs to D_2,2 withα1 =−1 and α2 = 1.However, this kernel can not be inD_2,1 since

1

X

i=0

(−1)ⁱ 2

i

a1−i,16= 0.

Corollary 2.13. Let κ_a ∈ D_1,ν for ν ≥ 2. Then there exists a positive number α and a complex polynomialpm of degree at mostm in one variable such that κa(z, w) =ακ1(z, w) +pν−2(hz, wi).

Corollary 2.14. Let κ_a ∈ D_ν,1.Then there exists a positive number α such thatκa(z, w) =ακν(z, w).

We now turn to proofs of Theorem 2.10 and Corollary 2.14, which involves several lemmas.

Lemma 2.15. We have the inclusion D_k,l ⊆ D_k+1,l+1. In particular, the U-invariant kernelκν belongs to D_ν+m,m+1 for any integer m≥1.

Proof. Consider the defect operator D_{T ,k} as defined in (2.3). Note that DT ,k+1 = DT ,k −QT(DT ,k), where QT is as given in (2.2). Suppose that κa is in D_k,l, that is, D_Mz^∗_,k =Pl−1

i=0αiEi for some scalars α0, . . . , αl−1. It follows that

D_Mz^∗_,k+1 =

l−1

X

i=0

α_iE_i−

l−1

X

i=0

α_iQ_Mz^∗(E_i).

However, QM_z^∗(Ei) = _a^ai

i+1Ei+1.Thus we obtain D_Mz^∗_,k+1 =

l−1

X

i=0

α_iE_i−

l−1

X

i=0

α_i ai

a_i+1E_i+1

=α₀E₀+

l−1

X

i=1

α_i−αi−1

ai−1

a_i

E_i−αl−1

al−1

a_l E_l.

Thus κa ∈ D_k+1,l+1. The remaining part is immediate from the fact κν ∈

D_ν,1, as recorded in Example 2.7.

Remark 2.16. In general, the inclusion D_k,l ⊆ D_k+1,l+1 is strict: For in- stance, takeκ_a(z, w) = _(1−hz,wi)¹ +_(1−hz,wi)¹ 2−1.Thena₀ = 1 anda_n=n+ 2 forn≥1.By Lemma 2.5,κ_a∈ D_2,3 butκ_a∈ D/ _1,2.

Lemma 2.17. Let kand lbe positive integers such that k≤l. Letκ_a be an U-invariant kernel of the form

κa(z, w) =

k

X

ν=1

ανκν(z, w) +pl−k−1(hz, wi)

for some real numbers α₁, . . . α_k, and a complex polynomial p_m of degree at mostm (with the interpretation that the termpm is absent if m <0). Then κ_a belongs toD_k,l.

Proof. Fix 1 ≤ ν ≤k. By Lemma 2.15, κ_ν belongs to D_ν+m,m+1 for any integerm≥1.Letting m:=k−ν, we obtain thatκν belongs to D_k,k−ν+1. Thus for n ≥ k−ν+ 1, the coefficient β_n,ν of E_n in D_Mz,ν^∗ _,k is zero. By Lemma 2.5, forn≥k, an−i,ν is a polynomial iniof degree at mostk−1.By hypothesis,an=Pk

ν=1ανan,ν for any integern≥l−k.Thus forn≥l≥k, an−i is a polynomial iniof degree at mostk−1.By another application of Lemma 2.5, forn≥l,the coefficientβn ofEn inDM_z^∗,k is zero. The desired

conclusion is immediate.

Lemma 2.18. Forν = 1, . . . , kand forl≥k,consideral−i,ν as anR-valued polynomial in ifrom{0,1, . . . , k−1}.Then the set{a_l−i,ν :ν = 1, . . . , k}is linearly independent in the following sense: If for real numbers α_ν (1≤ν≤ k), we have

k

X

ν=1

α_νa_l−i,ν = 0 (0≤i≤k−1), thenα1 = 0, . . . , α_k= 0.

