# Perturbations of principal submodules in the Drury-Arveson space

Mohammad Jabbari  Mohammad Jabbari, Centro de Investigacion en Matematicas,
A.P. 402, Guanajuato, Gto., C.P. 36000, Mexico<EMAIL_ADDRESS>and
Xiang Tang  Xiang Tang, Department of Mathematics and Statistics, Washington
University in St. Louis, St. Louis, Missouri, 63130, USA<EMAIL_ADDRESS>
###### Abstract.

We study the geometry in the perturbations of principal submodules in the
Drury-Arveson space. We show that the perturbations give rise to smooth vector
bundles of Hilbert spaces which are equipped with natural Hermitian
connections. We compute the associated parallel transport operators and
explore properties of the monodromy.

##  1\. Introduction

Let $A\mathrel{\mathop{\ordinarycolon}}=\mathbb{C}[z_{1},\ldots,z_{m}]$ be the
algebra of polynomials in $m$ variables, and $I$ be an ideal of $A$. The zero
variety of $I$ is the algebraic set
$V(I)\mathrel{\mathop{\ordinarycolon}}=\\{z\in\mathbb{C}^{m}\mathrel{\mathop{\ordinarycolon}}f(z)=0,\forall
p\in I\\}$. The quotient $Q_{I}\mathrel{\mathop{\ordinarycolon}}=A/I$ can be
viewed as the algebra of algebraic functions on $V(I)$. In algebraic geometry,
one studies the geometry of $V(I)$ by investigating the properties of the
ideal $I$ and the quotient $Q_{I}$.

At the beginning of this century, Arveson [4, 5, 8] and Douglas [9, 18]
introduced an intriguing connection between multivariate operator theory and
algebraic geometry. Let $H^{2}_{m}$ be the Drury-Arveson space, $\bar{I}$ be
the closure of $I$ in $H^{2}_{m}$, and $I^{\perp}$ be the orthogonal
complement of $I$ in $H^{2}_{m}$. The quotient Hilbert space
$\mathcal{Q}_{I}\mathrel{\mathop{\ordinarycolon}}=H^{2}_{m}/\bar{I}$, which is
isomorphic to $I^{\perp}$, can be viewed as the Drury-Arveson space
$H^{2}(\Omega_{I})$ associated to the set
$\Omega_{I}\mathrel{\mathop{\ordinarycolon}}=\mathbb{B}^{m}\cap V(I)$. Let
$P_{I^{\perp}}$ be the orthogonal projection in $H^{2}_{m}$ onto $I^{\perp}$.
For each $p\in A$, define

$T_{p}\mathrel{\mathop{\ordinarycolon}}=P_{I^{\perp}}M_{p}|_{I^{\perp}}$

to be the compression of the multiplication operator $M_{p}$ on $H^{2}_{m}$.
$T_{p}$ can be viewed as a Toeplitz operator on $H^{2}(\Omega_{I})$. Arveson
and Douglas in a series of articles proposed to investigate the operators
$\\{T_{p}\mathrel{\mathop{\ordinarycolon}}p\in A\\}$ on $\mathcal{Q}_{I}$ to
understand the geometry of $\Omega_{I}$. More precisely, they focused on the
following index problems.

######  Conjecture 1.1 (Arveson [5, 7]).

All commutators $[T_{z_{j}},T_{z_{k}}^{*}]$, $j,k=1,\ldots,m$ are compact.

Let $\mathfrak{K}$ be the ideal of compact operators on $I^{\perp}$ and
$\mathfrak{T}_{I}$ be the unital $C^{*}$-algebra generated by
$\\{T_{p}\mathrel{\mathop{\ordinarycolon}}p\in A\\}\cup\mathfrak{K}$. Suppose
that the conjecture above holds true. It follows that the quotient
$\mathfrak{T}_{I}/\mathfrak{K}$ is a commutative $C^{*}$-algebra, and can be
identified as $C(\sigma^{e}_{I})$, where $\sigma^{e}_{I}$ is the essential
Taylor spectrum of $(T_{z_{1}},...,T_{z_{m}})$. Furthermore, when $I$ is
homogeneous, $\sigma^{e}_{I}$ can be identified as
$X_{I}\mathrel{\mathop{\ordinarycolon}}=V(I)\cap\partial\mathbb{B}^{m}$, c.f.
[17, Corollary 3.10], [20], [21, Theorem 5.1]. In summary, the Arveson
conjecture 1.1 gives the following exact sequence of $C^{*}$-algebras,

$0\rightarrow\mathfrak{K}\hookrightarrow\mathfrak{T}_{I}\rightarrow
C(\sigma^{e}_{I})\rightarrow 0.$

By the Brown-Douglas-Fillmore theory [14, 15], such an extension of
$C(\sigma^{e}_{I})$ defines an odd $K$-homology class $\tau_{I}$ in the
K-homology group $K_{1}(\sigma^{e}_{I})$. Douglas asked for an explicit
computation of this element in the geometric realization of $K$-homology; more
specifically, he conjectured that:

######  Conjecture 1.2 (Douglas [18]).

Let $I$ be the vanishing ideal of an algebraic set $V\subseteq\mathbb{C}^{m}$
which intersects $\partial\mathbb{B}^{m}$ transversally. Then, Conjecture 1.1
holds true, and its induced extension class $\tau_{I}$ is identified with the
fundamental class of $X_{I}$, namely the extension class induced by the
$\text{Spin}^{c}$ Dirac operator associated to the natural Cauchy-Riemann
structure of $X_{I}$.

There have been many studies on Conjectures 1.1 and 1.2 in the last decades. A
survey of results about these conjectures is given in [1, Chapter 41] and
[22].

In this article, we propose to study the ideal $I$ and $\mathcal{Q}_{I}$ by
perturbation. Our study is inspired by the beautiful work of Brieskorn and
Milnor [13, 24, 34, 33] which connects singularity theory, algebraic geometry,
and differential topology. Let us take the following example of a polynomial,

$f_{j}\mathrel{\mathop{\ordinarycolon}}=z_{1}^{2}+z_{2}^{2}+z_{3}^{2}+z_{4}^{3}+z_{5}^{6j-1}\in\mathbb{C}[z_{1},\ldots,z_{5}].$

Consider the zero variety $V\big{(}f_{j}-\epsilon\exp(it)\big{)}$. Fix a
sufficiently small $\epsilon>0$. It is not a difficult fact to check in
differential geometry that the zero variety
$V\big{(}f_{j}-\epsilon\exp(it)\big{)}$ is smooth inside the unit ball
$\mathbb{B}^{5}$, and it intersects transversely with the sphere
$\mathbb{S}^{9}$ as the boundary of $\mathbb{B}^{5}$ in contrast to the
property that the zero variety $V(f_{j})$ has an isolated singularity at the
origin. The intersection
$X_{j}\mathrel{\mathop{\ordinarycolon}}=\mathbb{S}^{9}\cap
V\big{(}f_{j}-\epsilon\exp(it)\big{)}$ is a smooth manifold homeomorphic to
$\mathbb{S}^{7}$ but not necessarily diffeomorphic to the smooth structure on
the standard sphere. Actually, when $j$ runs through 1 to 28, $X_{j}$ gives
all distinct oriented smooth structures on the topological 7-sphere.

We hope to learn from the success above in differential topology to study the
Hilbert modules. As an experiment of this idea, we study a special class of
ideals in $H^{2}_{2}$ in this article. More precisely, take
$I_{k}\mathrel{\mathop{\ordinarycolon}}=\langle z_{1}^{k}\rangle$ to be the
principal ideal in $\mathbb{C}[z_{1},z_{2}]$ generated by the monomial
$f_{k}\mathrel{\mathop{\ordinarycolon}}=z_{1}^{k}$. Consider the perturbation
$I_{k}^{\epsilon,t}$ of the ideal $I_{k}$ by varying the generator
$f_{k}^{\epsilon,t}\mathrel{\mathop{\ordinarycolon}}=z_{1}^{k}-\epsilon\exp(it)$,
for a sufficiently small $\epsilon>0$. The collection
$\left\\{\big{(}I^{\epsilon,t}_{k}\big{)}^{\perp}\right\\}_{t\in[0,2\pi)}$
forms a family of closed Hilbert subspaces of $H^{2}_{2}$ parameterized by
$t\in\mathbb{R}/2\pi\mathbb{Z}$. In Section 2, we will present the following
results about the variation $\left\\{I_{k}^{\epsilon,t}\right\\}$.

  1. (1)

(Theorem 2.3) The family
$\left\\{\mathcal{Q}_{I^{\epsilon,t}_{k}}\right\\}_{t\in[0,2\pi)}$ forms a
smooth vector bundle $\mathcal{Q}_{k,\epsilon}$ of Hilbert spaces over the
circle
$\mathbb{S}^{1}\mathrel{\mathop{\ordinarycolon}}=\mathbb{R}/2\pi\mathbb{Z}$.

  2. (2)

(Theorem 2.4 and Theorem 2.8) The vector bundle $\mathcal{Q}_{k,\epsilon}$
naturally embeds in the trivial bundle $\mathbb{S}^{1}\times
H^{2}_{2}\to\mathbb{S}^{1}$, and is equipped with a natural connection. We
compute the parallel transport with respect to this connection, and the
associated monodromy operator $U$.

  3. (3)

(Theorem 2.11) Let $P_{t}$ be the orthogonal projection of $H^{2}_{2}$ onto
$\big{(}I^{\epsilon,t}_{k}\big{)}^{\perp}$. Though $P_{t}$ is not
differentiable with respect to $t$ as a family of operators on the Drury-
Arveson space, it is a differentiable family as maps between the Besov-Sobolev
spaces.

