THE CLOSURE PROBLEM FDR THE
BACKWARD SHIFT OPERATOR IN THE.
HARDY p-CLASSESI 5: R “RR,

Thesis for the. Degree Of Ph‘ D.
MICHIGAN STATE UNWERSITY
HAROLD ARTHUR ALLEN
1970

   
 

I‘HESIS

L I B 24’ 14 R Y
M;;1310 .. State
University

 

This is to certlfg that the

thesis entitled

THE CLOSURE PROBLEM FOR THE BACKWARD SHIFT
OPERATOR IN THE HARDY p-CLASSES 1 g p < co.

presented by

Harold Arthur Allen

has been accepted towards fulfillment
of the requirements for

Ph- 13- degree inmflics

Major professor?

Date 9-25-70

0-169

ABS TRACT

THE CLOSURE PROBLEM FOR THE BACKWARD SHIFT
OPERATOR IN THE HARDY p-CLASSES 1 s p < m.

BY

Harold Arthur Allen

The Hardy p-class of the disc is the set of all func-

tions f(z) holomorphic on the open unit disc for which

2R -
sup I |f(rele)|pde < a.
OSr<l O

For p fixed, 1 S p < a, the class HP becomes a separ-
able Banach space under the norm
2

1
”f” = SUP ——¢
Hp Osr<11277£3

7r - p 1/P
|f(re19)I d9] .

The left shift operator U* on HP is defined by

= f(z) - f(O)

*
(U f) (Z) z

, for [z] < 1.

A function f(z) 6 HP is said to be cyclic for U* if and
only if the linear manifold spanned by [U*nf}an is dense
in Hp. A function f(z) is said to be non-dSZlic for U*
iff f is not cyclic for U*. The closure problem1 for U*

is to characterize the cyclic or non-cyclic vectors for U*.

 

 

Harold Arthur Allen

A function f(z) holomorphic on the open unit disc is
said to have a pseudocontinuation across {2: I2] = 1} if
and only if f(z) has non-tangential limit f(eie) for al-
most all e E [0,2w], there existszafunction f(z) meromor-
phic on the complement of the closed unit disc With boundary
values f(eie) for almost all 9 6 [0,2W], and with
f(eie) = f(eie) almost everywhere.

Douglas, Shapiro and Shields2 were able to prove the
following theorem:

f(z) E H2 is non-cyclic for U* if and only if
(1) f(z) has a pseudocontinuation f(z)
and

(2) f(z) is of bounded Nevanlinna characteristic.

We have extended this theorem to the classes Hp, l:sp<:m.

 

lBeurling, A. On Two Problems Concerning Linear Trans-
formations in Hilbert Space. Acta Math., 81(1949), 239-255.

2Doug1as, R. G., Shapiro, H. S. and Shields, A. L.
Cyclic Vectors and Invariant Subspaces for the Backward Shift
Operator. Ann. Inst. Fourier, Grenoble, 20, 1(1970), 37-76.

 

THE CLOSURE PROBLEM FOR THE BACKWARD SHIFT
OPERATOR IN THE HARDY p-CLASSES 1 § p < m.

BY

Harold Arthur Allen

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Mathematics

1970

GO 7/061

To my wife

ii

ACKNOWLEDGMENTS

The author wishes to express his deep appreciation to
Dr. Gerald D. Taylor for his assistance and encouragement
during the preparation of this thesis. The author also
wishes to thank Dr. Allen Shields for his helpful comments
and suggestions.

This work was supported in part by NSF Grant GZl622.

iii

TABLE OF CONTENTS

CHAPTER I Introduction
CHAPTER II Classical Hp Spaces.
1. Basic Definitions and Theorems.
2. The J Operator. . . . . . . . . . . . . . .
3. Conjugate Harmonic Functions. . . . . . . .
4. The Shift Operators . . . . . . . . . .
5. Pseudocontinuations . . . . . . . . . . . .
6. Continuous Linear Functionals on HP. . .
CHAPTER III The Closure Problem for U*. . . . . .

1. Characterization Theorem. . . . . . . .

2. Applications of the Characterization Theorem.

BIBLIOGRAPHY

iv

.24
.28
.34
.38
.41
.46
.46
.65

.69

CHAPTER I

Introduction

We define the Hardy class H2(D) to be the class of
all functions f(z) holomorphic on D = [z:|z| < 1] such
that

2v .
sup £3 |f(rele)|2d9 < m.
Osr<l

It is well known [19] that f belongs to H2(D) if
and only if the sequence of Taylor coefficients of f be-
long to £2, the class of all square summable sequences of
complex numbers.

In his 1949 paper, "On Two Problems Concerning Linear
Transformations in Hilbert Space" Beurling [l] was concerned
with the shift operators on the Hilbert space H2(D). The
shift operators are u the forward (or right) shift and u*
the backward (or left) shift operator. The action of the

shift operators u and u* on functions in H2(D) is

given by
(1) u(f)(z) = zf(z). for f(z) 6 112(0).

and

(2) u*(f) (z) = flag—£191, for f(z) 6 112(1)).

The names forward and backward (or right and left)

shift come from the action of u and u* respectively

1

on functions in H2(D) represented by their Taylor series

about the origin, that is,

2 _ 2
u(a0+alz+a22 + ) — O+aoz+alz +a

3
22 +0.0,

and
2

* _ 2...
u (a0+alz+a22 + ) — al+azz+a3z + ,

or equivalently by the action of u and u* on the sequen-

ces of Taylor coefficients considered as elements of £2.

The problem with which.we are concerned was called the
closure problem by Beurling [1]. To describe the closure
problem, we first define a function f(z) E H2(D) to be
cyclic for a continuous linear operator A on H2(D) if
and only if the linear manifold spanned by {Anf;;£,is dense
in H2(D). A function f(z) 6 H2(D) is said to be non-
cyclic for an operator A on H2(D) if and only if f(z)
is not cyclic for A. The closure problem for an operator
A on H2(D) is to characterize the cyclic (or non—cyclic)
functions for A.

Beurling [l] was able to obtain an elegant characteri-
zation of the cyclic functions for the right shift u as
well as a description of all of the closed invariant sub-
spaces for u, namely,

(3) a closed subspace S of H2(D) is invariant under u

if and only if there exists an inner function m(z)

with S = wH2(D),

and

(4) a function f in H2(D) is cyclic for u if and only
if f is an outer function,
where an inner function m is a function holomorphic on the
unit disc with boundary values |m(eiefl = 1 almost every-
where on the circle [2: [Z]: 1], while an outer function
is a function F(z) holomorphic on the unit disc, of the
form .
F(z) = A exp [:%-£:W ggéig k(t)dt],
e -z
where k(t) is real valued and integrable on [0,2w], and
[XI = 1.

Since on the Hilbert space H2(D), u* is the adjoint
operator of u, Beurling's characterization of the invari-
ant subspaces and cyclic functions for u also gives a
characterization of the invariant subspaces for u* [3],
namely, the invariant subspaces for u* are the orthogonal
complements of the subspaces wH2(D), with m an inner
function, while f is non-cyclic for u* if and only if f
lies in a subspace of the form [mH2(D)}J- for some non-
constant inner function m.

Douglas, Shapiro and Shields [3] were able to give a
different characterization of the non-cyclic functions which
permits one to more readily identify classes of cyclic and
non-cyclic functions for the left shift on H2(D).

One can speak of the shift operators in spaces other

than H2(D). Indeed, the literature on these operators is

extensive, see for example, Wells and Kellog [25], Helson

and Lowdenslager [ll], Gamelin [8] and [9], deLeeuw and

Rudin [2], Hoffman [13], Hasumi and Srinivasan [10], and
Srinivasan and Wong [20].

The spaces with which we are concerned are the Hardy p-
classes of the disc. (We will formally state all defini-
tions and theorems in Chapters II and III.) On the Hardy
p-classes we can define the shift operators u and u* as
in (l) and (2) by merely replacing the condition f(z) 6 H2(D)
With f(z) 6 Hp. The operators so defined are continuous
linear operators on Hp(D) (p is considered to be fixed).

De Leeuw and Rudin [2] have shown that Beurling's
characterization of the cyclic vectors for u is the same
in H1(D) as in H2(D), namely, a function f(z) E H1(D)
is cyclic for the right shift u (either the norm topology
or the weak* topology on H1(D)) if and only if f(z) is an
outer function.

We will extend the theorem of Douglas, Shapiro and
Shields [3] characterizing the non-cyclic functions for the
left shift u* to the Hardy p-classes Hp(D), l s p < m.

In Chapter II we will present some of the well known
results on the structure of the Hardy p-classes. We will
state these theorems in forms which we can readily use in
the proof of the characterization theorem.

In Chapter III we prove the main theorem characterizing
the non-cyclic functions for the left shift and give a brief
discussion of applications of this theorem to certain classes

of functions.

CHAPTER II

Classical Hp Spaces
1. Basic Definitions and Theorems

The purpose of this chapter is to present some back-
ground material on the Hardy p-classes and related topics
for use in the subsequent chapters. We begin with some no-

tations and definitions.

Definition 2.1:
The following notation will be used throughout this
thesis:
C = set of all complex numbers
$=CU{w}
D = [2 E C: [2| < 1]

T=[z€C:|z| 1}

De: {2 e c: [z] > 1] U [on].

Let u(z) be a real valued continuous function on D,
possessing continuous second order partial derivatives with
respect to x and )r where z = x + iy, x and y real
numbers. u(z) is said to be harmonic on D if and only
if u(z) satisfies Laplace's equation on D,

uxx(z) + uyy(z) = O

5

in Cartesian coordinates, or

19 1 ie A ie
urr(re )+-r‘ur(re )+r2 ueg(re )

= O
in polar coordinates.
A complex valued function u(z) continuous on D is

said to be harmonic on D if and only if both the real and

imaginary parts of u(z) are harmonic on D.

For 0 < p s a and for u(rele) harmonic on D, we
define the Lp means of u on |z| = r by
2O ie p l/p

'5; £) [u(re )I def 1f 0 < p < m
m r,u =
p( ) .

sup [u(rele)| if p = m
O$9§2W

If f(z) is holomorphic, and hence harmonic, on D‘we
adopt the notation

Mp(r, f) = mp(r, f).