Proof. Note thatal−i,ν is a polynomial in iof degreeν−1.In particular,

∆^ν(al−i,ν) = 0 and ∆^ν−1(al−i,ν)6= 0,

where the difference operator ∆ is given by ∆γi = γi+1 −γi for a scalar sequence {γ_i}_i≥0.Let

γ_i :=

k

X

ν=1

α_νal−i,ν = 0 (0≤i≤k−1).

Note that ∆^k−1γ_i = Pk

ν=1α_ν∆^k−1(a_l−i,ν). Since ∆^k−1(a_l−i,ν) = 0 for 1≤ ν ≤k−1,and ∆^k−1(al−i,k)6= 0,it follows that α_k = 0. A finite inductive argument now gives the required linear independence.

Proof of Theorem 2.10. We discuss first the case in which k ≤ l. The easier half is precisely Lemma 2.17. We see the necessary part. To see that, fixn≥l.Asκ_a∈ D_k,l,the coefficientβ_nofE_nis zero. Hence by Lemma 2.5, an−i is a polynomial iniof degree less than or equals tok−1. We note that an−i,ν is a polynomial iniof degreeν−1 as noted in Example 2.7. It follows that{an−i,1, . . . , an−i,k}forms a basis for the vector space of polynomials in iof degree less than or equal tok−1.In particular, an−i belongs to the R- linear span of{an−i,1, . . . , an−i,k}.Thus there exist scalarsα_1,n, . . . , α_k,n ∈R such that

(2.4) an−i =

k

X

ν=1

αν,nan−i,ν (0≤i≤k, n≥l).

We claim that the sequence{α_ν,m:m≥l}is constant for any ν = 1, . . . , k.

We achieve this by verifying that α_ν,l+j = α_ν,l+j+1 for any integer j ≥ 0. Fix an integer j ≥ 0. If we take n = l +j in Equation (2.4) then we get al+j−i = Pk

ν=1αν,l+jal+j−i,ν for any 0 ≤ i ≤ k. Further, if we take n = l+j+ 1 and replace i by i+ 1 in Equation (2.4) then we get a_l+j−i = Pk

ν=1α_ν,l+j+1a_l+j−i,ν for any 0 ≤ i≤ k−1. Thus we obtain for any 0≤i≤k−1,

k

X

ν=1

(α_ν,l+j−α_ν,l+j+1)al+j−i,ν = 0.

But the set {a_l+j−i,ν :ν = 1, . . . , k} is linearly independent (Lemma 2.18).

This implies that αν,l+j =αν,l+j+1 for ν = 1, . . . , k. Thus the claim stands verified, and hence

a_n=

k

X

ν=1

α_ν,la_n,ν for any integern≥l−k.

To complete the proof, note that if l > k, κ_a(z, w) =

l−k−1

X

n=0

a_nhz, wiⁿ+

∞

X

n=l−k

a_nhz, wiⁿ

=

l−k−1

X

n=0

anhz, wiⁿ+

∞

X

n=l−k k

X

ν=1

α_ν,lan,ν

! hz, wiⁿ

=

k

X

ν=1

αν,l

∞

X

n=0

an,νhz, wiⁿ+

l−k−1

X

n=0

an−

k

X

ν=1

αν,lan,ν

! hz, wiⁿ

=

k

X

ν=1

α_ν,lκ_ν(z, w) +pl−k−1(hz, wi).

The same calculation yields the desired conclusion in casel=k as well.

Finally, we treat the case in which k > l.Clearly,D_k,l ⊆ D_k,k,and hence by the case k=l, there exist real numbersα1, . . . , αk such that

κ_a(z, w) =

k

X

ν=1

α_νκ_ν(z, w).

One may use Lemma 2.3 to deduce that κ_a in D_k,k belongs to D_k,l if and only if

k

X

ν=1

α_ν

n

X

i=0

(−1)ⁱ k

i

an−i,ν

!

= 0 forl≤n≤k−1.

Since κ_k∈ D_k,1 (Example 2.7),Pn

i=0(−1)^{i k}_i

an−i,k = 0 for anyn≥1.The required equivalence in case k > l is now immediate. This also completes

the proof of the theorem.

