Non cyclic functions in the Hardy space of the bidisc with arbitrary decrease

0.09 MB·English

by  Xavier Massaneda

NON CYCLIC FUNCTIONS IN THE HARDY SPACE
OF THE BIDISC WITH ARBITRARY DECREASE
3
1 XAVIER MASSANEDA AND PASCAL J. THOMAS
0
2
Abstract. We construct an example to show that no condition
n
of slow decrease of the modulus of a function is sufficient to make
a
J it cyclic in the Hardy space of the bidisc. This is similar to what
is well known in the case of the Hardy space of the disc, but in
1
1 contrast to the case of the Bergman space of the disc.

]
V
C
. 1. Background
h
t
a A cyclic vector f for a given operator φ from a topological vector
m space X to itself is one such that Span{φn(f),n ∈ Z } is dense in
+
[ X. In the classical framework of the Hardy space H2(D) on the unit

1 disk with the shift operator φ(f)(z) := zf(z), a cyclic function is one
v that verifies that C[Z]f is dense in H2(D), where C[Z] denotes the set
2
2 of holomorphic polynomials. The characterization of those as outer
6 functions goes back to Beurling, see [Ga] or [Ko] for details. For any
2
f ∈ H2(D),
.
1
0 2π eiθ +ζ
3 f(ζ) := B(ζ)exp (g(θ)dθ−dµ(θ)) ,
1 (cid:18)Z eiθ −ζ (cid:19)
0
:
v where B is a Blaschke product, eg ∈ L2(∂D) and µ is a positive sin-
i
X
gular measure. The factor containing only µ is called a singular inner
r
function. The function f is called outer when µ = 0 and B = 1 (or,
a
equivalently, log|f| is the Poisson integral of its boundary values).
We are interested in the situation in H2(D2), where D2 := D×D ⊂
C2, and
2π 2π
H2(D2) := f ∈ Hol(D2) : kfk2 := sup |f(reiθ1,reiθ2)|2dθ dθ < ∞ .
(cid:26) 2 Z Z 1 2 (cid:27)
r<1 0 0
2010 Mathematics Subject Classification. 32A35, 30H10, 46E15.
Key words and phrases. Hardy spaces, polydisk, cyclic vectors.
This work was carriedout while both autors were participating in the Semester
inComplex Analysis andSpectral Theoryofthe Centre de RecercaMatema`tica at
the Universitat Aut`onoma de Barcelona.
1
2 XAVIERMASSANEDAAND PASCAL J. THOMAS
A cyclic function is f ∈ H2(D2) such that C[Z ,Z ]f is dense in
1 2
H2(D2).
Rudin [Ru] proved that the analogous equivalence between being
outer (in an appropriate sense) and cyclic fails in the case of the poly-
disk. A lot of work on analogous problems was later carried out by
Hedenmalm, see for instance [He].
Since functions with the same modulus are cyclic or not simultane-
ously — because if f is cyclic and |f (z)| ≥ |f (z)|, for any z ∈ D2,
1 2 1
then f is cyclic — it is natural to look for conditions on the size of
2
f that would be necessary or sufficient for cyclicity. In general, |f|
cannot be allowed to vanish, nor decrease too fast near the boundary,
in order for f to be cyclic.
One type of necessary condition can be obtained by restricting func-
tions to the diagonal. Rudin [Ru, p. 53] noticed that the map f 7→
Rf(ζ) := f(ζ,ζ) is bounded and onto from H2(D2) to the Bergman
space A2(D) := Hol(D) ∩ L2(D) (this is a quite general phenomenon,
see [HoOb] for instance). Thus if f is cyclic in H2(D2), then Rf is
cyclic in A2(D), that is to say, C[Z]Rf is dense in A2(D).
Borichev [Bo] proved that if f ∈ A2(D) and |f| decreases slowly
enough, then f is cyclic in A2(D). More precisely, we call weight
function a non increasing function v from (0,1] to (0,∞) such that
limt→0v(t) = ∞ and v(t2) ≤ Cv(t) for some C > 0. The following is a
special case of [Bo, Theorem 2]:
Theorem 1.1. If |f(ζ)| ≥ exp(−v(1 − |ζ|)) for a weight function v