For 0 < p S m, we define the Hardy p-classes:

hp(D) = [u(z): u(z) is harmonic on D and sup m (r,u)<:m]
O$r<l
Hp(D) = [f(z): f(z) is holomorphic on D and sup M (r,f)<:m]

Osr<l
Hp(De)= [f(z): f(z) is holomorphic on De and if g(z) =

f(%) for [2] < 1, then 9(2) 6 Hp(D)]

where we have adopted the convention that (% = m.
We define the Nevanlinna classes:
2W
N(D) = [f(z):f(z) is meromorphic on D and sup [ 1n+
Osr<l O

[f(reig)|d9 < a}

N(De) = [f(z): f(z) is meromorphic on De and if g(z) = f(fi),
then 9(2) 6 N(D)}.
where ln+lt| = max [0,lnltl}.

We will also adopt the notations hp = hp(D), Hp = HP(D)
and N = N(D).

We point out that it follows from the definitions that
if p < q, then hq c hp and Hq c Hp. We also note that
H1 c hl.

For 0 < p < m the inequality ln|t| < Itlp immedi-

ately shows that Hp C N’ while f(z) E HOD implies that

lf(z)| s sup Ma(r,f) < m for each 2 E D and hence
. Osr<1

2O -
:Lf 1Jfl-|f(re19)|de g sup M (r,f) < m. Thus H? c N for
7To Osr<l "

O < p

M

m.

We will be working with functions defined on the bound-
ary of the unit disc and thus find it convenient to denote
such functions in the form f(eie) for O s e g 2W rather
than a notation of the form F(e) for O S 9 § 2w. We shall
always assume that functions of one variable 9, O s 9 s 2w
are periodic of period 2V and when we use the terms mea-
sure or almost everywhere we shall mean one dimensional
Lebesgue measure.

For f(ele) a measurable function of e, O S 6 § 2O

we define the Lp "norm" of f(elg) by

ess sup [f(ele)| if p = a

“f(e )H =
Lp 1 2n ie p 1/P

3H0 ”‘8 " “1 if °<P<°°‘

We say that f(ele) e Lp[O,2w] if and only if ”f(e19)n p< a.
L

For f(rele) 6 HP, we define the HP "norm" of f by

Hf” = sup M (r,f).
Hp Osr<l p

For u(rele) e hp, we define the hp "norm" of u by

= sup m (r,u).
Osr<l

HuH p
h
We note that for l s p g m, the HP, Lp and hp "norms"
are norms in the usual sense [6] and that the spaces HP,
Lp and hp are Banach spaces [6]. For 0 < p < l, the
Hp,Lp and hp "norms" are not norms since the triangle
inequality fails. We do know however, that for O < p < 1,
these spaces form translation invariant, complete metric
spaces [6] or Frechet spaces [4] under the metrics
P P P
p(f.g) = Hf-gH . p(f.g) = Hf-gH and o(f.g) = Hf-gH
P P P
H L h
respectively.
We observe that if f(rele) E Hp with r fixed, O<:r<<l,

then fr(ele) E f(rele) is a function of e, f (e19) E Lp[O,2w]

r
d Hf e19 = M , .
an r( )HLP p(r f)
We have defined hP as a class of complex valued func-

tions, while some authors restrict the class hp to be only

real valued functions.

 

Definition 2.2:
Let f(z) be defined and single valued on D. we say
that f(z) has non-tangential boundary value f(ele) at

e16 e T if and only if for any fixed a. O < a < w,

lim f(zn) = f(eig)

11-000
for any sequence [2 }on : S(a,e19) with lim 2 = e19
19 r1n=0 “*”
where S(a,e ) is the domain common to the unit disc and

the sector with vertex at e19, of angle a. symmetric

with respect to the radius from the origin to eie.

For f(z) defined and single valued on De' we say that
f(z) has non-tangential boundary value f(eie) at e19 E T
if and only if g(z) = f(hz) has non-tangential boundary
value f(eie) at 519.

We first state the Theorem of Lusin and Priwalow.

Theorem 2.1:

[14, p. 212] Let fl(z) and f2(z) be meromorphic on
D (or De) and both possess equal non-tangential boundary
values on the same set E c T with m(E) > 0. Then f1(z) =
f2(z) for each 2 E D (or De)'

Motivated by this theorem we identify any two boundary
functions which are equal for almost all ei9 G T.

It is well known that every function in HP, p > O
and every function in hq, q z I possess unique non-

tangential boundary values for almost all e16 E T [14],

 

 

10

and that given the boundary function we can reconstruct the
original function. For completeness we shall develop these
facts here. For the most part we shall follow the develop-
ment of Duren [5], Hoffman [13] and Priwalow [14].

The most natural place to begin this task seems to be
with harmonic functions in classes hg, q a l and then to
extend these results to the HP classes, even for O < p < 1.

We begin with a very important harmonic function, the

Poisson kernel, P(r,e).

Definition 2.3:
2

4L l-r
P(r,9) — 2%\l-2r cose + r2_)' 0 s r < 1, O s A 5 2R.

We list some of the more important and well known pro-

 

perties of the Poisson kernel. See, for example, Hille [12]

or Hoffman [13].

Q

(i) P0139) = Z r

n="w

Inleine' for O s r < l and

O s 9 S 2O with convergence of the series abso-
lute on D, and uniform on compact subsets of D:
(ii) P(r,e) > O, for O s r < l and any 9 E [0,2w]:

2W
(iii) f P(r,e)de = 1, for any r e [0,1];
0

(iv) For any real number a, O<€ < O, limsup [P(r,e)| = O.
r4I'(fl>e

Perhaps the most important property of the Poisson ker-
nel is furnished by the following theorem.which says, among

other things, that the Poisson kernel can be used to produce

11

a harmonic function on the disc, given an integrable function

defined on the unit circle.

Definition 2.4:
If f(ele) is a LebeSgue integrable function of e on
[0,2w] we define the Poisson integral of f to be
2w .
I f(elt) P(r,q-t)dt.
0

If u(t) is a complex Baire measure on [0,2w], the Poisson
integral of u is
2w

] P(r.e-t)du(t).
0

Theorem 2.2:
[13, pp. 33-34] Let f be a complex-valued harmonic

function on the open unit disc, and write

fr(eiq) = f(reig),

then
(i) If 1 < p é m, then f is the Poisson integral
of an Lp function on T if and only if
f(reie) 6 n9.
(ii) f is the Poisson integral of an integrable func-
tion m on T if and only if the functions fr

converge to m, as r41", in the L1 norm.

12

(iii) f is the Poisson integral of a continuous func-
tion on the unit circle if and only if the func-
tions fr converge uniformly.

(iv) f is the Poisson integral of a finite complex
Baire measure on the unit circle if and only if
f e hl.

(v) f is the Poisson integral of a finite positive

Baire measure if and only if f is non-negative.

Theorem 2.3:
(Fatou's Theorem) [13, p. 34] Let u be a finite com-
plex Baire measure on the unit circle and let f be the

harmonic function on D defined by

i9 2W
f(re ) = f P(r,e-t)du(t).
0
Let 90 be any point where u is differentiable with re-

spect to Lebesgue measure. Then

lim f(releo) = u’(O

).
r41' 0

18

and in fact lim f(re 19

) = u’(eo) as the point z = re
approaches eieo along any path in the open unit disc
which is not tangent to the unit circle.

There is much contained in these theorems. First of
all we wish to point out that Theorem 2.2 (iv) says that any

function u(z) E h1 can be represented as the Poisson inte-

gral of a finite Baire measure u on the unit circle T.

13

Theorem 2.3 says that u(z) = u(rele) 4 u’(eo) as
z = re16 4 e190 non-tangentially for almost all eo€E[O,2O].
When no confusion can result, we will denote the boundary

1 by the function u(ele) where it

6

values of u(rele) E h

is to be understood that u(e1 ) is unique up to sets of

Lebesgue measure zero on [0,2w].

We also note that since hp c h1 for p a 1, every

function u(z) 6 hp, p g 1 has non-tangential boundary

values almost everywhere on T. The same is true for H1

as H1 c h1 and thus for Hp, p g 1 since HP c H1 when-

ever p

(IV

1. Also it is a corollary to the proof of Theorems
2.2 and 2.3 that for 1 < p s a if f(z) e hp with boundary
function f(eie), then f(z) is the Poisson integral of
f(eie) [13]. This is not true in hl, but it is still true
for H1. [26].

We shall need some results for the classes HP, 0 < p < 1
which are not true for the corresponding hp classes, but
one of the classical methods of deriving these results is to

obtain them from the results true for Hp or hp, 1 < p g a.

To this end we state the Nevanlinna Theorem.

Theorem 2.4:

[5], [15] A function f(z) holomorphic on D is in
class N if and only if f(z) is the quotient of two func-
tions from H”. (We do not allow the function in the denomi-

nator of the quotient to be the zero function.)

14

Since every function in HQ has non-tangential boundary
values almost everywhere on T, Theorem 2.4 suggests the same
may be true for all of class N. However, the possibility
that the numerator and denominator functions from Theorem 2.4
could both have boundary value zero creates a problem. For-
tunately the class N is a class of "well behaved" functions

as the next theorem shows. First we state a Lemma.

Lemma 2.5:

 

(Jensen's Formula) [12, II, p. 189] If m(z) is holo-
2W -
morphic on D, then I lnlcp(re19)|de increases monotoni-
O
cally with r, O s r < l and

1 2R i6
PEI 1“ [Wm H59 = 1n Ico(0)| + Z(co.r).
0

where Z(m,r) = 2 1n ——', with [rn]0°

n=O the sequence of

modulae of the zeros of m repeated according to multi-
plicity.

Now we can proceed.

Theorem 2.6:

[5], [14] If f(z) E N, then f(z) has non-tangential
limit f(eie) almost everywhere. Furthermore, ln|f(eie)|
is integrable unless f(z) E O and f(z) 6 Hp for any

p > 0 implies that f(ele) 6 LP.