Our results above are only about a special class of ideals in $H_{2}^{2}$. The
success on these examples encourage us to seek a general theory of
perturbation of principal submodules. We briefly discuss in Section 3 a few
questions for the general cases and will study them more systematically in
future publications.

Acknowledgements: We would like to thank Ronald Douglas, Kai Wang and Guoliang
Yu for inspiring discussions. Both authors’ research are partially supported
by National Science Foundation. The works in this article were reported in the
first author’s Ph.D. thesis, [25].

##  2\. Perturbation of $\langle z_{1}^{k}\rangle$

Set

$I(t)\mathrel{\mathop{\ordinarycolon}}=\langle z_{1}^{k}-\epsilon
e^{it}\rangle,\quad\mathcal{I}^{\perp}\mathrel{\mathop{\ordinarycolon}}=\biguplus\left\\{\langle
z_{1}^{k}-\epsilon e^{it}\rangle^{\perp}\subseteq
H_{2}^{2}\mathrel{\mathop{\ordinarycolon}}t\in\mathbb{R}\right\\}\subseteq\mathbb{R}\times
H_{2}^{2}.$

Let

$p\mathrel{\mathop{\ordinarycolon}}\mathcal{I}^{\perp}\rightarrow\mathbb{R},\quad
p\left(I(t)^{\perp}\right)=\\{t\\},$

and

$P\mathrel{\mathop{\ordinarycolon}}\mathbb{R}\rightarrow
B\left(H^{2}_{2}\right),\quad P\mathrel{\mathop{\ordinarycolon}}=(P_{t})$

be respectively the assembly of Hilbert spaces $I(t)^{\perp}$ and orthogonal
projections $P_{t}\mathrel{\mathop{\ordinarycolon}}H^{2}_{2}\to\langle
z_{1}^{k}-\epsilon e^{it}\rangle^{\perp}$ into a smooth111The smoothness of
the vector bundle will be established in Theorem 2.3. Hilbert bundle and a
rough222Namely, we are momentarily putting aside continuity or smoothness
considerations. map between Banach spaces. Topologize
$\mathcal{I}^{\perp}\subseteq\mathbb{R}\times H_{2}^{2}$ with the subspace
topology.

###  2.1. Smooth vector bundle $\mathcal{I}^{\perp}$

We first find an explicit smooth orthonormal frame for our Hilbert bundle
$\mathcal{I}^{\perp}$.

######  Lemma 2.1.

Let $E$ be a complex number with $|E|<1$. Set

$F\mathrel{\mathop{\ordinarycolon}}=E^{\frac{1}{k}},$
$\zeta_{j}\mathrel{\mathop{\ordinarycolon}}=e^{i\frac{2\pi}{k}j},\quad
j=0,\ldots,k-1,$ $a_{j}\mathrel{\mathop{\ordinarycolon}}=1-\zeta_{j}F,\quad
j=0,\ldots,k-1.$

(a) We have

$\sum_{q\in\mathbb{N}}\binom{n+r+kq}{n}E^{q}=k^{-1}F^{-r}\sum_{j=0}^{k-1}\zeta_{j}^{-r}a_{j}^{-n-1},$
$\sum_{q\in\mathbb{N}}\binom{n+r+kq}{n}E^{q}q=k^{-2}F^{-r}\left(-r\sum_{j=0}^{k-1}\zeta_{j}^{-r}a_{j}^{-n-1}+F(n+1)\sum_{j=0}^{k-1}\zeta_{j}^{-r+1}a_{j}^{-n-2}\right),$
$\sum_{q\in\mathbb{N}}\binom{n+r+kq}{n}E^{q}q^{2}=k^{-3}F^{-r}\times\\\
\left(r^{2}\sum_{j=0}^{k-1}\zeta_{j}^{-r}a_{j}^{-n-1}+F(1-2r)(n+1)\sum_{j=0}^{k-1}\zeta_{j}^{-r+1}a_{j}^{-n-2}+F^{2}(n+1)(n+2)\sum_{j=0}^{k-1}\zeta_{j}^{-r+2}a_{j}^{-n-3}\right).$

(b) Given nonnegative integer $l$, we have the asymptotic formula333Two
sequences $a_{n}$ and $b_{n}$ are said to be asymptotically equivalent,
denoted by $a_{n}\approx b_{n}$, if there is a finite nonzero number $C$ such
that $\lim_{n\to\infty}a_{n}/b_{n}=C$.

$\sum_{q\in\mathbb{N}}\binom{n+r+kq}{n}E^{q}q^{l}\approx n^{l}(1-F)^{-n},$

as $n\rightarrow\infty$.

###### Proof.

(a) Note that the sequence of numbers

$\psi_{q}\mathrel{\mathop{\ordinarycolon}}=k^{-1}\sum_{j=0}^{k-1}\zeta_{j}^{q-r},\quad
q\in\mathbb{N}$

equals $1$ when $q$ has remainder $r$ modulo $k$, and zero otherwise.
Therefore, we have the following equation,

(1)
$\displaystyle\sum_{q\in\mathbb{N}}\binom{n+r+kq}{n}E^{r+kq}=\sum_{q\in\mathbb{N}}\binom{n+q}{n}\psi_{q}E^{q}$
$\displaystyle=k^{-1}\sum_{q\in\mathbb{N}}\sum_{j=0}^{k-1}\binom{n+q}{n}\zeta_{j}^{q-r}E^{q}$
$\displaystyle=k^{-1}\sum_{j=0}^{k-1}\zeta_{j}^{-r}\left(1-\zeta_{j}E\right)^{-n-1},$

where in the last line we have used the negative binomial formula

$\sum_{q\in\mathbb{N}}\binom{n+q}{n}G^{q}=(1-G)^{-n-1}.$

Equation (1) gives the first formula. The other two are followed by
differentiation with respect to $E$.

(b) A straightforward induction on $l$ shows that the left hand side is of the
form

(2)
$\sum_{m=0}^{l}A_{m}F^{-r}(n+1)\cdots(n+m)\left(\sum_{j=0}^{k-1}\zeta_{j}^{-r+m}a_{j}^{-n-m-1}\right),$

where each $A_{m}$ depends only on $k,l,r$ but not $E$, $F$ or $n$. When
$n\rightarrow\infty$, the dominant summand in each
$\sum_{j=0}^{k-1}\zeta_{j}^{-r+m}a_{j}^{-n-m-1}$ is the one with the smallest
$|a_{j}|$, namely the one with $j=0$. Therefore, the dominant term in (2) is
the one with $m=l$ and $j=0$. ∎

Having this lemma at hand, we construct an orthogonal frame for
$\mathcal{I}^{\perp}$.

######  Theorem 2.2.

Set

$F\mathrel{\mathop{\ordinarycolon}}=\epsilon^{\frac{2}{k}},$
$\zeta_{j}\mathrel{\mathop{\ordinarycolon}}=e^{i\frac{2\pi}{k}j},\quad
j=0,\ldots,k-1,$ $a_{j}\mathrel{\mathop{\ordinarycolon}}=1-\zeta_{j}F,\quad
j=0,\ldots,k-1,$
$J\mathrel{\mathop{\ordinarycolon}}=\left\\{(r,n)\in\mathbb{N}^{2}\mathrel{\mathop{\ordinarycolon}}0\leq
r\leq k-1\right\\}.$

(a) A smooth orthogonal frame for the Hilbert bundle $\mathcal{I}^{\perp}$ is
given by

(3)
$\alpha\mathrel{\mathop{\ordinarycolon}}=\left\\{\alpha_{r,n}(t)\mathrel{\mathop{\ordinarycolon}}=\sum_{q\in\mathbb{N}}\omega_{r+kq,n}^{-1}\epsilon^{q}e^{-iqt}z_{1}^{r+kq}z_{2}^{n}\
\mathrel{\mathop{\ordinarycolon}}\ (r,n)\in J,\ t\in\mathbb{R}\right\\},$

where

$\omega_{m,n}=\|z_{1}^{m}z_{2}^{n}\|^{2}_{H^{2}_{2}}=\binom{m+n}{m}^{-1}.$

(b) A smooth orthonormal frame for the Hilbert bundle $\mathcal{I}^{\perp}$ is
given by

(4)
$\beta\mathrel{\mathop{\ordinarycolon}}=\left\\{\beta_{r,n}(t)\mathrel{\mathop{\ordinarycolon}}=\frac{\alpha_{r,n}(t)}{\|\alpha_{r,n}(t)\|}\
\mathrel{\mathop{\ordinarycolon}}\ (r,n)\in J,\ t\in\mathbb{R}\right\\},$

where

(5)
$\|\alpha_{r,n}(t)\|^{2}=\sum_{q\in\mathbb{N}}\omega_{r+kq,n}^{-1}\epsilon^{2q}=F^{-r}k^{-1}\sum_{j=0}^{k-1}\zeta_{j}^{-r}a_{j}^{-n-1}.$

###### Proof.

(a) We first check the smoothness of the frame. For comparison purposes,
observe that any one-variable power series of the form

(6) $\displaystyle\sum_{q\in\mathbb{N}}R(q)\zeta^{q},\quad
R\in\mathbb{C}[\zeta]\ \text{a polynomial in single variable}\ \zeta$

has the radius of convergence equal to one, hence absolutely and uniformly
convergent on any compact subset of the open unit disk of the complex
$\zeta$-plane. The formal power series of term-by-term time derivative of each
$\alpha_{r,n}$ of order $l\in\mathbb{N}$, as well as its $H_{2}^{2}$-norm are
given by:

$\frac{d^{l}\alpha_{r,n}}{dt^{l}}\mathrel{\mathop{\ordinarycolon}}=\sum_{q\in\mathbb{N}}\binom{kq+r+n}{n}(-iq)^{l}e^{-iqt}\epsilon^{q}z_{1}^{r+kq}z_{2}^{n},$
$\left\|\frac{d^{l}\alpha_{r,n}}{dt^{l}}\right\|^{2}_{H_{2}^{2}}\mathrel{\mathop{\ordinarycolon}}=\sum_{q\in\mathbb{N}}\binom{kq+r+n}{n}q^{2l}\epsilon^{2q}.$

Comparison with (6) shows that for any $\epsilon<1$ and $t\in\mathbb{R}$, each
$d^{l}\alpha_{r,n}/dt^{l}(t)$ is an analytic function on $\mathbb{B}^{2}$ with
finite $H_{2}^{2}$-norm, hence it lives in $H^{2}_{2}$. That $\alpha_{r,n}$
lives in $I(t)^{\perp}$ is immediate from our derivation of $\alpha_{r,n}$ in
the next paragraph, but here is a direct verification. For each
$(M,N)\in\mathbb{N}^{2}$, $\alpha_{r,n}(t)$ and
$z_{1}^{M}z_{2}^{N}\left(z_{1}^{k}-\epsilon e^{it}\right)$ have no monomial in
common (hence orthogonal) except when $N=n$ and $r$ equals the remainder of
$M$ in division by $k$. For this exceptional case, assuming $M=kQ+r$,
$Q\in\mathbb{N}$, we have

$\left\langle\alpha_{r,n},z_{1}^{M}z_{2}^{N}\left(z_{1}^{k}-\epsilon
e^{it}\right)\right\rangle=\epsilon^{Q+1}e^{-i(Q+1)t}-\epsilon^{Q}e^{-iQt}\epsilon
e^{-it}=0.$

By Taylor’s theorem, we have

$\left\|\alpha_{r,n}(t+h)-\alpha_{r,n}(t)-h\frac{d\alpha_{r,n}}{dt}(t)\right\|^{2}=\sum_{q\in\mathbb{N}}\binom{kq+r+n}{n}\epsilon^{2q}\left|e^{-iq(t+h)}-e^{-iqt}+hiqe^{-iqt}\right|^{2}$
$\times\left(\sum_{q\in\mathbb{N}}\binom{kq+r+n}{n}\epsilon^{2q}\left(\frac{h^{2}}{2!}q^{2}\right)^{2}\right),$

which shows that
$\alpha_{r,n}\mathrel{\mathop{\ordinarycolon}}\mathbb{R}\rightarrow H_{2}^{2}$
is first-order differentiable. The same line of arguments proves the
smoothness as well.

Next, we show that the sections of $\mathcal{I}^{\perp}$ are linear
combinations of $\alpha_{r,n}$. A section of $\mathcal{I}^{\perp}$ has the
form

(7) $\xi(t)=\sum_{m,n\geq 0}x_{m,n}(t)z_{1}^{m}z_{2}^{n},$

and satisfies the following orthogonality equations:

$0=\left\langle\xi(t),z_{1}^{m}z_{2}^{n}\left(z_{1}^{k}-\epsilon
e^{it}\right)\right\rangle=x_{m+k,n}\omega_{m+k,n}-x_{m,n}\omega_{m,n}\epsilon
e^{-it},\quad\forall m,n\geq 0,$

or equivalently

(8) $x_{m+k,n}\omega_{m+k,n}=x_{m,n}\omega_{m,n}\epsilon e^{-it},\quad\forall
m,n\geq 0.$

Assuming

$X_{m,n}\mathrel{\mathop{\ordinarycolon}}=x_{m,n}\omega_{m,n},$

this latter recursive equation becomes

$X_{m+k,n}=X_{m,n}\epsilon e^{-it},$

hence

(9) $X_{r+kq,n}=X_{r,n}\epsilon^{q}e^{-iqt},\quad r=0,1,\ldots,k-1,\quad
q,n=0,1,2,\ldots.$

This shows that

$\left\\{X_{r,n}\mathrel{\mathop{\ordinarycolon}}(r,n)\in J\right\\}$

are basic Taylor coefficients of $\xi$ in the sense that they linearly
determine all the other coefficients, and there are no nontrivial linear
equations among them. Note that $\alpha_{r,n}$ is the section with
$X_{r,n}=1$, and all other basic coefficients vanish. Working backwards, one
can formally (neglecting convergence issues) write any element $\xi(t)$ of
$I(t)^{\perp}$ as a linear combination of elements (3). That the closed linear
span of (3) equals $I(t)^{\perp}$ can be easily proved by checking that
$\xi(t)=0$ is implied by assuming $\langle\xi(t),\alpha_{r,n}(t)\rangle=0$ for
all $(r,n)\in J$. Any two $\alpha_{r,n}$ and $\alpha_{r^{\prime},n^{\prime}}$
with $(r,n)\neq(r^{\prime},n^{\prime})$ are orthogonal because they have no
monomials in common, and we know that monomials constitute an orthogonal basis
for $H_{2}^{2}$.

(b) Lemma 2.1 gives Equation (5). Since $\left\|\alpha_{r,n}(t)\right\|$ does
not depend on $t$, the rest follows immediately from Part (a). ∎

######  Theorem 2.3.

The map
$p\mathrel{\mathop{\ordinarycolon}}\mathcal{I}^{\perp}\rightarrow\mathbb{R}$
defines a smooth bundle of Hilbert spaces over $\mathbb{R}$ and therefore on
$\mathbb{S}^{1}\mathrel{\mathop{\ordinarycolon}}=\mathbb{R}/\mathbb{Z}$.

###### Proof.

Assume $\mathcal{I}^{\perp}\subseteq\mathbb{R}\times H_{2}^{2}$ with the
subspace topology as a rough Hilbert bundle over $\mathbb{R}$. Recall that
$J\mathrel{\mathop{\ordinarycolon}}=\left\\{(r,n)\in\mathbb{N}^{2}\mathrel{\mathop{\ordinarycolon}}0\leq
r\leq k-1\right\\}$ is the index set of the orthonormal frame $\beta$ in
Theorem 2.2. Since each $\beta(t)$, $t\in\mathbb{R}$ is an orthonormal basis
for the fiber $I(t)^{\perp}$, the mapping

$\Phi\mathrel{\mathop{\ordinarycolon}}\mathbb{R}\times
l^{2}(J)\rightarrow\mathcal{I}^{\perp},\quad\left(t,(a_{r,n})_{(r,n)\in
J}\right)\mapsto\left(t,\sum_{(r,n)\in J}a_{r,n}\beta_{r,n}(t)\right)$

trivializes $\mathcal{I}^{\perp}$ as a topological vector bundle, namely,
$\Phi$ is a homeomorphism and the triangle

---
$\textstyle{\mathbb{R}\times
l^{2}(J)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\
\ \
\Phi}$$\scriptstyle{\mathrm{pr}_{\mathbb{R}}}$$\textstyle{\mathcal{I}^{\perp}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{\mathbb{R}}$

commutes. Since this trivialization is given by a single chart, it also gives
$\mathcal{I}^{\perp}$ the structure of a smooth vector bundle. ∎

###  2.2. Parallel transport $U_{t}$

We observe that $\mathcal{I}^{\perp}$ is a closed subbundle of the trivial
vector bundle $\mathbb{S}^{1}\times H^{2}_{2}\to\mathbb{S}^{1}$. Recall that
$P\mathrel{\mathop{\ordinarycolon}}\mathbb{R}\to B(H^{2}_{2})$ is the
orthogonal projection from $\mathbb{S}^{1}\times H^{2}_{2}$ to
$\mathcal{I}^{\perp}$.

The imitation of the standard construction of the Hermitian connection for
subbundles of Hilbert bundles [26, Example 1.5.14], [35, Volume II, Page 540]
defines a covariant derivative:

(10) $D\xi(t)=P_{t}\left(\frac{d\xi}{dt}\right),\quad\xi\in
C^{\infty}(\mathbb{R};\mathcal{I}^{\perp}),$

where $C^{\infty}(\mathbb{R};\mathcal{I}^{\perp})$ is the space of smooth
functions on $\mathbb{R}$ with value in $\mathcal{I}^{\perp}$.

The operator $D$ is called a covariant derivative because it satisfies the
Leibniz rule

$D(g\xi)(t)=g^{\prime}(t)\xi(t)+g(t)D(\xi)(t),\quad\forall g\in
C^{\infty}(\mathbb{R};\mathbb{C}),\quad\forall\xi\in
C^{\infty}(\mathbb{R};\mathcal{I}^{\perp}).$

A $D$-flat section of the bundle $\mathcal{I}^{\perp}$ is a section $\xi\in
C^{\infty}(\mathbb{R};\mathcal{I}^{\perp})$ which satisfies the evolution
equation

(11) $D\xi(t)=0,\quad\forall t\in\mathbb{R}.$

We prove below that Equation (11) has a unique solution operator for all
$t\in\mathbb{R}$ for any initial value $\xi(0)\in I(0)^{\perp}$. This leads to
the solution map

$U_{t}\mathrel{\mathop{\ordinarycolon}}=I(0)^{\perp}\rightarrow
I(t)^{\perp},\quad t\in\mathbb{R},$

which sends the initial value $\xi(0)$ of a flat section $\xi$ to its time-$t$
value $\xi(t)$.

######  Theorem 2.4.

Assume the notations in Theorem 2.3.

