Cyclicity in the Drury-Arveson space and other weighted Besov spaces
Authors: Aleman Alexandru , Perfekt Karl-Mikael , Richter Stefan , Sundberg Carl , Sunkes James 
Published: October 2023

Let

H

\mathcal {H}

be a space of analytic functions on the unit ball

B

d

{\mathbb {B}_d}

in

C

d

\mathbb {C}^d

with multiplier algebra

M
u
l
t

(

H

)

\mathrm {Mult}(\mathcal {H})

. A function

f
∈

H

f\in \mathcal {H}

is called cyclic if the set

[
f
]

[f]

, the closure of

{
φ
f
:
φ
∈

M
u
l
t

(

H

)
}

\{\varphi f: \varphi \in \mathrm {Mult}(\mathcal {H})\}

, equals

H

\mathcal {H}

. For multipliers we also consider a weakened form of the cyclicity concept. Namely for

n
∈

N

0

n\in \mathbb {N}_0

we consider the classes
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper C Subscript n Baseline left-parenthesis script upper H right-parenthesis equals left-brace phi element-of normal upper M normal u normal l normal t left-parenthesis script upper H right-parenthesis colon phi not-equals 0 comma left-bracket phi Superscript n Baseline right-bracket equals left-bracket phi Superscript n plus 1 Baseline right-bracket right-brace period">

C

n

(

H

)
=
{
φ
∈

M
u
l
t

(

H

)
:
φ
≠
0
,
[

φ
n

]
=
[

φ

n
+
1

]
}
.

\begin{equation*} \mathcal {C}_n(\mathcal {H})=\{\varphi \in \mathrm {Mult}(\mathcal {H}):\varphi \ne 0, [\varphi ^n]=[\varphi ^{n+1}]\}. \end{equation*}

Many of our results hold for

N
N

:th order radially weighted Besov spaces on

B

d

{\mathbb {B}_d}

,

H

=

B
ω
N

\mathcal {H}= B^N_\omega

, but we describe our results only for the Drury-Arveson space

H
d
2

H^2_d

here.

Letting

C

s
t
a
b
l
e

[
z
]

\mathbb {C}_{stable}[z]

denote the stable polynomials for

B

d

{\mathbb {B}_d}

, i.e. the

d
d

-variable complex polynomials without zeros in

B

d

{\mathbb {B}_d}

, we show that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row 1st Column Blank 2nd Column a m p semicolon if d is odd comma then double-struck upper C Subscript s t a b l e Baseline left-bracket z right-bracket subset-of-or-equal-to script upper C Subscript StartFraction d minus 1 Over 2 EndFraction Baseline left-parenthesis upper H Subscript d Superscript 2 Baseline right-parenthesis comma and 2nd Row 1st Column Blank 2nd Column a m p semicolon if d is even comma then double-struck upper C Subscript s t a b l e Baseline left-bracket z right-bracket subset-of-or-equal-to script upper C Subscript StartFraction d Over 2 EndFraction minus 1 Baseline left-parenthesis upper H Subscript d Superscript 2 Baseline right-parenthesis period EndLayout">

a
m
p
;
 if 
d
 is odd, then 

C

s
t
a
b
l
e

[
z
]
⊆

C

d
−
1

2

(

H
d
2

)
,
 and 

a
m
p
;
 if 
d
 is even, then 

C

s
t
a
b
l
e

[
z
]
⊆

C

d
2

−
1

(

H
d
2

)
.

\begin{align*} &\text { if } d \text { is odd, then } \mathbb {C}_{stable}[z]\subseteq \mathcal {C}_{\frac {d-1}{2}}(H^2_d), \text { and }\\ &\text { if } d \text { is even, then } \mathbb {C}_{stable}[z]\subseteq \mathcal {C}_{\frac {d}{2}-1}(H^2_d). \end{align*}

For

d
=
2

d=2

and

d
=
4

d=4

these inclusions are the best possible, but in general we can only show that if

0
≤
n
≤

d
4

−
1

0\le n\le \frac {d}{4}-1

, then

C

s
t
a
b
l
e

[
z
]
⊈

C

n

(

H
d
2

)

\mathbb {C}_{stable}[z]\nsubseteq \mathcal {C}_n(H^2_d)

.

For functions other than polynomials we show that if

f
,
g
∈

H
d
2

f,g\in H^2_d

such that

f

/

g
∈

H
∞

f/g\in H^\infty

and

f
f

is cyclic, then

g
g

is cyclic. We use this to prove that if

f
,
g

f,g

extend to be analytic in a neighborhood of

B

d

¯

\overline {{\mathbb {B}_d}}

, have no zeros in

B

d

{\mathbb {B}_d}

, and the same zero sets on the boundary, then

f
f

is cyclic in

∈

H
d
2

\in H^2_d

if and only if

g
g

is. Furthermore, if the boundary zero set of

f
∈

H
d
2

∩
C
(

B

d

¯

)

f\in H^2_d\cap C(\overline {{\mathbb {B}_d}})

embeds a cube of real dimension

≥
3

\ge 3

, then

f
f

is not cyclic in the Drury-Arveson space.