15

Proof:

Because of the importance of this theorem we include
the proof as found in Duren [5].

Let f(z) E N be given. Assume f(z) # 0. By
Theorem 2.4 there exist two functions u(z), 6(2) 6 HCO such
that f(z) =-QLEL , for each 2 E D.

6(2)
Without loss of generality we assume that Ha” a g l

and that ”B” m s 1 since if Ha” w > 1 or ”6“ a > 1 we
H H H

could replace u(z) and 6(2) respectively by

 

a'(Z) =

max {lla1]:.H,.HBH

and

 

6"“ max (Hall: ”min,

which are well defined since ”6” w+ O and which satisfy
H

HA

(i) “01’” a 1 and 115’” w—
H

and

.. _ g’(z)
(11) f(z) — B,(z) .

Now by Theorem 2.3 u(z) and 6(2) have non-tangential

limits u(ele) and 6(ele) respectively almost everywhere.

Noting that -Jxa|a(re19)| O and that

”V

-ln|a(eie)| a.e.

g -[-1n|a<reie)|]

16

we apply Fatou's lemma [15] obtaining
27F i6 . 2'” 19
f0 -ln|a(e )|de s 1im_ 1nf (- I 1n|a(re )ldef°
r41 0

2w
By Lemma 2.5 (Jensen's Theorem [ ln|a(rei e)Ide

2R
increases with r and hence - I ln|a(re i9)Ide is a non-
0

negative decreasing function of r. Thus

2W 2W
lim inf f - 1n|a(rei e)lde = lim ] - 1n|a(rei e)|de
r41” 0 r41 0
is finite. Hence

2 .
f Tr|1n|a(e19)||de < a,
O
that is,

ln|a(eie)l e L1[O,2n].

Similarly ln[6(ele)| E L1[O,2w]. In particular,
1g) nor 6(e19) can vanish on a set of positive

ie)I 6 L1 and ln|6(eie)l 6 L1. Thus

neither u(e
measure since ln|a(e
the non-tangential limit f(ele) exists almost everywhere,

and since ln|f(ei9 = 1n|a(eie)| - 1n|6(eie)| a.e. we

)I
have 1n|f(eie)| 6 L1.

Thus,in particular if 0 < p s m and if f 6 HP,

lim f(rele) = f(ele) exists a.e. and Fatou's lemma [15]
ral-
gives
2" 19 P 19 P
[ [f(e )I de é lim inf f: TT|f(re )l dei< a
O r41-

since f 6 HP,

17

Now there are times when zeros of an Hp function
cause difficulties in proofs, so we state some theorems

which enable us to in some sense factor out zeros.

Theorem 2.7:

[5] If f(z) # O is holomorphic on D with zeros

al,a2,--- repeated according to multiplicity, then
2W 16 a
sup I 1nlf(re )Ide < a if and only if 2 (l-Ianl) < a,
Osr<l 0 n=1
Proof:

The proof of this theorem follows from Jensen's Theorem

(Lemma 2.4).

Corollary 2.8:
Let f(z) E N, f(z) ¢ 0, then if [an] are the zeros

of f, 2(l-Ianl) < a.

Theorem 2.9:

[13] If [a I” is a sequence of non-zero complex

 

n n=l
on
numbers with no limit point in D and if 2 (l-Ian[) < w.
n=l
then the product
on [an] an-z
3‘2) = H a 1-5 2'
n=1 n n

converges uniformly in each disc |z| E R < 1. Each an is

a zero of B(z) with multiplicity equal to the number of

18

times it occurs in the sequence [an]. B(z) has no other
zeros in D. |B(z)| < l for each 2 6 D and |B(ele)|= 1
a.e.

Definition 2.5:
A function of the form

[an] an-z

 

elyzk H I

a l-a z
n n

where Y is a fixed real number, k is a non-negative inte-
ger and [an] is a sequence (finite or infinite) of complex
numbers in D satisfying 2 (l-Ianl) < w, is called a

Blaschke product.

Theorem 2.10:

[5] Any function f(z) 6 HP, 0 < p s m can be factored
in the form f(z) = B(z)g(z) where B(z) is a Blaschke pro-
duct and g(z) is an Hp function which has no zeros in

D.

Theorem 2.11:

 

[16] If f(z) 6 HP, 0 < p < a, then

2R ' -
lim_ I [f(rele) - f(ele)|pde = O,
r41 0

and /
. 2W - 1 p
1im M (r,f) = §%-f [f(ele)lpde] .
rel” p O

19

or
Hf (e19) - f(ele)H 4 O, as r41-,
r P
L
where
fr(ele) — f(rele).
and
“f(z)“ p = Hf<eie>n ,-
H L
We comment that Theorem 2.11 is a stronger result for
Hp than Theorem 2.2 (i). For 1 < p < m the conclusions

are the same. For 0 < p < 1, Theorem 2.11 tells us that

fr(e19) = f(rele) E Hp converges in Lp norm to the bound-
ary function f(ele), while for O < p < m an hp func-
tion need not have non-tangential boundary values. We now

wish to characterize the class of all boundary functions of
Hp.

Definition 2.6:

Let ©p(D) = 6p denote the set of all boundary functions
f(eie) of functions f(z) 6 HP, 0 < p g m.
We note that by Theorem 2.6 hp c LP, 0 < p E a.
Since the HP classes are linear spaces, it is clear that

Op is a linear manifold in LP. Also hp must contain all

polynomials in elne, n a O, that is, functions of the
N .
1k . .
form k2 ake 9 since each Hp class contains the polynomials
=0
§ k

az
k=0 k

20

Lemma 2.12:
[5] If f(z) 6 HP, 0 < p < m, then

P

1 1/
lf<z)| é HfHHp him)

Now a characterization of fip'will be given.

Theorem 2.13:

 

[5] Let p be fixed, 0 < p < m. 6p is the Lp

. . '9
closure of the set of polynomials in el’.

Proof:

We show first that 6p is closed in Lp. Let

0 o .e
[fn(e1°)] c 9p. Assume an(e1q) - cp(e1 )H p 4 O as n 4 m
L
for some m 6 LP. We must show m 6 3p.
i9 < P i9 in p.
Now an(e )H p - 2 an(e ) — m(e )H p + 2 LT” p
L L L
and hence there exists a constant M such that
19 < _ ...
an(e )HLp : M for n — 0,1,2,
Now fix R, O < R < 1. By lemma 2.12, if [2] < R,
1 1/p
lfn(z)| 2"; M kl-Rj
where fn(ele) is the boundary function of fn(z). Thus

the sequence [fn(z)] is uniformly bounded on [2: |z|:sR<<l].
Consequently {fn(z)] forms a normal family of functions

[12, II, p. 242]. Hence there exists a subsequence [f (2)]

“k

21

which converges uniformly to a holomorphic function f(z) on
compact subsets of D. By Fatou's lemma [15] f(z) 6 HP.
9)

It remains to be shown that cp(e1 is the boundary function

of f(z).
Let s > 0 be given. Choose N so that if m,n, g N,

an(z) - fm(2)H < e. Let r 6 (0,1) be fixed and m a N

be fixed.
2R . . p 2W . .
[ |f(rele) - f (rele)| d9 = lim I If (rele) - f (rele)[pde
. m n m
0 k4m (D k

HA

r].

since fn (z) converges uniformly to f(z) on [2:Izl
k
Now

27T- ' I p 27” O D P
9 -—7-
lim I Ifn (rel ) - fm(re19)| d9 s 11m [ lfk(rele) - fm(re19)|de.
k-oco O k 4m 0

But since m a N,

 

2R - .
lim [ lfk(re19) - f (rele)|pde g ep.2w
km» '0 m
Hence
2W . . p
[ [f(rele) - f [(re19)| d9 < ep.2n
.O m

Now by Fatou's lemma [15], letting r41- we obtain
2w . .

I If<e19) — f (e19)lpae s ep- 2w

0 m

or since 6 > 0 was arbitrary,

. '9 ie
11m ||f(el ) - f (e )H 4 0
III-Om m Lp

22

Hence

f(eie) = m(eie) a.e.

Thus 6p is closed in LP.
It remains to be shown that the polynomials are dense

in 6p. Let f(z) 6 HP and e > 0 be given. Choose an R,

O < R < 1 so that ”f(Reie) - f(eie)H p < 3&4. We can pick

such an R by Theorem 2.11. Let Sn(:) denote the nEh

partial sum of the Taylor series of f about the origin.

0n the compact set [2: [z] s R < l], sn(z) converges uni-

formly to f(z) so pick N such that n a N implies that

sup [f(z) - sn(z)| < 9%-.

[ZISR
Define p(eie), a polynomial in ei6 by p(eie) = sn(Rele).
Now
||p(e19) - f(elg)” p s Mme”) - f(Re19)+ f(Rele) - f(ele)“ p
L L
< 19 _ i9 i9 _ i6
-_2[Hp(e ) f(Re )HLp + ”f(Re ) f(e )HLpJ

<2-e/4+2-€/4

Since a > 0 was arbitrary we have shown that the polynom-
ials in e19 are dense in SP.
We comment that Theorems 2.11 and 2.13 enable us to

define a linear isometry between Hp and 6p, namely, the

23

correspondence between a boundary function in 6p and the

holomorphic function in Hp, 0 < p < m.

2. The J Operator

In Chapter III we will need to know something about the
relationship between functions holomorphic inside the unit
disc and functions holomorphic on the complement (with
respect to the Riemann Sphere) of the closed unit disc. The
method which we will employ was chosen primarily to simplify

notation.

‘Qefinition 2.7:

Let f(z) be holomorphic on D. We define

 

HA
8
§

(Jf)(2) = f(l/Z) for 1 < |z|
1

where we adopt the convention that -; = 0.

If f(z) is holomorphic on De' we define
(J-lf)(z) = f(g) for 0 g |z| < l,

with the convention '% = a.