  1. (a)

The solution operator
$U_{t}\mathrel{\mathop{\ordinarycolon}}I(0)^{\perp}\rightarrow I(t)^{\perp}$
acts diagonally by

(12) $U_{t}\left(\beta_{r,n}(0)\right)=e^{if_{r,n}t}\beta_{r,n}(t),$

where frequencies $f_{r,n}$ are given by

(13)
$f_{r,n}=\frac{\sum_{q\in\mathbb{N}}\omega_{r+kq,n}^{-1}q\epsilon^{2q}}{\sum_{q\in\mathbb{N}}\omega_{r+kq,n}^{-1}\epsilon^{2q}}=\frac{-r\sum\limits_{j=0}^{k-1}\zeta_{j}^{-r}a_{j}^{-n-1}+F(n+1)\sum\limits_{j=0}^{k-1}\zeta_{j}^{-r+1}a_{j}^{-n-2}}{k\sum\limits_{j=0}^{k-1}\zeta_{j}^{-r}a_{j}^{-n-1}}.$

  2. (b)

The operator $U_{t}\mathrel{\mathop{\ordinarycolon}}I(0)^{\perp}\to
I(t)^{\perp}$ is unitary, $\forall t\in\mathbb{R}$.

  3. (c)

When $n\rightarrow\infty$, $f_{r,n}$ varies asymptotically like

(14) $\frac{F}{k(1-F)}n+\frac{F}{k(1-F)}-\frac{r}{k}.$

  4. (d)

A smooth orthonormal parallel frame for the Hilbert bundle
$\mathcal{I}^{\perp}$ is given by

(15)
$\gamma\mathrel{\mathop{\ordinarycolon}}=\left\\{\gamma_{r,n}(t)\mathrel{\mathop{\ordinarycolon}}=e^{if_{r,n}t}\beta_{r,n}(t)\
\mathrel{\mathop{\ordinarycolon}}\ (r,n)\in J,\ t\in\mathbb{R}\right\\}.$

###### Proof.

(a) A flat section of $\mathcal{I}^{\perp}$ has the form $\eta(t)=\sum
y_{m,n}z_{1}^{m}z_{2}^{n}$ such that
$\dot{\eta}=\sum\dot{y}_{m,n}z_{1}^{m}z_{2}^{n}$ lives in
$(I(t)^{\perp})^{\perp}$ for each $t$. In other words, the inner product
$\langle\dot{\eta},\xi\rangle_{H^{2}_{2}}$ is zero for every section $\xi$ as
in (7). Equivalently, in terms of Taylor coefficients, we have

$\sum_{m,n}\dot{y}_{m,n}\overline{X}_{m,n}=0$

for all $X_{m,n}$ satisfying (9). Rewriting this in terms of basic Taylor
coefficients, we get

$\sum_{\begin{subarray}{c}0\leq r<k\\\ q,n\geq
0\end{subarray}}\dot{y}_{r+kq,n}\overline{X}_{r,n}\epsilon^{q}e^{iqt}=0.$

Since this is true for any choice of basic coefficients $X_{r,n}$, $(r,n)\in
J$, we should have

(16) $\sum_{q\in\mathbb{N}}\dot{y}_{r+kq,n}\epsilon^{q}e^{iqt}=0,\quad(r,n)\in
J.$

Since $\eta$ is a section, its Taylor coefficients satisfy

$y_{r+kq,n}=y_{r,n}\frac{\omega_{r,n}}{\omega_{r+kq,n}}\epsilon^{q}e^{-iqt},\quad(r,n)\in
J,\quad q\in\mathbb{N}.$

(Recall (8).) Plugging this into (16) yields

$\sum_{q\in\mathbb{N}}\left(\dot{y}_{r,n}-iqy_{r,n}\right)\frac{\omega_{r,n}}{\omega_{r+kq,n}}\epsilon^{2q}=0,\quad(r,n)\in
J.$

Therefore, we have the explicit evolution laws

(17) $\dot{y}_{r,n}=y_{r,n}if_{r,n},\quad(r,n)\in J,$

where

(18)
$f_{r,n}=\frac{\sum_{q\in\mathbb{N}}\binom{n+r+kq}{n}q\epsilon^{2q}}{\sum_{q\in\mathbb{N}}\binom{n+r+kq}{n}\epsilon^{2q}}.$

Evolution equations (17) are solved as

$y_{r,n}(t)=y_{r,n}(0)e^{if_{r,n}t},\quad(r,n)\in J,$

hence (12). Lemma 2.1 computes $f_{r,n}$.

(b) The linearity of $U_{t}$ is immediate from the properties that $D$ is
linear and the uniqueness of the solution of Equation (11). Let us first prove
the uniqueness of the solution. Suppose $\xi\in\mathcal{I}^{\perp}$ satisfies
$D\xi=0$ and $\xi(0)=0$. Then

$0=\langle
D\xi,\xi\rangle=\left\langle\frac{d}{dt}\xi,\xi\right\rangle=\frac{1}{2}\frac{d}{dt}\langle\xi,\xi\rangle,\quad\xi(0)=0.$

This implies that $\langle\xi,\xi\rangle\equiv 0$, and the solution of
Equation (11) is unique.

Assume that $\xi$ is a flat section. Since $\xi$ and $d\xi/dt$ are orthogonal,
we have

$0=2\left\langle\xi(t),\frac{d\xi}{dt}\right\rangle=\frac{d}{dt}\left\|\xi(t)\right\|^{2}=\frac{d}{dt}\left\|U_{t}\xi(0)\right\|^{2},$

hence

$\left\|U_{t}\xi(0)\right\|=\left\|U_{0}\xi(0)\right\|=\left\|\xi(0)\right\|.$

This shows that $U_{t}$ is an isometry. It remains to show that $U_{t}$ is
surjective. Given $\tau\in\mathbb{R}$, the inverse of
$U_{\tau}\mathrel{\mathop{\ordinarycolon}}\xi(0)\mapsto\xi(\tau)$ is given by
the parallel translation $\eta(0)\mapsto\eta(\tau)$ along the flat section
$\eta(t)\mathrel{\mathop{\ordinarycolon}}=\xi(\tau-t)$, where we have used the
uniqueness of the solutions of Equation (11).

(c) When $n\rightarrow\infty$, the dominant summands in the numerator and
denominator of $f_{r,n}$ in (13) are those with the smallest $|a_{j}|$, namely
those with $j=0$. Therefore, $f_{r,n}$ varies asymptotically like

$\frac{-ra_{0}^{-n-1}+F(n+1)a_{0}^{-n-2}}{ka_{0}^{-n-1}}=-\frac{r}{k}+\frac{F}{k(1-F)}(n+1).$

(d) By (12) we have

$U_{t}(\gamma_{r,n}(0))=\gamma_{r,n}(t),$

hence $\gamma$ is a parallel frame. The smoothness of $U_{t}$ is the result of
the smoothness of $\beta$ and the asymptotic formula (14) for $f_{r,n}$. ∎

Consider the Toeplitz algebra $\mathfrak{T}_{I(0)}$ associated to the ideal
$I(0)=\langle z_{1}^{k}-\epsilon\rangle\subseteq\mathbb{C}[z_{1},z_{2}]$. It
is the C*-algebra generated by
$\left\\{1,T_{z_{1}},T_{z_{2}}\right\\}\cup\mathfrak{K}(I(0)^{\perp})$, where
$T_{z_{j}}$, $j=1,2$ is the multiplication by the coordinate function $z_{j}$
compressed to $I(0)^{\perp}$. For brevity, we set
$T_{j}\mathrel{\mathop{\ordinarycolon}}=T_{z_{j}}$, $j=1,2$.

######  Proposition 2.5.

Assume the notations of Theorem 2.2, and set

$\beta_{k,n}\mathrel{\mathop{\ordinarycolon}}=e^{-it}\beta_{0,n},\quad\beta_{-1,n}\mathrel{\mathop{\ordinarycolon}}=e^{it}\beta_{k-1,n},\quad\beta_{r,-1}\mathrel{\mathop{\ordinarycolon}}=0$

for every $n\geq 0$ and $0\leq r<k$. Then, $T_{1}$, $T_{2}$ and their adjoints
are weighted shifts given by

$T_{1}\beta_{r,n}=F^{\frac{1}{2}}\frac{\left(\sum_{j=0}^{k-1}\zeta_{j}^{-r}a_{j}^{-n-1}\right)^{\frac{1}{2}}}{\left(\sum_{j=0}^{k-1}\zeta_{j}^{-r-1}a_{j}^{-n-1}\right)^{\frac{1}{2}}}\beta_{r+1,n},$
$T_{1}^{\ast}\beta_{r,n}=F^{\frac{1}{2}}\frac{\left(\sum_{j=0}^{k-1}\zeta_{j}^{-r+1}a_{j}^{-n-1}\right)^{\frac{1}{2}}}{\left(\sum_{j=0}^{k-1}\zeta_{j}^{-r}a_{j}^{-n-1}\right)^{\frac{1}{2}}}\beta_{r-1,n},$
$T_{2}\beta_{r,n}=\frac{\left(\sum_{j=0}^{k-1}\zeta_{j}^{-r}a_{j}^{-n-1}\right)^{\frac{1}{2}}}{\left(\sum_{j=0}^{k-1}\zeta_{j}^{-r}a_{j}^{-n-2}\right)^{\frac{1}{2}}}\beta_{r,n+1},$
$T_{2}^{\ast}\beta_{r,n}=\frac{\left(\sum_{j=0}^{k-1}\zeta_{j}^{-r}a_{j}^{-n}\right)^{\frac{1}{2}}}{\left(\sum_{j=0}^{k-1}\zeta_{j}^{-r}a_{j}^{-n-1}\right)^{\frac{1}{2}}}\beta_{r,n-1}$

for every $0\leq r<k$ and $n\geq 0$.

