Internat. J. Math. & Math. Sci.
(1986) 39-46
39
Vol. 9 No.
BANACH SPACES OF FUNCTIONS ANALYTIC IN A POLYDISC
LEON M. HALL
Department of Mathematics and Statistics
University of Missouri-Rolla
Rolla, Missouri 65401 U.S.A.
(Received March 8, 1985)
ABSTRACT.
in
This [aper is concerned wlth functions of several complex variables analytic
the unit olydlsc.
Certain Banach spaces to which these functions might
defined and some relationships between them are developed.
belong are
The space of linear
functlonals for the Banach space of functions analytic in the open unit polydisc and
continuous on the unit torus
is
then described in terms of analytic functions using
an extension of the Hadamard l)roduct.
KS/ WOiIDS
AND P,RASES.
Banach space, linear functional, analytic
lSO MATHE/4AICS L’UBJECT CLASCFICATION CODE.
].
function.
4615
INTRODUCTION.
In 1950 A. E. Taylor [I] studied Banach spaces of functions analytic in the unit
disc.
One of his principal results was a representation of linear functionals in
terms of functions analytic
In this paper, the results of Taylor are
the unit disc.
in
extended to functions analytic in the unit polydisc in n-dimensional complex space.
The goal
is
the representation theorem for linear functions, Theorem 4.5, in which the
functlonels are expressed in terms of a Hadamard product.
Taylor’s results have proved
to be very useful in work involving certain singular linear differential operators.
His representation theorem for linear functionals was a key part of the work of Grinur
and Hall in
[2], and of subsequent work of Hall in [3] and [4].
The following notation will be used extensively:
open unit disc in the complex plane C,
U
unit circle in
Iz e C:
+
U
n
I
n
Z+
C,
(i2)
I-6},
e
0
(1.3)
i,
(1.4)
onnegatlve integers,
U
n
n
IZl
(ii)
U
..
U, (n col)ies of U) the unit polydisc in C
.....
[
[
I
I
7
Z+
...
...
(n
I
colies
(n
+
of
copies
(n copies o
) the
of
I ),
Z + ).
unit torus in C
n
n,
(1.5)
(1.6)
(i 7)
(i 8)
L.M. HALL
40
,z
(Zl,Z 2,
If z
...,z w ).
Further, if z e C
el
z z
an
Zn and
I
on C
n n
2
2
n
lal
which are analytic in U
u (z)
g’ where x
(x
g(z)
w
f(ze
TM)
f(zle
e U
g, where
The operators ./
If f
n
U A
fa
by
n
A
x
,x n) e IR
ix
I
z2e
2
..,z e
n
n
and 7.
w
define z
e
A
An
by
An
n
e
(ZlW 1 ,z2w 2,
by z
the class of functions
Z n+
for each a e
A
n
by
by
and
(1.9)
iXn)
(l.10)
f(wz)
and g(z)
(l.ll)
and 7. are easily seen to be linear.
w
x
then f has the power series expansion
Z
f(z)
Denote
define zw by zw
and define the function u
n
ix
7. f
+
i
x2,.
I
n
e Z
(c1,2,...,c)
+
n
+
Denote
+
2
n"
and
n
are in C
n
n
Also define the operators
z
Ux f
n
II
by
,w
(Wl,W 2,
and w
n
y(f)
ya
where
is the functional defined by
(al!e2!’’an 1)
Ya (f)
II
-i
Zn
Zl
If f and g are in
A
n
(1.12)
f z
EZ n+
(1.13)
n
z--(0, ..,0).
with power series
Zzn
f(z)
ae
and g(z)
f z
aEZ n+
+
gez
(1.14)
define the Hadamard product of f and g by
Z
H(f,g;z)
H(f,g;z) is clearly in
A n,
and the following also can be proved:
y((7"w f)
H(f,g;z)
H(7. f,g;z)
w
where
Izil lil
(2#)n
for l=l
(1.16)
Ya (f)
w
(1.17)
H(g,f;z),
(1.18)
aH(f,h;z) + bH(g,h;z),
H(af+bg,h;z)
H(f,g;z)
(1.15)
fgz
Z n+
(1.19)
H(f,g;wz),
17.
Zl
n)g(1
f(l
E
Zn.) dl
n
i
dn
(i20)
"’’--n
n
n.
Equation (1.20) can also take the form
(2)
H(f,g;z)
where r.
1
_.
i@
n
0
f(rle
and z e U
n.
rne
n
z
g
I
(e
1
-i8
1
z
n
,----e
r
n
-i8
n
d@
I.
.dO
n
(1.21)
BANACH SPACES OF FUNCTIONS ANALYTIC IN A POLYDISC
2
SPACES OF TYPE
41
A k.
An
Let B be a nontrivial complex Banach space, each element of which belongs to
A.