such that
v(t)2
dt < ∞,
Z t(lnt)2
0
then f is cyclic in A2(D).
It has been known for a long time that no sufficient condition of
this type can hold for H2(D); indeed, no non-trivial singular inner
function may be cyclic, but one can construct them with arbitrarily
slow decrease. On the other hand, in view of the sufficient condition in
the Bergman space case, one may wonder whether a similar sufficient
condition could hold for H2(D2). The purpose of this note is to show
that it is not the case, as the example below shows.
We thank A. A. Aleksandrov for stimulating conversations on this
topic.
2. The example
For(z ,z ) ∈ D2, let δ(z) := max(1−|z |,1−|z |) (thisis comparable
1 2 1 2
to the distance to the distinguished boundary of the bidisk (∂D)2).
NON CYCLIC IN THE BIDISC 3
Proposition 2.1. For any weight function v, there exists f ∈ H2(D2)
such that log|f(z)| ≥ −v(δ(z)), but f is not cyclic for H2(D2).
Proof. Let µ be a singular positive measure on ∂D such that µ (I) ≤
v v
c |I|v(|I|), for c small enough, then the singular function
0 0
2π eiθ +ζ
f (ζ) := exp − dµ (θ)
0 (cid:18) Z eiθ −ζ v (cid:19)
0
is not cyclic for H2(D) and verifies log|f (z)| ≥ −v(1−|z|).
0
Now set f(z) := f (z z ). Since
0 1 2
max(1−|z |,1−|z |) ≤ 1−|z z | ≤ 1−|z |+1−|z |,
1 2 1 2 1 2
anappropriatemodificationoftheconstantc yieldslog|f(z)| ≥ −v(δ(z)).
0
If f (ζ) = α ζk, then f(z) = α zkzk = a zjzk, with
0 k k k k 1 2 j,k j,k 1 2
|a |2 =P |α |2 < ∞, so f ∈PH2(D2) (or inPthis case we just
j,k j,k k k
Pmight notice wPe have bounded functions).
In what follows, we freely use the fact that any function in H2(D)
or H2(D2) admits almost everywhere defined non-tangential boundary
values, and that its norm is equal to the L2 norm on the circle (resp.
torus).
Suppose there exists a sequence of polynomials P (z ,z ) such that
n 1 2
2π 2π
lim P (eiθ1,eiθ2)f(eiθ1,eiθ2)−1 2 dθ dθ = 0.
n 1 2
n Z Z
0 0 (cid:12) (cid:12)
(cid:12) (cid:12)
Using the change of variables θ = θ, θ = θ+α, we get
1 2
2π 2π
lim P (eiθ,ei(θ+α))f (ei(2θ+α))−1 2 dθ dα = 0,
n 0
n Z (cid:18)Z (cid:19)
0 0 (cid:12) (cid:12)
(cid:12) (cid:12)
which implies that there exists an increasing sequence n of integers
k
such that for almost any α,
2π
lim P (eiθ,ei(θ+α))f (ei(2θ+α))−1 2 dθ = 0.
k Z nk 0
0 (cid:12) (cid:12)
(cid:12) (cid:12)
For such an α, let g (ζ) := f (eiαζ2). This is again a singular function,
α 0
but we have a sequence of polynomials q (ζ) := P (ζ,eiαζ) such that
k nk
kqkgα −1kH2(D) → 0, which is a contradiction.
(cid:3)
References
[Bo] Alexander Borichev,Estimates from below and cyclicity in Bergman-Type
Spaces, Internat. Math. Res. Notices, 1996, no. 12, 603–611.
[Ga] John B. Garnett, Bounded analytic functions, Academic Press, New York,
1981.
4 XAVIERMASSANEDAAND PASCAL J. THOMAS
[He] H˚akan Hedenmalm, Outer functions in function algebras of the bidisc,
Trans. Amer. Math. Soc. 306 (1988), no. 2, 697–714.
[HoOb] Charles Horowitz, Daniel M. Oberlin, Restriction of Hp functions to the
diagonal of Un, Indiana Univ. Math. J. 24 (1974/75),767–772.
[Ko] Paul Koosis, Introduction to H spaces, London Math. Soc. Lecture Notes
p
40, 1980.
[Ru] Walter Rudin, Function theory in polydiscs, Benjamin, New York, 1969.
Facultat de Matema`tiques, Universitat de Barcelona, Catalonia
E-mail address: [email protected]
Universit´edeToulouse,UPS,INSA,UT1,UTM,InstitutdeMath´ematiques
de Toulouse, F-31062 Toulouse, France
E-mail address: [email protected]