We point out that if f(z) is holomorphic in D with
on
Taylor series about the origin f(z) = Z anzn, then
n=0
- -n
a z .

on

(Jf) (2) =
n

ll M8

If g(z) is holomorphic on De with Laurent series g(z) =

m -n
2 anz , then

n 0

(J‘lg><z) = 2 5 2
We also observe that if f(z) is holomorphic on D and if

24

25

f(z) has non-tangential boundary values f(ele) at
e19 E T, then (Jf)(z) has non-tangential boundary value

f(eig) = (Jf)(eie) at ej‘9 E T.

Definition 2.8:

Let p be fixed, 0 < p s m. We define the classes

Hp(De) [Jf: f e Hp(D)]

N(De) {sz f e N(D)].

We can now prove results for Hp(De) corresponding to most

of the theorems about Hp(D).

Theorem 2.3’:
A function f(z) holomorphic on D8 is in class N(De)
if and only if f(z) is the quotient of two functions from

Hw(De).

Page;
Let f(z) e N(De), then there exists a function

h(z) G N(D) such that (Jh)(z) = f(z) for each 2 6 De'

By Theorem 2.3, there exists two functions u(z), 6(2) 6 H”(D)

such that

11(2) = (1(2).

 

Thus

26

 

_I_
(Jh)<z) = h(l/Z) = 91-1121]
Bhfl)

_E112_)_I_1£o1_1a)_
_§-(_$)-- (JEHZ)

Theorem 2.6’:

If f(z) e N(De), then f(z) has non-tangential bound-
ary values f(eie) almost everywhere. Furthermore,
ln|f(eie)| is integrable unless f(z) = 0 and f(z) E Hp(De)

for any p > 0 implies that f(ele) 6 LP.

Proof:
Let f(z) E N(De). Let h(z) 6 N(D) be such that
(Jh)(z) = f(z) for any 2 with [2| > 1. Let h(ele)

be the boundary function of h(z). Then f(z) has non-

 

tangential boundary values f(ele) = h(ele) a.e. Also by
Theorem 2.5, ln|h(ele)| is integrable and hence
ln|h(elq)| = ln|f(ele)| is integrable unless h(z) E O,

that is, f(z) s 0. Furthermore, f(z) E Hp(De) implies
that h(z) E Hp(D). By Theorem 2.5, h(z) 6 HP(D) implies

that h(elq) = f(ele) 6 LP.

Definition 2.9:
Let p be fixed, 0 < p s a. We define the class
$p(De) to be the class of all boundary functions of func-

tions in class Hp(D ).
e

27

Theorem 2.12’:

 

Let p be fixed, 0 < p < m. 69(De) is the Lp clo-
sure of polynomials in éie. (Polynomials in e-19 means

I O I I O _i
linear combinations of non-negative integer powers of 1e 9).

Proof:

By Theorem 2.5’ we have bpwe) c Lp[0,2‘lT]. Since each

polynomial in powers of e“16 is in ©p(De), it remains to

be shown that ©p(De) is closed in Lp and that the poly-

nomials in e—19 are dense in OP(De). Now from the defini-

tion of the J operator and the discussion following that
definition, @p(De) contains exactly those functions which
are complex conjugates of functions in bp(D). By Theorem
2.12 ©p(D) is closed in Lp[0,2w] and hence hp(De) is

also closed in LP.

. ie . .
in e are polynomials in

Since complex conjugates of polynomials
e-le, by Theorem 2.12 we can

conclude that polynomials in e_19 are dense in ©p(De).

3. Conjugate Harmonic Functions

If u(z) is harmonic on D, we say that a function
v(z) harmonic on D is a harmonic conjugate of u(z) if
and only if u(z) + iv(z) is holomorphic on D. Any given
u(z) harmonic on D has many harmonic conjugates all differ-

ing by constants.

Definition 2.10:

If u(z) is harmonic on D we say that v(z) is the
normalized harmonic conjugate of u(z) if and only if v(z)
is harmonic on D, v(O) = O and u(z) + iv(z) is holomor-
phic on D.

The problem with which we are concerned is the following:
Given u(z) 6 hp can we claim that v(z), the normalized
harmonic conjugate of u(z), is in any hp class? The

question is answered in part by the following theorems.

Theorem 2.14;

 

[26, p. 253] (Theorem of Riesz) If u(z) is real val-
ued, u(z) 6 hp, 1 < p < m, then v(z), the normalized
harmonic conjugate of u(z) is in hp and there exists a
constant AP depending only on p such that Hv(z)||hp s
Apuu<z> uhp.

The Theorem of M. Riesz is false for p = l, a counter-
example being the Poisson kernel [13]. In the case p = 1

we do have the following theorem.
28

29

Theorem 2,15:

[5] or [26, p. 254] (Theorem of Kolomogrov)

If u(z) is real valued and u(z) E hl, then v(z)
the normalized harmonic conjugate of u(z) is in hp for
any p 6 (0,1) and there exists a constant Bp depending

only on p such that

Hv<z>uhp e spuu<z>uh1.

Harmonic functions are very closely related to Fourier
series. We observe that if f(ele) 6 L1[0,2R], then for

each integer n,

2W .
'1 19 -in9
271 I0 f(e ) d9 — Cn
exists, and Icnl 4 O as |n| 4 m; however, it is suffici-

ent for our purposes to know the cn's are well defined

and uniformly bounded.

Definition 2.11:

If f(elq) E L1[O,2R] we define the Fourier series of

(D

f to be the formal power series 2 cnelne, where
n=-m
2R . .
_ 1 19 -ine
cn =‘5; £3 f(e )e dB,

and we write

ie °
f(e ) ~ 2 c e
n

n=-m

ine

30

The complex number CD is called the Fourier coeffici-
A
ent of index n, and will be denoted by CD = f(n).
Now the connections between Fourier series and harmonic

functions with which we are concerned are the following:

Theorems 2.16:

 

If f(eie) E L1[O,2R] has the Fourier series 5 cneine,
then u(z) = u(reie) = E cnr|nleine is harmonic EH-QD.
n=—m
Theorem 2.17:
If u(eig) E L1[O,2R] has the Fourier series E cdeing,
then, n='°
. i9 2” it . .
(i) u(re ) = fl) P(r,9-t) u(e )dt 15 harmonic on D;
(ii) u(rei“) e hl;
(iii) u(reie) = E cnrlnleine, 0 s r s l, 6 E [0,2R];
n=-m

a
(iv) g(z) =n§0 cnzn is holomorphic on D:

(v) 9(2) 6 Hl/Z:

and

(vi) 9 § C u(eiq) , where C is a constant
u RBI/2 u 1L1

independent of u(eie).

Proof:

(i) and (ii) are restatements of Theorem 2.2 (ii).

Q .
Since P(r,9) = 2 rlnlelngo

we have by substitution

31

. 2w 0 . _ .
u(rele) = I 2 rln'e1n(9 t)u(elt)dt.
0 n='°°

Fix r, 0 S r < 1. Then, by uniform convergence,
w rInIeine

. U(e1t)dt = Z
O n=-m n=-a 0

2” -int it
e u

J..27T E rInleinM-t) (e )dt

m In] ine
= Z r e C

n="m

n.
Thus (iii) holds.

Since [on] S ”u(eie)H g(z) is holomorphic on D.

Ll,
(v) will follow from (vi)
To prove (vi) we treat the case where u(ele) is real

valued first, and then extend to the complex valued case.

Case I:
19) 9) is real

. . i
Assume u(e is real valued. Since u(e

valued cn =‘E_ Let v be the normalized harmonic conju-

gate of u(rele). Then 29(re19) = u(rele) + iv(re19

n'
) + co.
Now for O < p < 1, Theorem 2.15 implies that

sup __]__ 2TT ie p l/p sup J— 2'TT i9
33 .- n 33) 3...... .... We

is _ ie
= BpHu(re )th — BpHu(e )HLl.
Since
_.J- 2" 19
c0 — 2w £3 u(e )de,
we also have ICOI S ”u(e16)H 1. Thus, if we fix r, 0 g r < 1.

L

- -
“a
.

32

2W

2W ' - ' '
l i. P l . P
3; £) |g(re q)| d9 S EEC) [u(rele) + iv(rele) + col de
2w . 2'IT . p
g 31- |3<3319>IP39 + —1—- |v<re19)| 33
“-0 2W<D
P
+ IcOI .
s [LL.2qu(reig)Ide]p + B pHU(eie)H p .
— 2W0 p L1 I
. p ..
+ ”u(elGIH 1 3
3 I
P i9 P
= C .
where cp is a constant depending on p alone. Hence
2 I S l i9 .
II9( )IHp - cp|u(e )HLl
Case II:
i9 . i9
If u(e ) is complex valued, then set u1(e ) =
Re u(elg) and u (e16) = Im u(elg), and thus u(rele) =

2
i9 i9
u1(re ) + u2(re )-

Let g(z) = 91(2) + ig2(z) where 91(2) and g2(2)

are the holomorphic functions in (iv) corresponding to

u (e19) ie)

and u2(e respectively.

1

Now by case I, for any r, O s r < l ‘we have

P :1 2w i - P
—_ 9 - 16
S 2” fl) Igl(re ) + igz(re )I de

2‘”. ' 2n- 0
p
—1 f |g1(re1°)|pde + if Ig2 (re19)| d9
O O

M

2w 3 2R

33

HA

' p 19 P:
cppHul(ele)HL§ + Cp Hu2(e )HLl

2W -
277' 'q p l ]p
CPp 2: .0 Iu1(el )Ideip + Cp [zvnrO [112(e’9)|dg

HA

2W - p
P .;L 19 d9] ,
ch [-277- .O |u(e ) I

Since r was arbitrary, taking l/p powers and a supremum

we obtain

< l/P I ( i6) .
”9(Z)HHP = 2 CPI“ 6 ”L1

4. The Shift Operators

We are primarily concerned with the Hardy p-classes
for l S p < m. On these spaces we define the left and

right shifts as follows:

Definition 2.12:
Let p be fixed, 1 s p < m. Let f(z) 6 HP be given.
We define

(Uf)(z) = 2of(z), for z 6 D

and

(U*f)(z) = f(zl; f(0) , for z e D.