###### Proof.

We do the computations for $T_{1}\beta_{r,n}$ when $0\leq r<k-1$. Each
$\beta_{r,n}$ is a sum of monomials $z_{1}^{r+kq}z_{2}^{n}$, $q\geq 0$. Since
distinct monomials are orthogonal to each other in $H_{2}^{2}$,
$z_{1}\beta_{r,n}$ is orthogonal to all elements
$\beta_{r^{\prime},n^{\prime}}$ of our orthonormal basis except for
$\beta_{r+1,n}$. Therefore, $T_{1}\beta_{r,n}$ is just the orthogonal
projection of $z_{1}\beta_{r,n}$ onto $\beta_{r+1,n}$, namely

$T_{1}\beta_{r,n}=\langle z_{1}\beta_{r,n},\beta_{r+1,n}\rangle\beta_{r+1,n}.$

According to Theorem 2.2 and Lemma 2.1, the weight $\langle
z_{1}\beta_{r,n},\beta_{r+1,n}\rangle$ equals

$\frac{\sum_{q\in\mathbb{N}}\omega_{r+kq,n}^{-1}\epsilon^{2q}}{F^{-r-\frac{1}{2}}k^{-1}\left(\sum_{j=0}^{k-1}\zeta_{j}^{-r}a_{j}^{-n-1}\right)^{\frac{1}{2}}\left(\sum_{j=0}^{k-1}\zeta_{j}^{-r-1}a_{j}^{-n-1}\right)^{\frac{1}{2}}}=F^{\frac{1}{2}}\frac{\left(\sum_{j=0}^{k-1}\zeta_{j}^{-r}a_{j}^{-n-1}\right)^{\frac{1}{2}}}{\left(\sum_{j=0}^{k-1}\zeta_{j}^{-r-1}a_{j}^{-n-1}\right)^{\frac{1}{2}}}.$

The rest is similar. ∎

It is immediate from Proposition 2.5 that:

######  Proposition 2.6.

Assume the notations of Theorem 2.2, Theorem 2.4 and Proposition 2.6. Set

$f_{k,n}\mathrel{\mathop{\ordinarycolon}}=f_{0,n},\quad
f_{-1,n}\mathrel{\mathop{\ordinarycolon}}=f_{k-1,n},\quad
U\mathrel{\mathop{\ordinarycolon}}=U_{2\pi}.$

for every $n\geq 0$. Then, we have

$U^{\ast}T_{1}U\beta_{r,n}=e^{i2\pi\left(f_{r,n}-f_{r+1,n}\right)}T_{1}\beta_{r,n},$
$U^{\ast}T_{1}^{*}U\beta_{r,n}=e^{i2\pi\left(f_{r,n}-f_{r-1,n}\right)}T_{1}^{*}\beta_{r,n},$
$U^{\ast}T_{2}U\beta_{r,n}=e^{i2\pi\left(f_{r,n}-f_{r,n+1}\right)}T_{2}\beta_{r,n},$
$U^{\ast}T^{\ast}_{2}U\beta_{r,n}=e^{i2\pi\left(f_{r,n}-f_{r,n-1}\right)}T_{2}^{*}\beta_{r,n}$

for every $0\leq r<k$ and $n\geq 0$.

The operator $U$ is called the monodromy operator associated with the
connection $D$ introduced in Equation (10).

We study the asymptotic behavior of the factors appearing in Proposition 2.6
when $n$ grows large. Recalling the asymptotic formula (14) for $f_{r,n}$, one
expects:

######  Lemma 2.7.

Assume the notations of Theorem 2.2 and Theorem 2.4. Then

$f_{r,n}-f_{r-1,n}\rightarrow-\frac{1}{k},\quad\quad\quad
f_{r,n}-f_{r,n-1}\rightarrow\frac{F}{k(1-F)}$

for every $0\leq r\leq k$ as $n\rightarrow\infty$.

###### Proof.

By (13), $k\left(f_{r,n}-f_{r,n-1}\right)$ equals

$\frac{Fn\sum\limits_{j,l=0}^{k-1}\zeta_{j}^{-r+1}a_{j}^{-n-2}\zeta_{l}^{-r}a_{l}^{-n}-Fn\sum\limits_{j,l=0}^{k-1}\zeta_{j}^{-r}a_{j}^{-n-1}\zeta_{l}^{-r+1}a_{l}^{-n-1}+F\sum\limits_{j,l=0}^{k-1}\zeta_{j}^{-r+1}a_{j}^{-n-2}\zeta_{l}^{-r}a_{l}^{-n}}{\sum\limits_{j,l=0}^{k-1}\zeta_{j}^{-r}a_{j}^{-n-1}\zeta_{l}^{-r}a_{l}^{-n}}.$

We need to find the dominant terms in the numerator and denominator of the
latter fraction when $n$ grows large. The dominant summand in the denominator
is the one with the smallest $|a_{j}||a_{l}|$, which is the one with $j=l=0$,
namely

$\zeta_{0}^{-r+1}a_{0}^{-n-1}\zeta_{0}^{-r}a_{0}^{-n}=(1-F)^{-2n-1}.$

We have three summations in the numerator with dominant terms

$Fn(1-F)^{-2n-2},\quad n(1-F)^{-2n-2}\quad\text{and}\quad F(1-F)^{-2n-2},$

respectively. The first two cancel each other, and all the remaining summands
in the first two summations are dominated by the dominant term of the
denominator $(1-F)^{-2n-1}$. Therefore, the dominant term of the numerator is
$F(1-F)^{-2n-2}$, so

$\lim\limits_{n\rightarrow\infty}k\left(f_{r,n}-f_{r,n-1}\right)=\lim\limits_{n\rightarrow\infty}\frac{F(1-F)^{-2n-2}}{(1-F)^{-2n-1}}=\frac{F}{1-F}.$

Using (13), $k\left(f_{r,n}-f_{r-1,n}\right)$ equals

$\frac{-\sum\zeta_{j}^{-r}\zeta_{l}^{-r+1}(a_{j}a_{l})^{-n-1}+F(n+1)\left(\sum\zeta_{j}^{-r+1}a_{j}^{-n-2}\zeta_{l}^{-r+1}a_{l}^{-n}-\zeta_{j}^{-r}\zeta_{l}^{-r+2}(a_{j}a_{l})^{-n-1}\right)}{\sum\zeta_{j}^{-r}a_{j}^{-n-1}\zeta_{l}^{-r+1}a_{l}^{-n-1}}.$

When $n$ grows large, the dominant terms in the numerator and denominator of
the latter fraction are

$-(1-F)^{-2n-2}+F(n+1)\times\big{(}\text{exponentially smaller than
}(1-F)^{-2n-2}\big{)}$

and

$(1-F)^{-2n-2},$

respectively. Therefore, $k\left(f_{r,n}-f_{r,n-1}\right)\rightarrow-1$. ∎

Proposition 2.6 and Lemma 2.7 together give:

######  Theorem 2.8.

As before, $F\mathrel{\mathop{\ordinarycolon}}=\epsilon^{\frac{2}{k}}$. The
unitary operator $U\mathrel{\mathop{\ordinarycolon}}=U_{2\pi}$ acts by
conjugation on the Toeplitz algebra $\mathfrak{T}_{I(0)}$ in the sense that
$U^{\ast}\mathfrak{T}_{I(0)}U\subseteq\mathfrak{T}_{I(0)}$. In more details,

$U^{\ast}T_{1}U-e^{i\frac{2\pi}{k}}T_{1},\quad
U^{\ast}T_{1}^{\ast}U-e^{-i\frac{2\pi}{k}}T^{\ast}_{1},\quad
U^{\ast}T_{2}U-e^{-i\frac{2\pi F}{k(1-F)}}T_{2},\quad
U^{\ast}T^{\ast}_{2}U-e^{i\frac{2\pi F}{k(1-F)}}T_{2}^{\ast}$

are all compact.

######  Remark 2.9.

We observe that the intersection of the zero variety of $z_{1}^{k}-\epsilon$
with the unit ball $|z_{1}|^{2}+|z_{2}|^{2}<1$ is a disjoint union
$\\{(\epsilon^{\frac{1}{k}}\exp(\frac{2j\pi
i}{k}),z_{2})\mathrel{\mathop{\ordinarycolon}}|z_{2}|^{2}<1-\epsilon^{\frac{2}{k}},0\leq
j\leq k-1\\}$. Let $\mathfrak{T}(\mathbb{D})$ be the algebra of Toeplitz
operators on $\mathbb{D}$ with continuous symbols. The Toeplitz algebra
$\mathfrak{T}_{I(0)}$ can be identified as a direct sum of $k$ copies of
$\mathfrak{T}(\mathbb{D})$, i.e.

$\mathfrak{T}_{I(0)}\cong\oplus_{j}\mathfrak{T}(\mathbb{D})_{j},$

which is indexed by the eigenvalue of $T_{z_{1}}$. With Theorem 2.8, we can
directly compute that the conjugation of operator $U$ is a permutation mapping
the component $\mathfrak{T}(\mathbb{D})_{j}$ to
$\mathfrak{T}(\mathbb{D})_{j+1}$. This picture is compatible with the Milnor
fibration and the open book decomposition associated to the polynomial
$z_{1}^{k}$ near the isolated singularity $0$, c.f. [34]. We plan to explore
this phenomenon and its applications in the future.

###  2.3. Smoothness of $P$

Recall the projection assembly map
$P\mathrel{\mathop{\ordinarycolon}}\mathbb{R}\rightarrow B(H_{2}^{2})$ acting
between Banach spaces. We now prove what we mentioned earlier:

######  Proposition 2.10.

$P$ does not have a bounded derivative.