Such a space will be called a space of type
B may also have one or more of the
following properties.
PI:
The least such constant will be denoted
P2:
for all e.
P3:
P4:
U
Zn
B for all e
U
f
X
Tr f
will be denoted
B if f e B and r
denoted
Z n+.
AI(B)
B and x is a real n-tuple.
(rl,r2,..,r)
n
ll[rfll Allfll.
constant A such that
if f e B and e E
There exists a constant A such that
The least such constant
B if f
AI Ifl
Iye(f)
There exlsts a constant A such that
Also
< A
A2(B).
llUxfll
r.
with 0
lu
Ifl
There exists a
i.
l
The least such constant will be
A4(B).
Ak
B will be called a space of type
if B is of type
A
and also satisfies axioms
Pl,...,Pk.
Let B* denote the space of continuous linear functionals on B.
Yn
Then by
PI’
e B* and t can be shown that
supl
A (B)
1
(B)
Other relations satisfied by the constants
and
(2.1)
are
supllTrll,
r
A4(B)
1
IyeIl,
<_
A
(2.3)
I(B)A2(B),
(2.4)
1 < A (B)
4
Let B be a space of type
A I.
(2.5)
For any fixed g e
N(g;z)
An
IH(f,g;z)
sup
and z
U
n
define
I.
(2.6)
The following are clear from the properties of the Hadamard product.
N(g+h;z) < N(g;z) + N(h;z)
(2.7)
N(ag;z)
N(T g;z)
N(g;wz),
W
THEOREM 2.1.
(28)
A 3.
Let B be a space of type
lwl
(2.9)
i.
Then the function N(g;z) has the
properties:
N(g,z)
N(g;(r
N(g,z), where g e
N(g;(r
I
z e U
n
and z
r.
(2.10)
n
r )) is a nondecreasing function of each r., i=l
n
I
where 0
n
,m,
(2.11)
i,
,rn ))
0 for all
(rl,
rn
if and only if g
0.
(2.12)
L.M. HALL
42
A{
Let | denote the set of all nonzero elements of B and define, for f
PROOF.
(2.13)
A
where g is fixed in
m
Also define
(2.14)
M(z)
n
x
n
be fixed.
n
M(f;zeXX).
M(U f;z)
Let z e U
and z e U
N(g;z) and also, for x e R
Observe that M(z)
(2.15)
Then
f
1
I%1 Izln,
Hence, for fixed z e U
IM(f;z) is bounded as f varies over 4.
n
U f varies over all of as f varies over all of
Now for x e R
U (U f),
x x
proved.
M(z) if x is chosen carefully, and (2.10) is
and so by (2.15) we get M(z)
i and 0
for k=l,...,n, k
the points on the tori
IM(f;z)
{z:
.
_<
r < i
k
and
z be
Then let z
Also suppose f e
r. < R < 1
r
R
i
l
respectively, at which
and {z:
(RI’’’’’Ri’’’’Rn)
Next, let R
since f
IZkl
(rl,...,ri,...,rn)
and r
IZkl
}
assumes its maximum value.
with 0
rk},
The maximum principle for polydisc functions now
yields
I(f;Zr)
But
IM(f;zR)
M(z
R)
M(ZR)
Finally, if g
I;zl.
IM(f;Zr)
e
< M(R),
(2.18)
M(R), and (2.11) is proved.
0, N(g,r)
0 for all r.
If g
e
Let f= u
(2.17)
M(R) and so
IM(f;r) <_
which implies M(r)
!
for this e
Then H(f,g;r)
.
ye(g)r I
I
0, then
a
...r
n
n
ye(g)
@
0 for some
which is nonzero if r
k
M
e.
0
k=l,...,n, and the proof is complete.
3.
THE SPACES B’ AND B
Suppose B is a space of type
A 3.
Two related spaces which will be used later in
characterizing B* will be defined as follows:
B’
{FeAn:
N(F;r) is uniformly bounded in r,
0<_rk<l
k=l,...,n},
(3.1)
BANACH SPACES OF FUNCTIONS ANALYTIC IN A POLYDISC
B
0
{FEAn:
For F e B
1
43
H(f,F;r) exists for each f e B}.
lim
r(l
(3.2)
,i)
sup{N(f;r)
define N(F)
r
0
N(F)
k=l,...,n}
< i,
k
By (2.11) we have
N (F;r)
lim
r (i,
(3.3)
I)
If B is a space of type A then B’ is a Banach space with norm N(F).
3
From the properties of the Hadamard product and N(F;r), and from theorem
THEOREM 3.1.
PROOF.
2,1, B’ is clearly a normed linear space with norm N(F).
To prove completeness
proceed as follows.
Let {F
N(F.)