Remark 2.19. The coefficients of the polynomial pl−k−1, as appearing in (1), are real.

Proof of Corollary 2.14. This is the case in which k > l= 1. By Theo- rem 2.10, there exist real numbersα₁, . . . , α_k such that

κa(z, w) =

k

X

ν=1

ανκν(z, w),

k−1

X

ν=1

αν n

X

i=0

(−1)ⁱ k

i

an−i,ν

!

= 0 for 1≤n≤k−1.

If cn,ν := Pn

i=0(−1)^{i k}_i

an−i,ν (1 ≤ n, ν ≤ k−1) then we have a system AX = 0 of k−1 equations ink−1 variablesα₁, . . . , αk−1,where A is the (k−1)×(k−1) matrix (cn,ν) and X is the column vector [α1. . . αk−1]^T. Since the system AX= 0 admits a trivial solution, it suffices to check that it has a unique solution. Note that c_n,ν is precisely the coefficient of E_n in D_Mz,ν^∗ _,k. Since κν belongs to D_k,k−ν+1, cn,ν = 0 for n ≥ k−ν + 1. In particular, the matrix A is a lower triangular matrix. Also, since κ_ν does not belong toD_k,k−ν,the off-diagonal entries ofAare nonzero. Thus we get α1 = 0, . . . , αk−1 = 0,and hence κa=α_kκ_k as desired.

We discuss some applications of the classification result.

Corollary 2.20. Let κa∈ D_k,k.Then there exist finite sequences {ν₁, . . . , νn} and {µ₁, . . . , µm}

of positive integers such that any f ∈H(κa) admits the decomposition f =

n

X

i=1

α_νif_i+

m

X

j=1

α_µjg_j

for some finite sequences {α_νi}ⁿ_i=1 ⊆ (0,∞), {α_µj}^m_i=1 ⊆ (−∞,0), f_i ∈ Hνi (i= 1, . . . , n) and g_j ∈Hµj (j = 1, . . . , m). Moreover,

kfk² ≥min nXⁿ

i=1

|α_νi|²kf˜ik²_H

νi :

n

X

i=1

ανif˜i=

n

X

i=1

ανifi

o (2.5)

−minnX^m

j=1

kα_µj|²|˜g_jk²_H

µj :

m

X

i=1

α_µjg˜_j =

m

X

i=1

α_µjg_jo .

Proof. By Theorem 2.10, there exist real numbers α1, . . . , αk such that κ_a(z, w) =

k

X

ν=1

α_νκ_ν(z, w).

Consider the finite sequence {ν₁, . . . , νn} for which the corresponding coef- ficients α_ν1, . . . , α_νn are positive, and also the finite sequence {µ₁, . . . , µ_n} for which the corresponding coefficients αµ1, . . . , αµn are negative. Then κ_a(z, w) satisfies

n

X

i=1

ανiκνi(z, w) =κa(z, w) +

m

X

i=1

(−α_µi)κµi(z, w).

This yields the first part. Further, by Aronszajn’s Theorem on sum of reproducing kernels [2, Section 6],

min ( _n

X

i=1

|α_νi|²kf˜ik²_H

νi :

n

X

i=1

ανif˜i =

n

X

i=1

ανifi

)

= min (

kf˜k²_H(κ

a)+

m

X

j=1

|α_µj|²k˜gjk²_H

µj

: ˜f+

n

X

j=1

(−α_µj) ˜g_j =f +

n

X

j=1

(−α_µj)g_j )

≤ kfk²_H(κ

a)+ min ( _m

X

j=1

|α_µj|²k˜g_jk²_H

µj :

n

X

j=1

α_µjg˜_j =

n

X

j=1

α_µjg_j )

,

which gives the desired norm estimate.

Remark 2.21. The case in which the finite sequence{µ₁, . . . , µ_m}is absent, equality holds in (2.5) (refer to [2]).

We conclude this section with one application of Theorem 2.10 to operator theory.

Recall that and-tupleTof commuting bounded linear operatorsT₁, . . . , T_d isp-essentially normalif the cross-commutators [T_i^∗, Tj] belong to the Schat- tenp-class for all i, j= 1, . . . , d.