We call. U the forward or right shift on Hp and U*
the left or backward shift. As we noted in the introduction
the action of U on a function f(z) represented by its
Taylor series about the origin is to shift the Taylor coeffi-
cients forward or to the right, that is, if

” n
f(z) = 2 anz , for IzI < 1

then

(Uf)(z) = Z anz , for IzI < l.

The action of U* is similar, that is, if

f(z) = 2 anz“, for |z| < 1,

n=0
34

35

then

m
(U*f)(z) = Z a z , for IzI < 1.

Definition 2.13:

 

Let X and Y be two complex Banach spaces with norms
H’HX and "."Y respectively. A mapping T: X 4 Y is
called a linear operator or operator if and only if

T(ax + BY) = OT(X) + BT(Y)

for each X,y 6 X and each a.B complex numbers.

An operator T is said to be continuous if and only if
T is continuous in the norm topologies on X and Y. An
operator from X to Y is said to be bounded if and only
if there exists a real number M such that

sup ”Ht“ S M.
11311.31
It is well known [4, pp. 59-60] that a linear opera-

tor is bounded if and only if it is continuous.

Theorem 2.18:
Let p be fixed, 1 s p < m. The mappings U and U*
of definitions 2.12 are continuous linear operators from Hp

to HP.

36

Proof:
Linearity follows immediately from the definition of U

and U*. For continuity, let f(z) 6 HF be given. Then

HUfHHp Hz-f(z)HHp =

2n . . p
lim [.jL' Irelef(rele)I d9]
r41" w "0

2H - P
lim {-fL I [f(re19)| d9]
r41 0

 

= ”f” -
HP

Thus sup HUfH = l or “U” = l.

Hf S1 Hp

HP
Similarily,
f(Z) - f(O)
HU*fH = H H
Hp 2 HP

2n . . p 1/P
. 1 -1 ~16 19
= 11m [—— r e (f(re ) - f(0)) d9]
r41_ 27TJaO | I

2w . p l/p
. l 19
= lim '-— f(re ) - f(0) d9
r41_[27r‘r0 I | ]

= ”f(z) - f(O)
|al

HA

1131al + lf(0)|

2v .
1 is
+ |2Tr IO f(e )del

HfH
Hp

37

p'

g uanp + HfHHp = 2HfHH

Thus,

Hffiu:§lnu*fngz' or HU*H § 2.
H

Theorem 2.19:

Let p be fixed, 1 s p < a. Let f(z) 6 HP be given.

0
Let n be a positive integer. If f(z) = 2 akzk, then
k=0
(U*“f)(z) = [f(z) - pn<z)]z"“. o < |z| < 1.
n—l k
where pn(z) = Z akz .

k=O

5. Pseudocontinuations

From the theorem of Lusin and Priwalow, Theorem 2.1, we
know that if f(z) is meromorphic on D and if f(z) has
non-tangential boundary.values f(eie) on a set E C T
with m(E) > 0, then f(z) is uniquely determined by these
boundary values. Now if there is a function f(z) meromor-
phic on De' which also has non-tangential boundary values
‘f(eig) on the same set E c T with f(eie) = f(eie) then
in some sense f(z) and ‘f(z) uniquely determine one anot-
her. We formulate this in a more precise manner in the

following definition [21].

Definition 2.14:

If f(z) is meromorphic in D we say that f(z) is
pseudocontinuable across T onto De if and only if the
following hold:

(1) f(z) has non-tangential boundary values f(eie)

for almost all g e [0,2n],

(ii) there exists a function f(z) meromorphic on De’

(iii) ‘f(z) has non-tangential boundary values ‘f(eie)
for almost all 9 6 [0,2w], and
(iv) he”) = f(eie) a.e.
We remark that we have defined a pseudocontinuation

across T onto De and in this definition we require that

the pseudocontinuation be meromorphic on all of De' The

38

39

reason for requiring De to be the domain for a pseudocontin-
uation for our purposes will be made clear in Chapter III.
In general one could define a pseudocontinuation across a sub-
arc of T (with positive one-dimensional Lebesgue measure)
onto a subarc of De having the arc as part of its boundary.
See Shapiro [21] for a discussion of Pseudocontinuations.

We note that if f(z) is holomorphic on D and if
f(z) can be continued analytically across T onto De 'with
the continuation meromorphic on De' then the analytic
continuation is a pseudocontinuation across T.

A pseudocontinuation may exist even though the original
function is nowhere analytically continuable [3] or [21],

To see this we first consider inner functions.

Theorem 2.20:

 

[3] If f(z) is an inner function, then f(z) is

pseudocontinuable across T.

Proof:

Let f(z) be an inner function, that is, [f(ele)| = l

 

a.e. The function (Jf)(z) = f(£2) has boundary values
f(ele) = f(ele) of modulus l for almost all e16 e T, hence
f(ele) =‘lé=——a.e. Thus the function f(z) = -——1—-—
Hana) . . (Jf) (Z)
has boundary values f(elg) = f(ele) a.e. Finally, f(z)

is holomorphic on De except at the zeros of (Jf)(z), that

40

is 'f(z) is meromorphic on De' We also note that if f(z)
is a singular inner function (no zeros on D), then the

pseudocontinuation f(z) is holomorphic on De.

Theorem 2.21:

[13, p. 68] If Su(z) is the singular inner function
determined by the positive singular measure u, then S
is analytically continuable everywhere in the complex plane
except at those points which are in the closed support of LL
The function Su (or even ISuI) is not continuable from
the interior of the disc to any point in the closed support
Of u.

Now take a measure n which is positive and singular
with respect to Lebesgue measure on T and with the closed
support of u all of T. The singular inner function
Su(z) is pseudocontinuable across T onto De' the pseudo-
continuation is holomorphic on De and Su(z) is not analy-
tically continuable across any subarc of T onto any sub-
domain of De'

A question which arose when Shapiro [21] defined
pseudocontinuations was: Are there any functions which do
not admit a pseudocontinuation? The answer given by

Shapiro was: The function

M8

2n
f(z) = 6:57 . M s 1.

is not pseudocontinuable across any subarc of T.

n=O

41

6. Continuous Linear Functionals on Hp.

We shall have need of an integral representation of the

continuous linear functionals on the HP classes.

Definition 2.15:

For 0 < p E a, a mapping @: Hp 4 C such that
¢(af + g) = a¢(f) + 2(9)

for each a E C and f.g E Hp for which there exists a

real number M satisfying

sup |i> (f)I§M
nfn pél
H
is called a bounded linear functional on HP.

For 1 g p 5 m. HP is a Banach space,,and it is well
known that a linear functional is bounded if and only if it
is continuous. For 0 < p < 1, HP with the metric
p(f,g) = Hf—ngp is a Frechet space as we have noted. It
is known [6] tgat a linear functional on HP, 0 < p < l,
is continuous in the Frechet space topology if and only if it
is bounded.

For 0 < p < a, we can regard HP as a subspace of
Lp [0,2w] by identifying f(z) with the corresponding
boundary function f(eie) E @p c Lp. For 1 s p < a this
approach is quite fruitful for considering the continuous

linear functionals on Hp since DP is a closed subspace

 

42

of Lp and for l s p the spaces Lp have many continuous
linear functionals. In the case 0 < p < 1, however, only
the zero functional is continuous on Lp [6] while for Hp

we still have enough continuous linear functionals to sepa-

rate the points.

Theorem 2.22:

[22] If ¢ is a continuous linear functional on HP,
1 < p < m, then there exists a function 9(2) 6 HQ, q = 59$.
such that

 

2 o o
@(f) = jL-f ”f(ele)g(ele)de, for each f 6 HP,
"' o

and conversely, each 9 6 H9 so defines a continuous linear
functional on HP.
We restate this theorem in a form which we will find

more useful.

Theorem 2.23:
If ¢ is a continuous linear functional on HP,
1 < p < m, then there exists a function 6(2) 6 Hq(De),

_ .2.
q — p-l , such that

__L
4)“) — 27T IO

2w . .
f(ele)G(e19)de, for each f 6 HP

and conversely each G e H9(De) so defines a continuous

linear functional on Hp.

43

££22£=

Hg(De) was defined in terms of the J operator on
Hq(D). Given p e (Hp)*, take the g(z) E Hg(D) guaranteed
to exist by Theorem 2.22. Define G(z) = (Jg)(z). Then
G(z) e Hg(De) and G(eie) = g(eie). Similarily for the

converse.

Theorem 2.24:

If ¢ is a continuous linear functional on H1, then
there exist two functions G(eie), g(z) such that

(i) G(eie) 6 Lco [0,2w];

(ii) 9(2) 6 Hp, for any p < m;

 

-..L 2W 19‘1“? ,
(111) ¢(f) — 2w I f(e )G(e )de.
0
2V . .
(iv) ¢(f) = 1im_-§; I f(rele)g(e19)de:
r41 0
and
(v) if G(ele) has the Fourier series 2 cnelne, then
a n=-m
g(z) has the Taylor series chzn about the
. n=O
origin and g(ele) has the Fourier series
Q o
2 Cne lne.
n=0

We shall need to know something about the linear func-

tionals on Hp for O < p < 1. We first define several

classes of functions.

 

J

44

Definition 2.16:

 

[26, p. 42] Let A denote the class of functions
holomorphic on D and continuous on the closed unit disc.
Let f(ele) be defined for e E [0,2v]. We define the

modulus of continuity of f by

m(h7f) = sup [f(eit) - f(eis)|
It-sl g h

tl S 6 [0; 2”].

For f 6 A, we say that f 6 Ad (0 < a s 1) if and only
if
w(h;f(e19)) = (0(ha ) as h 4 o,

and we say that f E A* if and only if

ei(t+h))

- 2f(eit) + F(e

i(t-h)” = 0(h).

|f<

uniformly in t as h 4 0.