B’o
be a Cauchy sequence in
N(F
J
conslder, for
i, and N(F.)
m
Let 0 < r
-F
m
)rel
H(u
’
note that convergence
a
A
2(B)E.
Consder
{a }, e e
A2(B)E
n
Z,
and
is bounded.
which yields
(3.5)
a z
+
and it will be shown that F e B’ and lim F.
for fxed
IH(f,F
Since
_<
ae=lim, y(Fj)
3
Next deflne
Z
A,
Now define
+
’((Fj)r
0 we obtain
F(z)
Clearly F e
Z
m
n
and that the science
is uniform in
Fm
(34)
m
is also a Cauchy sequence for each
Using the above argument with
_<
k
F -F ;r)
m
e
{ya (F3)}
1 for k=l,...,n and
Z+,
g
17(F
Thus,
> J,
Then there is a J > 0 such that if
bounded by, say, Eo
is
r
F.
and f e
<
;r) -H(f,F;r)
ZIy(f)resuply(F
(3.6)
a
0 uniformly in s,
lim[7 (F0)-a
e
e
j+
lim H(f,F.;r)
j
(3.7)
H(f,F;r).
Further, for f e B
which implies
IH(f,F;r)
< E
Ifl
Finally, for e > 0, choose
or N(F;r) < E.
Hence, F e B’.
such that if j,m >
jo(e)
jo(e)
then N(F.-F
m
< e.
For
f e B (3.5) yields
IH(f
As m
(3.4) ylelds
F -F ;r)
m
IH(f,Fj--F;r)
< N(F -F
_<
ellfll
m
)I Ifll
if
<
El Ifll
__> j0(e),
(3.9)
which means
N(F.-F)3
< e
and the proof is complete.
THEOREM 3.2
If B is a Banach space of type
and is a Banach space of type
A4
with norm N(F)t,
A3
then B
0
is a linear subset of
B’
44
L.M. HALL
PROOF:
Clearly B
0
is a linear subset of
H(f,F;r) deflne a functional of f over B.
For f e B and F e B
0
let
r(f)
Then
lr(f)
sup
and since F e B
A.
N(F;r),
(3.10)
0
then N(f;r) is bounded as a functlon of r by the uniform boundedness
0
is a linear subset of B’o
0
0
In order to show that B is closed in B’, let {F0}
B and F e B’ be such that
principle.
Thus, B
lm N(F -F)
j
For f
0
6
B
IH(f,F;r)-H(f, F ;0)I<IH(f,F-F.;r)I+IH(f,F.;r)-H(f,F.;0)
For f
B flxed and
(3.11)
If I+IH(f,F. ;r)-H(f,F. ;0)
<2N(F.-F)
large enough so that 2N(F.-F)
0 choose
Ilfll
< /2 and choose
r and Q close enough to (i,i,...,I) so that
IH(f,F.;r)-H(f,F.;0)<e/2
4.
Therefore lim
r, 0+ (i,... ,i)
IH(f,F;r)-H(f,F;p)
0 and F
B
0.
.REPRESENTATION OF LINEAR FUNCTIONALS.
In thls section a series of results which culminate in a representation of the
space B* and ts relationship to the spaces B’ and B
THEOREM 4.1.
Z
G(z)
eZ
and
IGI
B’ and
Then G
PROOF:
n
e U
A
G
n
<_
N(G)
since
IGII
IYl IA2(B)-
E f
y(z)w
(-
sup
N(G)
SUp
IH(f,G;r)
Iy(Trf)
[H(f,G;r)
A
N(G;r)
<_
r
The passage from
y(u)
Also
y(G)
G
Let f e
An
I f G w
H(f,G;w).
(4.2)
4(B) [y[
A
4(B)
and G
IYI
B’.
Ifl I.
(4.3)
Also,
4(B) II l-
A
to G given by (4.1) defines an operator F: B*
B’ where
G.
THEOREM 4.2.
by {u
(4.1)
+
B then
llfll--
and
z e U
B* let
Then [ f e B and
w
Hence N(G;r)
F(7)
y(u )z
If y
n
4(B) Iyl I.
IY(ue)
y(T f)
If f
A
n
will be proved.
A 4.
Let B be a Banach space of type
e
0
IFI
},
PROOF
A
e
4(B).
n
Z+
A 4. Then F is a linear operator
F has an inverse if and only if the linear subspace of B spanned
Let B be a Banach space of type
is dense n B
By Theorem 4.1 F is linear and
IY(Irf)
NOW choose y so that
I1tl
<_
IFI
A4(B)
By (4.2)
IIFII I111"
Ilrrfll so that ((rrf[(
I[fll IIGtl < Ilfll
i and
(</rf)
(4.4)
_<
Ilfll Ilrll.