###### Proof.

According to Theorem 2.4.(d), each
$\delta_{r,n}\mathrel{\mathop{\ordinarycolon}}=e^{if_{r,n}t}\alpha_{r,n}$,
$(r,n)\in J$ is a flat section of $\mathcal{I}^{\perp}$, namely, it satisfies
the following equations

$P_{t}\delta_{r,n}(t)=\delta_{r,n}(t),\quad P_{t}\dot{\delta}_{r,n}(t)=0.$

Suppose by contradiction that $P$ has a bounded derivative. Differentiating
the first equation and plugging from the second gives

$\dot{P}_{t}\delta_{r,n}(t)=\dot{\delta}_{r,n}(t).$

However, the ratio

(19)
$\frac{\left\|\dot{\delta}_{r,n}(t)\right\|}{\left\|\delta_{r,n}(t)\right\|}=\frac{\left\|if_{r,n}\alpha_{r,n}(t)+\dot{\alpha}_{r,n}(t)\right\|}{\left\|\alpha_{r,n}(t)\right\|}=\left(\frac{\sum_{q\in\mathbb{N}}\omega_{r+kq,n}^{-1}\epsilon^{2q}(f_{r,n}-q)^{2}}{\sum_{q\in\mathbb{N}}\omega_{r+kq,n}^{-1}\epsilon^{2q}}\right)^{\frac{1}{2}}\\\
=\left(f_{r,n}^{2}-2f_{r,n}\frac{\sum_{q\in\mathbb{N}}\omega_{r+kq,n}^{-1}\epsilon^{2q}q}{\sum_{q\in\mathbb{N}}\omega_{r+kq,n}^{-1}\epsilon^{2q}}+\frac{\sum_{q\in\mathbb{N}}\omega_{r+kq,n}^{-1}\epsilon^{2q}q^{2}}{\sum_{q\in\mathbb{N}}\omega_{r+kq,n}^{-1}\epsilon^{2q}}\right)^{\frac{1}{2}}$

asymptotically behaves like $n^{\frac{1}{2}}$ as $n\rightarrow\infty$, hence
$\dot{P}_{t}$ would be unbounded. Here are more details. By Lemma 2.1 and the
asymptotic formula for $f_{r,n}$ in (14), the three consecutive terms $a$, $b$
and $c$ in the last expression $\left(a-b+c\right)^{1/2}$ in (19)
asymptotically behave like $a_{2}n^{2}+a_{1}n$, $b_{2}n^{2}+b_{1}n$ and
$c_{2}n^{2}+c_{1}n$, where $a_{j}$, $b_{j}$ and $c_{j}$ are nonzero constants
(with respect to $n$) satisfying $a_{2}-b_{2}+c_{2}=0$ and
$a_{1}-b_{1}+c_{1}\neq 0$. (Here by saying that $a$ behaves asymptotically
like $a_{2}n^{2}+a_{1}n$, we mean that $a\approx n^{2}$, $a-a_{2}n^{2}\approx
n$ and $a-a_{2}n^{2}-a_{1}n\ll 1$. Likewise for $b$ and $c$.) This shows that
$\left(a-b+c\right)^{1/2}$ asymptotically behaves like $n^{1/2}$. This
contradiction shows that $P$ does not have a bounded derivative. ∎

We discuss a possible way to fix this problem by using weights to compensate
for differentiation [10].

More precisely, viewing the Drury-Arveson space
$H_{2}^{2}=\mathcal{H}^{(-2)}_{2}$ as a member of the Besov-Sobolev scale
$\mathcal{H}^{(s)}_{2}$, $s\in\mathbb{R}$ of Hilbert spaces444Defined in
Appendix A., we have:

######  Theorem 2.11.

(a) The modification
$\widetilde{P}\mathrel{\mathop{\ordinarycolon}}\mathbb{R}\rightarrow
B\left(\mathcal{H}^{(-2)}_{2},\mathcal{H}^{(4)}_{2}\right)$ of $P$ is first-
order differentiable, where $\widetilde{P}_{t}$ is the composition of $P_{t}$
with the inclusion
$\mathcal{H}^{(-2)}_{2}\hookrightarrow\mathcal{H}^{(4)}_{2}$.

(b) Given a positive integer $l$ and a positive real $\sigma$, the
modification

$\widetilde{P}\mathrel{\mathop{\ordinarycolon}}\mathbb{R}\rightarrow
B\left(\mathcal{H}^{(-2)}_{2},\mathcal{H}^{(2l+1+\sigma)}_{2}\right)$

of $P$ is $l$-th order differentiable, where $\widetilde{P}_{t}$ is the
composition of $P_{t}$ with the inclusion
$\mathcal{H}^{(-2)}_{2}\hookrightarrow\mathcal{H}^{(2l+1+\sigma)}_{2}$.

###### Proof.

(a) We have a corresponding version of Theorem 2.2 for $\mathcal{H}^{(4)}_{2}$
instead of $H_{2}^{2}=\mathcal{H}^{(-2)}_{2}$, where $\omega_{r+kq,n}$ is
replaced by

$\widetilde{\omega}_{r+kq,n}\mathrel{\mathop{\ordinarycolon}}=\left\|z_{1}^{r+kq}z_{2}^{n}\right\|_{\mathcal{H}^{(4)}_{2}}=\frac{(r+kq)!n!6!}{(r+kq+n+6)!}=S(n)\omega_{r+kq,n+6},$

and

$S(n)\mathrel{\mathop{\ordinarycolon}}=\frac{6!n!}{(n+6)!}\approx n^{-6}.$

Let

$\left\\{e_{m,n}\mathrel{\mathop{\ordinarycolon}}=\omega_{m,n}^{-\frac{1}{2}}z_{1}^{m}z_{2}^{n}\
\mathrel{\mathop{\ordinarycolon}}\
(m,n)\in\mathbb{N}^{2}\right\\}\quad\text{and}\quad\left\\{\widetilde{e}_{m,n}\mathrel{\mathop{\ordinarycolon}}=\widetilde{\omega}_{m,n}^{-\frac{1}{2}}z_{1}^{m}z_{2}^{n}\
\mathrel{\mathop{\ordinarycolon}}\ (m,n)\in\mathbb{N}^{2}\right\\}$

be the time-independent standard orthonormal bases of $\mathcal{H}^{(-2)}_{2}$
and $\mathcal{H}^{(4)}_{2}$, respectively. We first compute the matrix
coefficients of $\widetilde{P}_{t}$ with respect to these bases. Note that for
each $(m,n)\in\mathbb{N}^{2}$, $e_{m,n}$ is orthogonal to all members of the
orthonormal frame $\widetilde{\beta}$ except for $\widetilde{\beta}_{r,n}$,
where

$m=kQ+r,\quad Q,r\in\mathbb{N},\quad 0\leq r<k$

is the division of $m$ by $k$. Therefore,
$\widetilde{P}_{t}\left(e_{m,n}\right)=\left\langle
e_{m,n},\beta_{r,n}\right\rangle\beta_{r,n}$ is given by

$\frac{\epsilon^{Q}e^{iQt}\sum_{q\in\mathbb{N}}\omega_{r+kq,n}^{-1}\widetilde{\omega}_{r+kq,n}^{\frac{1}{2}}\epsilon^{q}e^{-iqt}\widetilde{e}_{r+kq,n}}{\omega_{m,n}^{\frac{1}{2}}\left\|\alpha_{r,n}\right\|^{2}}=\frac{\sum\limits_{q}\omega_{m,n}^{-\frac{1}{2}}\omega_{r+kq,n}^{-1}\widetilde{\omega}_{r+kq,n}^{\frac{1}{2}}\epsilon^{Q+q}e^{i\left(Q-q\right)t}\widetilde{e}_{r+kq,n}}{\sum\limits_{q}\omega_{r+kq,n}^{-1}\epsilon^{2q}}\\\
=\sqrt{S(n)}\frac{\sum\limits_{q}\omega_{m,n}^{-\frac{1}{2}}\omega_{r+kq,n}^{-1}\omega_{r+kq,n+6}^{\frac{1}{2}}\epsilon^{Q+q}e^{i\left(Q-q\right)t}\widetilde{e}_{r+kq,n}}{\sum\limits_{q}\omega_{r+kq,n}^{-1}\epsilon^{2q}}.$

This shows that the formal matrix $\dot{\widetilde{P}}_{t}$ of entry-by-entry
differentiation of $P_{t}$ is given by

$\dot{\widetilde{P}}_{t}\left(e_{m,n}\right)\mathrel{\mathop{\ordinarycolon}}=\sqrt{S(n)}\frac{\sum\limits_{q}\omega_{m,n}^{-\frac{1}{2}}\omega_{r+kq,n}^{-1}\omega_{r+kq,n+6}^{\frac{1}{2}}\epsilon^{Q+q}i(Q-q)e^{i\left(Q-q\right)t}\widetilde{e}_{r+kq,n}}{\sum\limits_{q}\omega_{r+kq,n}^{-1}\epsilon^{2q}}.$

From this expression, the Hilbert-Schmidt norm of $\dot{\widetilde{P}}_{t}$,

$\left\|\dot{\widetilde{P}}_{t}\right\|_{\mathrm{HS}}^{2}\mathrel{\mathop{\ordinarycolon}}=\sum\limits_{n,r}S(n)\frac{\sum\limits_{Q,q}\omega_{r+kQ,n}^{-1}\omega_{r+kq,n}^{-2}\omega_{r+kq,n+6}\epsilon^{2Q+2q}(Q-q)^{2}}{\left(\sum\limits_{q}\omega_{r+kq,n}^{-1}\epsilon^{2q}\right)^{2}},$

is bounded above by

$\sum\limits_{n,r}\frac{S(n)\sum\limits_{Q,q}\omega_{r+kQ,n}^{-1}\omega_{r+kq,n}^{-1}\epsilon^{2Q+2q}\left(Q^{2}+q^{2}\right)}{\left(\sum\limits_{q}\omega_{r+kq,n}^{-1}\epsilon^{2q}\right)^{2}}=\sum\limits_{n,r}\frac{2S(n)\sum\limits_{q}\omega_{r+kq,n}^{-1}\epsilon^{2q}q^{2}}{\sum\limits_{q}\omega_{r+kq,n}^{-1}\epsilon^{2q}}\approx\sum\limits_{n}S(n)n^{2},$

a finite number. This shows that $\dot{\widetilde{P}}_{t}$ is bounded (in
operator norm) [6, 2.8.4]. With the same line of arguments along with Taylor’s
theorem, for any $h\in\mathbb{R}$, we have