Theorem 2.21:

 

[6] Let A 6 (Hp)*, O < p < 1. Then there is a unique

function g E A such that

2W - _-
(1) 9(f) = 119-§% ] f(re19)g(e 19)de. f 6 HP.
r41 0
..l. .1 _ ... (n-l)
If n+1 < p < n (n—l,2, ), then 9 6 Ad' where
_.l _
a — p n.

(n-l)

Conversely, for any 9 with g E Aa' the limit (1)

45

exists for all f E Hp and defines a functional 9 6 (HP)*.

In the case p = -l- 9(n-1) E A*: and conversely, any 9

n+l'
with g‘n—l) e A* defines through (1) a bounded linear

functional on Hp.

Corollary 2.2&:

a 2“, then for
n

n 0

any fixed n (n = O,l,2,°-°), the mapping Pnf = an is a

IIMS

If f E Hp has Taylor series f(z) =

bounded linear functional on Hp.

This corollary implies that if we have a sequence of
functions [fn] C HP (0 < p < l) with fn 4 O in Hp
metric as n 4 m, then the Taylor coefficients converge to

zero. We state this in a corollary.

‘Qorollary 2.27:

 

Let p be fixed, 0 < p < 1. Let Mn]co c Hp. If
n=l
there exists a g 6 Hp such that an-gH p 4 O as n 4 m,
H

then if
fn(z) = 2 an 2k, |z| < l,
k=0 ’k
and
” k
g(z) = 2 bkz , |z| < l,
k=0
then lim a = bk' for n = 1,2,3,
n4co 'k

 

CHAPTER III

The Closure Problem for U*

1. Characterization Theorem.

In this chapter we will present a characterization of

the non-cyclic vectors for the left shift in the Hp Spaces,

1 g p < m. The main result of Douglas, Shapiro and Shields

E3 is a characterization of the non-cyclic vectors in H2

for the left shift in terms of pseudocontinuations across T.

The specific techniques used by Douglas, Shapiro and

Shields involved identifying H2(D) with 12 (the space of

all square summable sequences of complex numbers [an] with

2

2
”[an}H 2 = E [an[2) and the dual space of L with itself.
i

There is little difficulty in extending their result to the

Hp spaces, 1 < p < a. The extension to H1 is quite

difficult due to several factors. First of all the dual space

of H1 is not as neatly described as that of H2 or HP,

1 < p < a. Secondly, in the H1 case one has problems with

sequences of L1 functions which converge in L“, O < u < l.

but perhaps not in L1. This latter convergence causes prob-

l

lems as we will have a sequence of L functions for which

the Fourier coefficients of index n = O, $1,12,--- converge

46

I.
W.
l

47

to zero but the sequence may only converge in the L1/2

metric. We will want to be able to conclude that the limit
function is the zero function but we will need to use much
of the structure of the Hp spaces as Fourier coefficients
LP

do not make much sense in the spaces when 0 < p < 1.

We shall first state and prove the characterization
theorem for Hp with l < p < a, basically repeating the
methods of [3], but changing the notation.

We recall that the left shift operator U* is defined

by

= f(z) - f(O)

*
(U f) (2) z

* if and

and that f(z) is said to be non-cyclic for U
only if span [U*nf]:=0 is not dense in Hp. It is to
be understood that when speaking of the left shift operator
one has a fixed space (p) in mind.

In the proofs of Theorems 3.3 and 3.4 we will need some
relatively straightforward results whose proofs are simply

computations. In order to keep the notation somewhat reason-

able we state these results in the form of two Lemmas.

Lemma 3.1:
Let H(z) e Hp(De), l s p s a have boundary values

H(ele). Then 9 (n) = O for n = +l,+2,°°-, that is, the

positive Fourier coefficients of H(ele) vanish.

 

48

2322::
Since Hp(De) C H1(De) for l s p it is sufficient to
prove the Lemma for H1(De).
Let H(z) E H1(De) be given. Denote the boundary val-

ues of H(z) by H(ele). Let c > 0 be given. By Theorem

 

2.12’ there exists a polynomial Q(z) such that

 

T
”H(ele) — Q(e-lg)” 1 < e. Let n be any positive integer.
L
Now .-i
A 2W - .
l -1nA
|H(n)l = lg; [ e 'H(e19)de|
0
2V . . .
1 -1n 1 -i
= [3; f e e[H(e 9) - Q(e eflkel
0
2V . .
l 1 —
.5 g [me E’) - Q(e 16)Ide
O
= “H(e”) — Q(e-19)“ 1 < e.
L
A
Since 6 > O was arbitrary, H(n) = 0.

Lemma 3.2:

 

 

Let G(ele) and g(ele) be any two non-zero functions

. 2 . . . 19 ".8. ing
in L [0,2w] With Fourier series G(e ) ~1 2 (n)e
i9 ” 4 ing n=_m a
and g(e ) ~ 2 g(n)e respectively. Let G(n) = g(n),
n=-w .
for n = -l,-2,--:. Let f(z) = f(rele) be a holomorphic

function on D. For each value of r, O s r < 1 define
hr<e19) = f(relg) g(elg)

and

49

kr(eig) = f(reie) G(ele).

/\ /\
Then hr(n) = kr(n), for n = -1'—2'-3'...

Proof:

For any negative integer n,

/\ co/\ kA
hr(n) = Z f(k)r g(n-k)
k=O

on A kA
= z f(k)r G(n-k) =
k=O

= fir(n).

Theorem 3.3:

Let p be fixed, 1 < p < m. A necessary and sufficient

*
condition for f(z) 6 HP to be non-cyclic for U is that

the following two conditions hold:

(1)

(ii)

Remark:

f(z) has pseudocontinuation, f(z), across T,
with ‘f(z) meromorphic on De'
f(z) 6 N(De), i.e., f(z) is of bounded Nevanlinna

characteristic or type on De.

Note that this proof of sufficiency is also valid for

the case where p = l.

50

Proof:
(Sufficiency) Let f(z) 6 HP be given. Assume f(z)
has pseudocontinuation f(z) across T, where ‘f(z) is

meromorphic in De and of bounded Nevanlinna characteristic

on De. Let E anzn be the Taylor series about the origin
n=0

for f(z), and let G(z), H(z) be two functions in H”(De),
. c' _ G(z)
Wlth H(z) # O for z E De' such that f(z) - H(z)‘

Without loss of generality we may assume that G(a) = 0.

Indeed, if G(m) + O, we replace H(z) and G(z) with

H*(z) =-%H(z) and G*(z) = %G(z), respectively. This

 

gives
ism *
H(z) l H (Z) e
H(z)
z
and

* * a *
G (2) , H (z) E H (De) with G (on) = 0,

since i’ is bounded and holomorphic for [2] > 1.
Define a continuous linear functional ¢ on HP by

2w

¢(k) = :%-f k(ele)H(eie)de, for any k(z) 6 HP.

0

where k(ele) and H(ele) denote the boundary functions of
k(z) and H(z) respectively.

Now by Theorem 2.6 and Theorem 2.5’ we have

k(ele) e Lp[0,2w] since k(z) 6 HP, and

H(elq) 6 Lq[0,2w], for %'+ i = l,

51

since

H(z) e H°°(De ) c: que).

Thus by Theorem 2.23 ¢ is a continuous linear functional
on HP. Observe that ¢ is not the zero functional since
H(z) # 0.
We claim that ¢ annihilates f and all of its left

shifts. We proceed by induction.

27T . . 27T .

_ 4L. 19 19 _.JL 18
¢(f) — 2” f f(e )H(e )de — 27 f G(e )de.

0 0

since G(ele) = f(ele)H(ele) a.e. [0,2w].

. 277' .
Now since G(z) E H1(De), lim [ZWG(re19)d9 = I G(e19)de,
O

r41- 0

2" 19
but ( G(re )d9 = G(m) = 0. Thus ¢(f) = 0. Let n be

0
a fixed non-negative integer. Assume that p((U*)kf) = O,
for k = O,l,°°°,n. We wish to show that ¢((U*)n+1f) = O.

* n n-l k
Set K(z) = [(U ) f](z), p(z) = Z akz . Thus,
k=0

an(z) = f(z) - p(z) or f(z) = an(z) + p(z).

We must show that @(U*K) = 0.

NOW

2n_- . .
4><U*K) = if e ”mew-mm] H(eleme

ZTT' O
2W . - . 2w . .
__ 1 -19 16 19 _ 4L, ‘19 1
_ 2w [0 e K(e )H(e )dt) 27le0 e K(O)Hé%d9.

We claim that both of the integrals above are zero.
2W . .

By lemma 3.1, f e 16H(e19)d9 = O.
O

52

Now fix r, O < r < l and set 2 = rele. From the defini-

tions of K(z) and p(z) it follows that
f(2)H(ei9)==[an(2)4-p(2)]H(eie)==an(2)H(eie)4'p(2)H(eiq).

Since 2 + O we obtain

z"“’1[f(z)H(eie)] = z-lK(z)H(eie) + (z-n-lp(2))H(eie).
and hence

2” 2W

] r‘“'1e(’“'l)19f(re19)H(e19)de==j r-le-i9K(re19)H(ele)d9
o o
2” -n-1 (—n-l)° i is
+ I r e 19p(re e)H(e )de
0

-n-le-(n+l)ie

Now r is fixed, 0 < r < 1 so that r p(nele)

is a finite linear combination of negative powers of e19

since p(z) is a polynomial in z of degree at most n-l.
2w _ _ _ -

J- r n 1e (n+1)19

O
nation of Fourier coefficients of positive index of H(e

H(e19)de is a finite linear combi-
i
9).