BANACH SPACES OF FUNCTIONS ANALYTIC IN A POLYDISC
litr II
Then
Ilrll
Suppose F(y)
A4(B)
0
Then y(u
if and only if (u
if and only if
Ilrll,.
and
Ilrll
Hence
Z n+
0 for all a e
e
0 for all
2n+
45
and hence saying that F
is equivalent to saying y
0.
-I
Thus F
-I
exists
exists
is total in B which is equivalent to the subspace spanned by {u
{u
being dense in B.
For f e B, H(f,F;r) defines a linear functional with norm N(F;r), so if F e B
(4.5)
H(f,F;r)
llm
y(f)
0
r+(l,...,l)
The passage from F to y given by (4.5)
defines an element of B* corresponding to F.
defines an operator .%
B
0
B* where A(F)
y
A 4.
Let B be a Banach space of type
THEOREM 4.3.
FA(F)
IA()I <_
(F)
A
Then
0, and
1^() I, for
is a linear operator,
(4.6)
F, for F e B
<_
N ()
F e B
O.
A deflnes an injective mapping, with bounded inverse, of B
Thus
PROOF:
y(G)
y(u
Also, since
A(F).
Let F s B
0,
H(u
lim
r >(i
IH(f,F;r)
A(F)
i)
y and F(y)
,
F;r)
(4.7)
onto a subspace of B*.
Then
y (F)r
lim
r+(l,
lfll IFII
<
G.
0
y(F),
i)
Iy(f)
it follows that
<
and so F(A(F))
lfll IFII
F.
where
Thus, from (4.6)
(4.8)
If the space B satisfies an additional axiom, viz.,
If f e B then
P5
B’ and B
then B*,
0
THEOREM 4.4.
T
r
f
B and
lim
r+(l,...,l)
llTrf-fll__
0
turn out to be isometrically isomorphic.
A4
Let B be a Banach space of type
B
0
also satisfying
P5"
Then
B’
(4.9)
A is bijective and isometric, and
(4.10)
-i
PROOF:
(4.11)
By (1.19) H(T f,F;p)
r
H(f,F;r) -H(f,F;0)
H(T f,F;r) and so
p
H(f-[ f,F;r) + H(] f-] f,F;0) + H(T f-f,F;0)
p
(4.12)
;)1
(4.13)
r
p
p
Therefore, if f e B and F e B’
IH(f,Ftr)
and by P
H(f
[-I If-r fll
5
lim
r,p+ (i,... ,i)
so F g B
Irrf-rpfl
+
0
Since B
0
IH(f,F;r)
B’ already, this means B’
H(f,F;p)
B
o
O,
(4.14)
L.M. HALL
46
From
P5’
lim
r+(l
analytic in w then
IITrll I1=11. if Iwl
ITwfll lTrfll by P3" Hence
(see
r
i)
This yields A
r.
B*.
Now suppose f e B and
< 1
i=l
..,n, and ]" f is
w
from the maximum modulus theorem
ITrfll
[3], p. 211)
for functions with values in a Banach space
continuous function of
i
is a nondecreasing
1 and by Theorem 4.3 .’ is isometric.
4(B)
Then
[A(F())] (f)
H(f,F(y);r)
lira
r(l,
i)
(4.15)
y(T f)
lim
r(l,
(f).
r
,i)
As a corollary of Theorem 4.4 the following theorem provides a representation
of B* in terms of the Hadamard product_
Under the hypotheses of Theorem 44 every linear functional y e B*
THEOREM 4.5.
is representable
in
the form
6.
,i)
I, 0 < r i <
T
e
i
is uniquely determined by y and
1
i)
)n
lim
r(l
where, 0
H(f,F;r)
lira
r+(l
y(f)
x...xT
I
< 1
f(l
e
If
(n)F
--,
() i n
n
i=l,
Y
(4.16)
,n and F
B
0
F uniquely determines and
N(F).
Equation (4.16) can be expressed in terms of real integrals using (1.21).
REFERENCES
I.
TAYLOR, A. E.
2.
3.
4.
5.
Banach Spaces of Functions Analytic in The Unit Circle, I, II,
II (1950), pp. 145-170; Studia. Math. 12 (1951), pp. 25-50.
GRIMM, L. J. and HALL, L. M. An Alternative Theorem for Singular Differential
Systems, J. Diff. Equa. 18 (1975), 411-422.
HALL, L. M. A Characterization of The Cokernel of a Singular Fredholm Differential Operator, J. Diff. Equa. 24 (1977), I-7.
HALL, L. M. Regular Singular Differential Equations Whose Conjugate Equation
Has Polynomial Solutions, Siam J. Math. Anal. 8 (1977), 778-784.
TAYLOR, A. E. Introduction To Functional Analysis, John Wiley and Sons, Inc.,
Studia. Math.
New York, 1967.