$\left\|\widetilde{P}_{t+h}-\widetilde{P}_{t}-h\dot{\widetilde{P}}_{t}\right\|_{\mathrm{HS}}^{2}\leq\sum\limits_{n,r}S(n)\frac{\sum\limits_{Q,q}\omega_{r+kQ,n}^{-1}\omega_{r+kq,n}^{-2}\omega_{r+kq,n+6}\epsilon^{2Q+2q}(Q-q)^{4}\left(\frac{h^{2}}{2!}\right)^{2}}{\left(\sum\limits_{q}\omega_{r+kq,n}^{-1}\epsilon^{2q}\right)^{2}}$
$\ll\sum\limits_{n,r}S(n)h^{4}\frac{\sum\limits_{q}\omega_{r+kq,n}^{-1}\epsilon^{2q}q^{4}}{\sum\limits_{q}\omega_{r+kq,n}^{-1}\epsilon^{2q}}\ll\sum_{n}S(n)h^{4}n^{4}=h^{4}\sum\limits_{n}n^{-2}.$

This finishes the proof that $\widetilde{P}$ is first-order differentiable.

(b) Imitating the proof in Part (a), we set

$\widetilde{\omega}_{r+kq,n}\mathrel{\mathop{\ordinarycolon}}=\left\|z_{1}^{r+kq}z_{2}^{n}\right\|_{\mathcal{H}^{(2l+1+\sigma)}_{2}}=\frac{(r+kq)!n!(2l+3+\sigma)!}{(r+kq+n+2l+3+\sigma)!}=S(n)\omega_{r+kq,n+2l+3+\sigma},$

where

$S(n)\mathrel{\mathop{\ordinarycolon}}=\frac{(2l+3+\sigma)!n!}{(n+2l+3+\sigma)!}\approx
n^{-2l-3-\sigma}.$

Given $j=1,\ldots,l$ and $h\in\mathbb{R}$, we have estimates

$\left\|\frac{d^{j}\widetilde{P}_{t}}{dt^{j}}\right\|_{\mathrm{HS}}^{2}\ll\sum\limits_{n}S(n)n^{2j}\approx\sum\limits_{n}n^{-3-\sigma-2(l-j)}<\infty,$
$\left\|\frac{d^{j-1}\widetilde{P}_{t+h}}{dt^{j-1}}-\frac{d^{j-1}\widetilde{P}_{t}}{dt^{j-1}}-h\frac{d^{j}\widetilde{P}_{t}}{dt^{j}}\right\|_{\mathrm{HS}}^{2}\ll\sum\limits_{n}S(n)h^{2j+2}n^{2j+2}\approx
h^{2j+2}\sum\limits_{n}n^{-1-\sigma-2(l-j)},$

which implies that $\widetilde{P}$ is $l$-th order differentiable. ∎

######  Remark 2.12.

Recall the identification
$\mathcal{H}^{(s)}_{m}=W^{-\frac{s}{2}}_{\mathrm{hol}}(\mathbb{B}^{m})$,
$s\in\mathbb{R}$, between Besov-Sobolev and Bergman-Sobolev spaces mentioned
in Appendix A. Theorem 2.11.(b) says that by taking $l$-th order derivative of
$P$, we lose differentiability by order no worse than $l+2$. We do not know
whether this estimate of differentiability loss is optimal.

######  Remark 2.13.

Suppose a section $\xi\in C^{\infty}(\mathbb{R},\mathcal{I}^{\perp})$.
Proposition 2.11.(b) shows that $\frac{d^{l}P_{t}}{dt^{l}}\left(\xi(t)\right)$
lives in $\mathcal{H}^{(2l+1+\sigma)}_{2}$. Similar arguments show that for
each $s\leq-2$, if $\xi(t)\in\mathcal{H}^{(s)}_{2}$, then
$\frac{d^{l}P_{t}}{dt^{l}}\left(\xi(t)\right)$ in fact lives in
$\mathcal{H}^{(s+2l+3+\sigma)}_{2}$.

We are now ready to conclude with the smoothness of the connection $D$.

######  Corollary 2.14.

Let $\mathcal{S}$ denote the set of all sections $\xi\in
C^{\infty}(\mathbb{R};\mathcal{I}^{\perp})$ such that for each $t$, $\xi(t)$
and all its time derivatives live in
$\bigcap_{s\in\mathbb{R}}\mathcal{H}^{(s)}_{2}$. The connection $D$ defined by
(10) maps $\mathcal{S}$ to itself.

###### Proof.

Suppose that $\xi$ is from $\mathcal{S}$. Then $D\xi=P_{t}(\frac{d}{dt}\xi)$
by Equation (10). By the definition of $\mathcal{S}$, $\frac{d}{dt}\xi$
belongs to $\mathcal{S}$. It follows from Theorem 2.11 that
$P_{t}(\frac{d}{dt}\xi)$ is in
$\bigcap_{s\in\mathbb{R}}\mathcal{H}^{(s)}_{2}$.

We compute the derivative of $P_{t}(\frac{d}{dt}\xi)$, i.e.

$\frac{d}{dt}\left(P_{t}\Big{(}\frac{d}{dt}\xi\Big{)}\right)=\frac{dP_{t}}{dt}\Big{(}\frac{d}{dt}\xi\Big{)}+P_{t}\Big{(}\frac{d^{2}}{dt^{2}}\xi\Big{)}.$

It follows from Remark 2.13 that both
$\frac{dP_{t}}{dt}\big{(}\frac{d}{dt}\xi\big{)}$ and
$P_{t}(\frac{d^{2}}{dt^{2}}\xi)$ belong to
$\bigcap_{s\in\mathbb{R}}\mathcal{H}^{(s)}_{2}$. Therefore,
$\frac{d}{dt}\big{(}P_{t}(\frac{d}{dt}\xi)\big{)}$ lives in
$\bigcap_{s\in\mathbb{R}}\mathcal{H}^{(s)}_{2}$.

By induction, one easily proves that
$\frac{d^{l}}{dt^{l}}\big{(}P_{t}(\frac{d}{dt}\xi)\big{)}$ belongs to
$\bigcap_{s\in\mathbb{R}}\mathcal{H}^{(s)}_{2}$. And $D(\xi)$ belongs to
$\mathcal{S}$. ∎

##  3\. Outlook

It is natural to ask whether Theorems 2.3, 2.4, and 2.8 have a natural
extension to more general principal ideals. In the following, we propose a
series of problems attempting to generalize the results in Section 2.

Suppose a polynomial $f\in A=\mathbb{C}[z_{1},\ldots,z_{m}]$ which vanishes at
the origin, and it has the origin as an isolated critical point. In geometric
terms, the origin is an isolated singularity of the hypersurface
$V(f)\subseteq\mathbb{C}^{m}$. Consider the family of principal ideals

$I(t)\mathrel{\mathop{\ordinarycolon}}=\langle f-\epsilon
e^{it}\rangle\subseteq A,\quad t\in\mathbb{R},$

where $\epsilon$ is a fixed, sufficiently small, positive real number. We
think of $t$ as the time variable. Let $P_{t}$ be the orthogonal projection in
$H_{m}^{2}$ onto $I(t)^{\perp}$. Let

$p\mathrel{\mathop{\ordinarycolon}}\mathcal{I}^{\perp}\rightarrow\mathbb{R},\quad\mathcal{I}^{\perp}\mathrel{\mathop{\ordinarycolon}}=\biguplus\left\\{I(t)^{\perp}\subseteq
H_{m}^{2}\mathrel{\mathop{\ordinarycolon}}t\in\mathbb{R}\right\\}\subseteq\mathbb{R}\times
H_{m}^{2},\quad p\left(I(t)^{\perp}\right)=\\{t\\},$

and

$P\mathrel{\mathop{\ordinarycolon}}\mathbb{R}\rightarrow
B\left(H^{2}_{m}\right),\quad P\mathrel{\mathop{\ordinarycolon}}=(P_{t})$

be respectively the assembly of Hilbert spaces $I(t)^{\perp}$ and orthogonal
projections $P_{t}$ into a rough Hilbert bundle and a rough map between Banach
spaces. Topologize $\mathcal{I}^{\perp}\subseteq\mathbb{R}\times H_{m}^{2}$
with the subspace topology.

We would like to ask the following question about $p$.

######  Problem 3.1.

Is $p$ a smooth Hilbert bundle555$C^{2}$-smoothness is enough for our purposes
[26, 28, 30].?

At the moment, we can only prove that $p$ is a smooth Hilbert bundle for the
special case in Section 2 (see Theorem 2.3). Note that since the base space of
$p$ is contractible, even the weaker property that $p$ is a topological vector
bundle implies that it is trivial [23, IV.2.5], [29, Corollary 1], hence
automatically smooth, and this smooth structure is unique up to smooth vector
bundle isomorphisms [23, IV.3.5]. Let us proceed with assuming that Problem
3.1 has a positive answer. Denote the set of all (smooth) sections of $p$ by
$C^{\infty}(\mathbb{R};\mathcal{I}^{\perp})$.