Corollary 2.22. Ifκ_a∈ D_k,l then the multiplicationd-tuple M_z,aacting on H(κ_a) is p-essentially normal for any p > d.

Proof. Consider the weight multi-sequence {w⁽ⁱ⁾_α : α ∈ N^d, i = 1, . . . , d}

given by

w_α⁽ⁱ⁾= β¯_|α|+1

β¯_|α|

s αi+ 1

|α|+d (α ∈N^d,1≤i≤d), where the scalar sequence {β¯_k} is given by

β¯_k² = (d−1 +k)!

(d−1)!k!

1

a_k, k≥0.

Note that δ_k := ¯β_k+1/β¯_k = qd+k

k+1

q _a

k

ak+1 (k ∈ N). It is easy to see that M_z,a is a weighted multi-shift: For i = 1, . . . , d and α ∈ N^d, M_zikz^zααk = w⁽ⁱ⁾α z^α+i

kz^α+ik,where_iis thed-tuple with 1 in theith place and zeros elsewhere.

In view of [9, Theorem 4.2], it suffices to check that (2.6)

∞

X

k=1

δ_k^2pk^d−p−1+

∞

X

k=1

δ²_k−δ²_k−1

pk^d−1 <∞.

LetN = max{l−k,0}.By Theorem 2.10, for anyn≥N, a_nis a polynomial of degree, saym.Note that for any polynomialq(x) of degreen, the degree ofq(x+1)−q(x) is at mostn−1.It follows thatδ_n²−δ²_n−1 = ^r(n)_s(n),wherer(n) is a polynomial in n of degree at most 2m+1 ands(n) is a polynomial in n of degree 2m+ 2.Thus there exists a scalarC >0 such that|δ_n²−δ²_n−1| ≤C/n forn≥N.It is now easy to see that (2.6) holds for any integerp > d.

Remark 2.23. LetM be a z-invariant subspace of H(κ_a) such that M^⊥ is finite dimensional. Then for each i, the self-commutator of Mzi|_M is a finite rank perturbation of the self-commutator ofM_zi.In particular,M_z|_M isp-essentially normal for any integerp > d.

We will see in the last section that for any graded submoduleM ofH(κν), the defect operatorD_(Mz,ν|_M)^∗,ν is of finite rank if and only ifM^⊥is of finite dimension. Consequently, Mz,ν|_M is p-essentially normal for any integer p > din this case. This supports Arveson–Douglas conjecture [12].

3. Pure row ν-contractions

In this section, we introduce a notion of pure rowν-contraction. We then combine the theory of weighted symmetric spaces as developed in [17] with the powerful techniques from [3] to show that certain row ν-contractions correspond to a unique A-morphism in a natural way. This can be used to obtain a version of von Neumann inequality for tuples which are row k-contractions for 1 ≤ k ≤ ν. The latter one is certainly known as a con- sequence of a dilation theorem of M¨uller and Vasilescu [18] (The reader is referred to [11] for a ball analog of von Neumann inequality in case ν = 1, and also to [13], [17] and [19] for some interesting variants of von Neumann inequality). These results also form basis for our analysis of finite rank graded submodules ofH(κν) as carried out in Section 4.

Let T = (T₁, . . . , T_d) be a commuting d-tuple on H. Recall that Q_T is given by

QT(X) :=

d

X

i=1

T_i^∗XTi (X ∈B(H)).

We also recall that thedefect operator D_{T ,k} of order kis given by DT ,k:=

k

X

l=0

(−1)^l k

l

Q^l_T(I).

Definition 3.1. Let T be a commuting d-tuple of operators T1, . . . , T_d in B(H).We say thatT is a row ν-contractionifD_T^∗_,ν is a positive operator.

We say that T is a row ν-hypercontraction if T is a row k-contraction for k= 1, . . . , ν. In caseν = 1 then we refer toT as a row contraction. We say thatT is arow ν-contraction of finite rankifDT^∗,ν is a positive operator of finite rank.