Thus,

Since H(z) E H°(De), Lemma 3.1 says that the Fourier coef-

ficients of positive index of H(ele) are all zero, hence,

2W _ _ _ . . .
I r n 1e (n+l)lep(rele)H(ele)de = O, for O < r < l.
0

Next, a straight forward application of the Lebesgue dominated

convergence theorem gives

27 . . . 2W . . .
lim f r-n-le-(m'l)lef(re19)H(ele)d9=I e-(n+l)lef(e1%H(ele)d9
r41" 0 O

53

2W _ . .
=‘[ e (n+1)leG(ele)de=O,

0

since the last integral above is a Fourier coefficient of

positive index of G(ele), and G(z) 6 Hm(De). Hence,

2W . _ - ~ 2W_. - -
0: lim_[ e'ler 1K(rele)H(ele)d9=J' e leKele)H(ele)de = ¢(U*K)
r41 0 O

*)n+l

Therefore ¢((U f) = ¢(U*K) = O which completes the

induction step. Hence under the hypothesis that f has a
pseudocontinuation across T, of bounded Nevanlinna charac-
teristic on De, we have shown that there exists ¢ 6 (Hp)*,
¢ # 0 such that ¢[(U*n)f] = O, for n = O,l,2,°--. Thus
span (U*nf]:=o c Kernel (¢). Since ® is continuous,

Hp. Since ¢ 4 O, Kernel (¢) + Hp.

Kernel (D) is closed in

Thus span [U*nf]co O is not dense in Hp or equivalently
n:

f is non-cyclic for U*.

(Necessity):

Let p be fixed, 1 < p < e and let f(z) e HP(D)

*. Let 2 anzn be the

n=0
Taylor series about the origin for f(z). We must show

be a given non-cyclic vector for U

that f has a pseudocontination across T of bounded

Nevanlinna characteristic on De'

If f is the zero function, or indeed any constant

function we are clearly done, so assume f is non-constant.

From the definition of non-cyclic, the closure in Hp

of span [U*nf]n=O is a closed proper subspace of Hp.

54

Hence by the Hahn-Banach theorem there exists a continuous

Q
linear functional p + O on Hp such that ¢[span [U‘rnflnuO

= 0.
By the Riesz Theorem [19] there exists 9(2) 6 Hg,
___1_
q — p—l such that
1 2F 19 ie p
(3.1) ¢(h) =-§- I h(e )g(e )de, for any h E H . ‘
"' 0

Now define 6(2) 2:. (J9) (z) -=- 902) for on g |z| > 1. ' 1
By definition of Hq(De), we have G(z) e H9(De), G(ele) =
g(ele) and G(ele) E L9[O,2w], where G(ele) is the bound-

ary function of G(z). Now define

(3.2) H(eie) = f(eie)G(ei9) a.e.

 

By the Holder inequality [15] H(elg) E Ll[0,2w]. We claim
that H(ele) E 91(De). This will follow from the hypothesis
that f is non-cyclic for U* and a theorem of F. and M.

Riesz. We first show that the non-negative Fourier coeffi-

 

cients of H(ele) are all zeros. Now note that

2w . 2w . . 2w . .
] H(ele)d9 =]“ f(ele)G(ele)d9 =f f(e19)g(e19)d9 = 2w ¢(f) = o,
O O O

or H(O) = 0. We now show by induction that the positive
Fourier coefficients of H are all zero. Let n be a non-
negative integer and assume that

2

7r_. .
I e1k9H(ele)de = O for k = O,l,°°°,n.
O

55

zk. Recall that (U*n+lf)(z) = f(z)-p(z)
O k zn+1

Also note that

19 19 19 -i(n+1)9
f)(e ) = [f(e ) - p(e )]e a.e.
Now by hypothesis,

2’IT . . _. ——-.—-
0 = 4>(U*“+lf) = 517] we”) - p(ele) 1e l""'1"’<_:;(e19.)<ile =
O

 

 

 

 

2W . . . n 271’ .
l -1(n+1)9 19 19 l 1(k-n-1)9
=-—— e f(e )g(e )de-- 2 a I e 19
27f .16 2TTk=O k0 g(e )de
Now g(elq) 6 ©q(D) so that
2” i(k-n—l)9 19 2" i(n+1-k)9 19
f e g(e )d9 = j e g(e )d9 = o,
O O
for k = 0:102: In
Thus
_jh 2” —i(n+1)9 i9 i9
0 = 2” £) e f(e )g(e )de
1 2” -i(n+l)9 19
=‘5‘ [ e H(e )de.
71' .0

Hence by induction all of the non-negative Fourier coeffi-
cients of H(ele) are zero.
. 19 19 . . 1
Now the function h(e ) = H(e ) is in L [0,2w] and
has negative Fourier coefficients all zero, and thus by the
theorem of F. and M. Riesz [l3], h(ele) is the boundary
function of h(z) E H1(D). But H(ele) is the boundary

function of H(z) = (Jh)(z), that is H(eie) E H1(De).

56

 

Thus H(elg) = f(ele)G(ele) a.e., where G(ele) =

g(ele) E ©q(D), with g(ele) not the zero function and thus

g(ele) = O at most on a set of measure zero. Thus
. i
f(ele) =-Ei§-—l a.e. Now H(z) and G(z) are both in

i9
G(e )
H1(De) and thus each is the quotient of two functions from

H”(De); that is, 'fiigl 6 N(De) has the boundary function

G(z)
Elsie.) ___ “em,
19 a.e.
G(e )
This completes the proof.

In an attempt to generalize the proof of the previous
theorem to the case H1, the first place that trouble begins
iijiline(3.l): the function g(eie) is not necessarily the
boundary function of an H°(D) function. About the most
we can conclude is that g(eie) E ©p(D) for each p < a,
which is an unfortunate consequence of fact that there is no
bounded projection from Lca onto Hm. Now in line (3.2)
we can define H(eie) = f(ei9)G(eie) a.e., however, H(eie)
is the product of an L1 function with a function not neces-
sarily in L". The resulting H(eie) is in L“, for any
u 6 (0,1), but this does not help if one wishes to speak of
i9).

the Fourier coefficients of H(e We can get around

these problems, but the proof becomes a bit complicated.

Theogem 3.4:

A necessary and sufficient condition for f(z) e H1 to
be non-cyclic for U* is that the following two conditions

hold:

 

 

57

~

(1) f has pseudocontinuation f across T with f

meromorphic (M1 De

(11) f e N(De).

Proof:
As we have noted the proof of sufficiency in Theorem

3.3 holds for the case p = 1.

Necessity:

Assume f(z) E H1 is non-cyclic for U*. We wish to

1")

CD
show that conditions (i) and (ii) hold. Let Z a 2 be

n=O
the Taylor series about the origin of f(z). Assume
f(z) # 0 since if f(z) E O the theorem is trivially

true. By definition of non-cyclic the closure in H1 of

span [U’mf]::O is a proper closed subspace of H1. Thus,
by the Hahn-Banach theorem there exists a continuous linear
functional ¢ + O on H1 such that @ [span [U*nf]::o] = 0.

By the Riesz representation theorem [15] and Theorem

. . 19
2.24 there eXist two functions G(e '), g(z) such that

G(elq) E Lm[0,2w], g(z) E Hp, for all p < m,

1 2” 19 19 1
(3.3) Wk) =377i k(e )G(e )d9, for all kEH
o
and
2w . ———.——
(3.4) ¢(k) = lim 'g; k(rele)g(ele)d9, for each k E H1

r41“ 0

58

We remark that G(eie) can be obtained by extending ¢
from H1 to Q on L1 by the Hahn Banach theorem, and tak-
ing G(eie) to be the "Riesz representation theorem func-
tion" for the integral representation of Q in 3.3. G(eie)

is by no means unique. We also note that it was shown in

Theorem 2.24 that

A A
G(n) = g(n), for n = 0,1,+2,+3'...

Now by Theorem 2.23, g(ele) is the boundary function of

 

(Jg)(z) = g(%§). Also (Jg) E Hp(De) for any p < m since
9 E Hp(D). Since lim f(reie) = f(eiq) exists for almost
all 9 6 [0,2w], wgaian define a function h(eie) =

lim f(reie)g(;zg; a.e.

r41” 1
In Theorem 3.3 we defined a function similar to h(e
In that theorem we had f E Hp and g 6 Hg, 1 < p < a,

q = 5?: and thus h(eie) was in L1 by the Holder ine-
quality [15]. The most we can get from the Holder inequal-
ity here is that h(eie) 6 LLl for any u 6 (0,1). In sub-
sequent portions of the proof we will need to know that
h(eie) is in some fixed Lu class, so we pick Ll/Z.
There is nothing special about this choice of u as any

other number in(0,1) will work just as well.

Now for each value of r, O s r < 1 define

 

hr<ei9) = f(reie)g<eie)

and

 

kr(eie) = f(reie)G(ei9)

9).

 

 

59

We stress that we are regarding hr(ele) and kr(ele)

 

as functions of 9 and we want to know what happens to these

functions of 9 as r4l-. That 11m hr(e19) = h(ele)
r41-
exists a.e. has already been noted as has the fact that

h(ele) E L1/2[O,2w]. Similarly 11m kr(ele) = lim f(rele)
r41‘ r41“ |

 

 

G(eie) = f(eie)G(ei9) = k(eie) exists a.e. since f E H1.
Furthermore,
Hkr(e19) - k(ele)H 1 = ”f(rele)G(ele) — f(e19)G(elg)H 1
L L
s ue<e19>n mnf<re19> - f<e19>n 1
L L

But by Theorem 2.11, “f(reig) - f(eig)H l 4 O as r41"
. . L
and thus Hk (e16) — k(elq)H 4 O as r4l‘.
r L1

We now use the hypothesis that f is non-cyclic for

U* to show that the non-negative Fourier coefficients of
hr(e1q) converge to zero as r41-

If we denote the Fourier coefficient of index n of

 

i9
hr(e ) by hr(n), then
. 1 2” 19 19
O = ¢(f) = 11m '5; I f(re )g(e )d9
r41’ 0
27 .

= lim 5%] hr(ele)d9
r41' 0

= 11m fir(0)

r41"

 

 

60
Next,
2W . .
= ¢(U*f) = lim -§= ] (U*f)(re19)g(e19)d9

_ w

r41 0

. 1 2” 19 -1 -1 i
=lim Z—I [f(re )-f(0)]r e age %d9

— w

r41 0

 

= lim_[-21—r1[:Treigf(reig)g(eie)d9 -
r41
_ 2w _. .
-§; r 1f(0)[) e leg(elq)d9]

. -lA
= 11m [r h (+1) - O]
r4l’ r

A l
= hr(+)

since by Theorem 2.24 and Lemma 3.2 the Fourier coefficient

of index +1 of g(elg) is zero.
We proceed by induction. Assume that lim hr (k) = O
r41

for k = O,l,"',n-l.