Unfortunately, the family of projections $P$ is not smooth in general (see
Section 2.3, Proposition 2.10). Thinking of $P$ as a rough connection between
nearby fibers $I(t)^{\perp}$, the imitation of the standard construction of
the Hermitian connection for subbundles of Hilbert bundles [26, Example
1.5.14],[35, Volume II, Page 540] gives us a rough covariant derivative:

(20) $D\xi(t)=P_{t}\left(\frac{d\xi}{dt}\right),\quad\xi\in
C^{\infty}(\mathbb{R};\mathcal{I}^{\perp}).$

Note that $D$ is called a covariant derivative because it satisfies the
Leibniz rule

$D(g\xi)(t)=g^{\prime}(t)\xi(t)+g(t)D(\xi)(t),\quad\forall g\in
C^{\infty}(\mathbb{R};\mathbb{C}),\quad\forall\xi\in
C^{\infty}(\mathbb{R};\mathcal{I}^{\perp}).$

The $D$-flat sections of $p$ are those $\xi\in
C^{\infty}(\mathbb{R};\mathcal{I}^{\perp})$ which satisfy the evolution
equation

(21) $D\xi(t)=0,\quad\forall t\in\mathbb{R}.$

This leads us to the problem about solving Equation (21).

######  Problem 3.2.

Does the parallel transport equation (21) has a unique solution on
$t\in\mathbb{R}$ for each initial value $\xi(0)\in I(0)^{\perp}$?

Suppose that the operator
$U_{t}\mathrel{\mathop{\ordinarycolon}}I(0)^{\perp}\to I(t)^{\perp}$ is the
solution operator to Equation (21). We are especially interested in

$U\mathrel{\mathop{\ordinarycolon}}=U_{2\pi}\in B(I(0)^{\perp}),$

which can be viewed as a noncommutative analogue of the Milnor monodromy map
$h\mathrel{\mathop{\ordinarycolon}}F_{0}\rightarrow F_{0}$ in [34, Page 67].
We would like to know the answer to the following problem.

######  Problem 3.3.

Does $U$ act by conjugation on the Toeplitz algebra $\mathfrak{T}_{I(0)}$ in
the sense that $U\mathfrak{T}_{I(0)}U^{\ast}\subseteq\mathfrak{T}_{I(0)}$?

A positive answer to Problem 3.3 will induce a map
$K\left(\mathfrak{T}_{I(0)}\right)\rightarrow
K\left(\mathfrak{T}_{I(0)}\right)$ at the $K$-homology level. Such a map
should be viewed as the analytic analogue of the monodromy operator introduced
and studied by differential topologists, e.g. [34, 33].

Besides the examples $z_{1}^{k}$ in Section 2, we are also able to solve
Problems 3.1, 3.2 and 3.3 for monomials $z_{1}^{k}z_{2}^{l}$ with similar
methods. Unfortunately, our current methods in Section 2 are not sufficiently
developed to solve Problems 3.1, 3.2 and 3.3 for more general ideals.

We end this paper by pointing out that it is interesting to study the
asymptotic behavior of the unitary operator $U$ when $\epsilon\rightarrow 0$.
More specifically, note that in Theorem 2.8 appears the phase factor
$\exp\frac{2\pi iF}{k(1-F)}$, $F=\epsilon^{\frac{2}{k}}$. When
$\epsilon\rightarrow 0$, this factor varies like

$\exp\left(\frac{2\pi i}{k}\epsilon^{\frac{2}{k}}\right).$

For the example
$f\mathrel{\mathop{\ordinarycolon}}=z_{1}z_{2}\in\mathbb{C}[z_{1},z_{2}]$, our
computations (with similar methods as in Section 2) show that the phase factor
equals

$\exp\left(2\pi
i\frac{1-\sqrt{1-4\epsilon^{2}}}{\sqrt{1-4\epsilon^{2}}}\right)=\exp\left(4\pi
i\epsilon^{2}+O\left(\epsilon^{4}\right)\right).$

It is desirable to understand these phase factors in the general case and
study their connections to the corresponding polynomials.

##  Appendix A Some analytic Hilbert spaces

In this appendix, we review the definitions of some famous analytic Hilbert
spaces used in this paper.

  1. (1)

Let $\Omega\subseteq\mathbb{C}^{m}$ be a bounded domain with smooth boundary.

     * •

$W^{s}_{\mathrm{hol}}(\Omega)$, $s\in\mathbb{R}$ is the Bergman-Sobolev space
consisting of all holomorphic functions in the $L^{2}$ Sobolev space
$W^{s}(\Omega)$. (See [19, 31, 10].) These are also known as the holomorphic
Sobolev spaces.

     * •

$W^{s}_{\mathrm{hol}}(\partial\Omega)$, $s\in\mathbb{R}$ is the Hardy-Sobolev
space consisting of all functions in the $L^{2}$ Sobolev space
$W^{s}(\partial\Omega)$ whose Poisson extension to $\Omega$ is not only
harmonic but also holomorphic. Alternatively, it is the closure in
$W^{s}(\partial\Omega)$ of the boundary values of holomorphic functions on
$\Omega$ which are continuous up to the boundary [12, 19]. Note that
$W^{s}_{\mathrm{hol}}(\partial\Omega)$ is isometrically isomorphic to the
Bergman-Sobolev space $W^{s+\frac{1}{2}}_{\mathrm{hol}}(\Omega)$ through the
Poisson extension and the trace map [19].

     * •

$H^{2}(\partial\Omega)\mathrel{\mathop{\ordinarycolon}}=W^{0}_{\mathrm{hol}}(\partial\Omega)$
is the Hardy space [27, stein, 36, 16].

     * •

$L_{a,s}^{2}(\Omega)$, $s>-1$ is the weighted Bergman space consisting of
holomorphic functions $f$ on $\Omega$ such that
$\int_{\Omega}|f(z)|^{2}\rho(z)^{s}dV(z)<\infty$, where $\rho(z)$ is a
positively signed smooth defining function for $\Omega$ (equivalently, the
distance function $\mathrm{dist}(z,\partial\Omega)$), and $dV(z)$ is the
Lebesgue measure on $\Omega$ normalized such that
$\int_{\Omega}\rho(z)^{s}dV(z)=1$. (See [10, 19, 38].) Note that
$L_{a,s}^{2}(\Omega)=W^{-\frac{s}{2}}_{\mathrm{hol}}(\Omega)$ as sets with
equivalent norms [32, 19, 10].

     * •

$L_{a}^{2}(\Omega)\mathrel{\mathop{\ordinarycolon}}=L_{a,0}^{2}(\Omega)$ is
the (unweighted) Bergman space [27, 36, 35].

  2. (2)

$H_{m}^{2}$ is the Drury-Arveson space of analytic functions on
$\mathbb{B}^{m}$, the one with the reproducing kernel $\left(1-\langle
z,w\rangle\right)^{-1}$. (See [3], [1, Chapter 41].) It has the standard
orthonormal basis $(n!/|n|!)^{-1/2}z^{n}$, $n\in\mathbb{N}^{m}$. It is also
known as the $m$-shift or symmetric Fock space.

  3. (3)

$\mathcal{H}^{(s)}_{m}$, $s\in\mathbb{R}$ is the Besov-Sobolev space of
analytic functions on $\mathbb{B}^{m}$, the one with the reproducing kernel

$K_{s}(z,w)\mathrel{\mathop{\ordinarycolon}}=\begin{cases}(1-\langle
z,w\rangle)^{-s-m-1},&\quad s>-m-1,\\\ (-s-m)^{-1}F\left(1,1;1-s-m;\langle
z,w\rangle\right),&\quad s\leq-m-1,\end{cases}$

where
$F(a,b;c;\zeta)\mathrel{\mathop{\ordinarycolon}}=\sum_{q\in\mathbb{N}}\frac{(a)_{q}(b)_{q}}{(c)_{q}q!}\zeta^{q}$
is the hypergeometric function, and
$(x)_{y}\mathrel{\mathop{\ordinarycolon}}=\Gamma(x+y)/\Gamma(x)$ is the
Pochhammer symbol. (See [11, 37, 19, 2, 38]; our parameter $s+m+1$ is $q$ in
[11], $\alpha+m+1$ in [37, 19], and $2\sigma$ in [2]; [38] only studies the
$s=-m-1$ case.) $\mathcal{H}^{(s)}_{m}$ has the standard orthonormal basis
$\omega_{s}(n)^{-1/2}z^{n}$, $n\in\mathbb{N}^{m}$, where

$\omega_{s}(n)\mathrel{\mathop{\ordinarycolon}}=\begin{cases}\frac{n!(s+m)!}{(|n|+s+m)!},&\quad
s>-m-1,\\\ \frac{n!(-s-m)_{|n|+1}}{\left(|n|!\right)^{2}},&\quad
s\leq-m-1.\end{cases}$

Note that

$\omega_{s}(n)\approx\frac{n!}{|n|!\left(|n|+1\right)^{s+m}}$

for each $s\in\mathbb{R}$. (We will not need the reproducing kernel, but this
equivalent norm is enough for our purposes.) We have the identifications:

$\mathcal{H}_{m}^{(s)}=\begin{cases}\text{the Bergman-Sobolev space}\
W^{-\frac{s}{2}}_{\mathrm{hol}}(\mathbb{B}^{m})\ \text{(as sets with
equivalent norms)},&s\in\mathbb{R},\\\ \text{the Hardy-Sobolev space}\
W^{-\frac{s+1}{2}}_{\mathrm{hol}}(\partial\mathbb{B}^{m})\ \text{(as sets with
equivalent norms)},&s\in\mathbb{R},\\\ \text{the Drury-Arveson space}\
H_{m}^{2}\ \text{(as sets with equal norms)},&s=-m,\\\ \text{the Hardy space}\
H^{2}(\partial\mathbb{B}^{m})\ \text{(as sets with equal norms)},&s=-1,\\\
\text{the weighted Bergman space}\ L^{2}_{a,s}(\mathbb{B}^{m})\ \text{(as sets
with equal norms)},&s>-1.\end{cases}$

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