Set
n-l .
p(z) = 2 a 23
n=0 3
Then, by Theorem 2.19:
n _
(U* fl(z) = f z n z , for O < |z' < 1.

so that
2W '_‘_-_‘
o = ¢(U*“f) = lim 21] (U* nif)(re e)g(eigme
r41-

 

 

! 1

‘- :

 

61

 

 

2W . . _ _. .
= lim 21i0[f(relg) - p(re19)]r ne lneg(emmg
_ w
r41
. l -n 2” -in9 19 19 1 -n 2” -in9
=11m-2-Erj'e f(re )g(e )-'§'T'Fr[e
r41- 0 O
p(re ol)g(e fide]
-qin- -l . 2w . . -———r-
= lim [r rlhr (n) -'§— 2 a ray el(J n)eg(ele)de]
r4l- W j=0 j o
=lim_[r"r1 hr(,n)]-O
r41
since for j-n < 0,
2w . . -———r-
l 1 - i
EFJ‘ e (j n)eg(e 9)de
0
is a Fourier coefficient of positive index of g(ele) and

thus is zero by Theorem 2.24. We have thus shown that

O: limhr (n), for n=O,l,2,
r41

We now restate the functions which we have defined and

some of what we have shown about them.

 

kr(ei9) f(reie)G(eie)

 

hr(eie)

f(rei 9i)g(e 6)

For r fixed, 0 < r < l,

62

where convergence of the two series above is in the L
sense.

Thus, by the Riesz-Fischer Theorem [13]

. —1A . A o
k (e19) = z k (n)elne+ 2 k (n)e1ne
r r r
n--m n=0
and
. -J_A . m A .
h (e19) = 2 h (n)ein9+ Z hr(n)elne,
r n=-m r n=0

again r is still fixed and convergence of each series is
in the L2 sense, to an L2 function.

Now by Lemma 3.2 we have
fi(—)—1’§ 1
r n — r( n), for n — + ,+2,+3,'°

Thus
on A _ o
2 hr(-n)e 1n9 =

Qr(-n)e-1ne a.e.
n=l n

"MB

1

Now we have shown that “kr(eie) - k(eie)H l 4 O as
r4l‘, and by Theorem 2.17 for any fixed r, OL< r < l we
have

1/2

H 2 Q (n)zn - 2 k(n)zn g k (e ) - k(e )
n=o r n=0 ”HI/2 H r N

1/2
L1

Now by the Riesz-Fischer Theorem [Hoffman, p.14],

2 Ikr(n)|2 < w. Hence by the Riesz-Fischer Theorem [Rud1n,
“=0 °° A n 2 1/2
p. 332], we have ngo kr(n)z e H (D) c H (D) for O:sr<<l.

1/2

Since H is a complete space we have Eok(n)zn E H1/2(D).
n:

 

63

Q .
Denote the boundary function of 2 k(n)zn by K(ele) where
- n=O
K(ele) e 91/2(D).

0A
Also, note that 2 kr(n)zn E H2(D) has boundary func-
n=O

 

tion of Qr(n)elne E 142 [0, 27r]. Now we have just shown that
n=O
a a 1/2
H 2 Qr(n)zn - 2 Q(n)zn” 1/2 4 O as r41-
n=O n=O H

Hl/Z

But by Theorem 2.11, convergence in the metric implies

convergence of the corresponding boundary functions in the ]

a . .
Ll/2 metric. Thus 2 kr(n)elne converges to K(ele) in
n=O
the L1/2 metric as r41". ‘
Also kr(e19) converges in the L1 metric to the L1
. o . .6 . . ‘
function k(elg) = f(e19)G(el ). Since L1 converges im- ‘
plies Ll/2 convergence and L1 c Ll/2 we have that
-1 ,
Z Qr(n)elne converges in the L1/2 metric, as r4l‘, to
n=-w
the L1/2 function k(ele) - K(ele).

1/2

"1 in9 . .
Now 2 hr(n)e converges in the L metric to

n=-e°

the L1/2 functions k(elg) - K(ele). Also hr(e19) con-

verges to h(elg) = f(elg)g(ele), in the 1.1/2 metric r4l-,

1/2

T h in9 . - -
and thus 2 r(n)e converges in the L metric as r41

n=O
to an L1/2 function, say E(ele) s h(ele) - [k(ele)-K(ele)].

Now for each fixed r, O < r < l, E hr(n)elne is the
n=0

2(1)) c 111/2 (D). Thus

00A
boundary function of 2 hr(n)zn 6 H
n=O
since 91/2(D) is a closed subspace of Ll/z, we have

1/2

E(ele) e b1/2(D). Let E(z) be the H function whose

64

boundary function is E(ele). Now by Theorems 2.11 and

_ ° A ine 19
2.13 the convergence, as r41 of 2 hr(n)e to E(e )
n=0
. 1/2 . . ° A n .
in 9 implies that 2 hr(n)z converges to E(z) 1n
1 11:0 A
H /2 as r41". But, for n = o,1,2,---, lim hr(n) = o,

r41-

A
by Corollary 2.26 the Taylor coefficients of I hr(n)zn
n=0
converge to those of E(z). Hence E(z) has all of its
Taylor coefficients zero and thus E(z) E 0. Therefore

E(ele) is the zero function.

-1 .
We now note that 2 h (n)elne E 92(D ) c 51/2(D ) for
n=-a r e . e
any r, O s r < l, and thus as r4l', hr(e19) converges
to
. -l A .
h(ele) = lim 2: hr(n)elne e 51/2(De).
r41" n=-o

Thus hr(ele) = f(rele)g(eie) converges in L1/2 metric

 

to h(eiG) e sl/Z(De). Now lim hr(eie) = f(eie)g(eie) and
r4l'

1/2

therefore f(e19)g(ele),= h(eie) e 9 (De) and

g(ele) 6 D2(De). Also, g(ele) is not the zero function

and hence is zero only on a set of measure zero. Thus

 

. 19 . .
f(ele) =-El§;5) a.e. and since both h(ele) and g(e‘e)
g(e )

are quotients of bounded functions, f(z) has a pseudocontin-
uation f(z) across T, ‘f(z) is meromorphic on D6 and of

bounded Nevanlinna characteristic on De'

 

 

2. Applications of the Characterization Theorem.

Theorems 3.3 and 3.4 enable us to determine certain
classes of cyclic and non-cyclic functions. Since Theorems
3.3 and 3.4 are extensions of the H2 result of Douglas,
Shapiro, and Shields [3], the results of [3] for H2 whose
proofs depend only on pseudocontinuations carry over to
HP, 1 s p < m with the proofs unchanged.

Theorem 3.5:

[3] If f 6 HP, 1 s p < m and f is analytically

continuable across all points of T with the exception of

an isolated winding on T, then f is cyclic for U*.

Proof:

It follows from the definition of pseudocontinuations
that if f has an analytic continuation across any subarc
of T that this analytic continuation must be the same as

any pseudocontinuation across T.

Examples:
f(z) = (z-l)1/2, for |z| < 1 (either branch)
g(z) = ln(z-l), for |z| < 1 (any branch)

Both f(z) and g(z) defined above are in H1(D). However,

it is impossible to define a pseudocontinuation across all of

T onto De for either f(z) or g(z).
65

 

 

66

Theorem 3.6:

 

[3] All Rational functions in Hp, 1 s p < m, are

, *
non-cyclic for U .

Proof:

Rational functions are meromorphic on s, and of bounded
Nevanlinna characteristic on De'
Corollary 3.7:

The non-cyclic functions for U* are dense in HP,

1 s p < w.

Theorem 3.8:
[3] If f is holomorphic in |z| < R for some R > 1.
then f is either cyclic or a rational function (and hence

non-cyclic).

Proof:

By Theorem 3.6 rational functions are non-cyclic. If
f(z) is non-cyclic and holomorphic in |z| < R. with R > 1,
then the pseudocontinuation of f, f, is an analytic con-
tinuation of f across T. Since f can be continued to
be meromorphic on the Riemann Sphere, f is a rational

function.

 

67

Theorem 3.9:
[3] Let f and g be non-cyclic and h be cyclic

for U*.

Then f + g is non-cyclic and f+h is cyclic for
U*. Furthermore, fg and €®3 are non-cyclic while fh
and $41 are cyclic for U* insofar as any of these are

in Hp, 1Sp<co.

Proof:

Theorems 3.3 and 3.4.

Theorems 3.10:

[3] The set of non-cyclic vectors is a dense linear
manifold in Hp, while the set of cyclic vectors is dense
in HP, 1 s p < m.
2222£=

Since the polynomials are non-cyclic and dense, the
first part of the theorem follows from Theorem 3.9. For the
second part let f be a fixed cyclic vector and p an
arbitrary polynomial. [f+p] where p ranges over all
polynomials is dense in HP, 1 S p < a and by Theorem 3.9
this set consists only of cyclic vectors.

We conclude with an observation about non-cyclic vectors

in HP, 1 s p < m.

 

68

Theorem 3.11:

Let f 6 HP, 1 s p < a, have pseudocontinuation f
across T with ‘f meromorphic on De and of bounded
Nevanlinna characteristic on De' Then the closed linear

span of [U*nfLF4)is a proper closed invariant (under U*)

subspace of Hp.

2355!?

Theorems 3.3 and 3.4.

We remark that Douglas, Shapiro and Shields [3] have
shown that in H2 every closed invariant subspace for U
is the closed linear span of [U*nf}:=0 for some f E H .
The proof of this theorem uses the Beurling theory for the
invariant subspaces of U in H2 as well as the fact that
U* is the adjoint of U (in H2). Since the Beurling

theory is not available in the pre-dual of H1, it is not

. . 1
known whether the U* invariant subspaces of H ,

are'byclidfi

 

——-
\

 

BIBLIOGRAPHY

 

10.

11.

12.

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