Some finite-dimensional backward-shift-invariant ... - Springer Link

Integr. equ. oper. theory 42 (2002) 1-21 0378-620X/02/010001-21 $1.50+0.20/0 9 Birkh~iuserVerlag, Basel, 2002

I IntegralEquations and OperatorTheory

SOME FINITE-DIMENSIONAL BACKWARD-SHIFT-INVARIANT S U B S P A C E S IN T H E B A L L A N D A R E L A T E D I N T E R P O L A T I O N PROBLEM DANIEL ALPAY* and H. TURGAY KAPTANOGLU

We solve Gleason's problem in the reproducing kernel Hilbert space with reproducing kernel 1 / ( 1 - ~ zjw~). We define and study some finite-dimensional resolvent-invariant subspaces that generalize the finite-dimensional de BrangesRovnyak spaces to the setting of the ball.

1

Introduction

In this paper we study a family of finite-dimensional Hilbert spaces of rational functions analytic in the ball =

(zl,...,zN)

l

lzjl 2 < 1

,

(1.1)

1

whose reproducing kernel is of the form

I,~ - B(z)S(w)* 1 --

(1.2)

ZW*

In this expression, z and w are in the unit ball, zw* = ~ z~w;, and the function B is rational and matrix-valued. To provide motivation and set the framework, we first consider the case N = 1, that is, the case of the open unit disk D. First recall that the Hardy space of the open unit disk H2(D) consists of power series f(z) = ~ o fn zn with complex coefficients such that [Ifll~I2(D) def. f i

lfnl 2 < (:~

(1.3)

0 *This research was supported by a grant from the United States-Israel Binational Science Foundation (BSF), Jerusalem, Israel, and by the Israeli Academy of Sciences.

2

Alpay, Kaptano~lu

1 We recall that and is the reproducing kernel Hilbert space with reproducing kernel 1--G~w." this means the following: For every w 6 ]3 the function z ~-+ 1 belongs to H2(D) and

f(z),

H2(ID)

1 - zw*

for every f E H2(D). The space H2(D) is invariant under the backward shift (also called resolvent-like) operators Ra defined as z E D.

Raf(Z) = f ( z ) - f ( a ) ,

(1.4)

z--a

We recall that these operators satisfy the resolvent equation a, b E D,

Ra - Rb = (a - b)R~Rb,

(1.5)

hence their name. Furthermore, the adjoint of the operator Ra in H2(D) is given by

1, the natural analogues of these operators would seem to be the operators (1.11), but they are unbounded, as we showed above. On the other hand, the adjoints of the operators Tj(a) are of course bounded. These will be our backward shift operators here. Definition 3.1 The operator Tj(a)* will be called the j-th backward shift operator in H(I~N) at the point a.

8

Alpay, Kaptanoglu

These operators in general do not satisfy the resolvent equation (1.5). They do when one restricts a to have only one fixed nonzero component. We cannot give a closed-form expression for the backward shift operators acting on arbitrary f E H(~N). On the other hand, as is usual in reproducing kernel spaces, we can compute the adjoint on the kernels. This is a special case of a general theorem oa multipliers in reproducing kernel Hilbert spaces (see e.g. [4, formula (2.3.5), p. 34]), but we give a proof for completeness. L e m m a 3.2 W e have (Tj(a)*)

1 ,) w; 1 1 --'z-~-w* -- 1 - aw* I - zw*

(3.1)

P r o o f : Let v be in the ball. Then

--

(Tj(

) (;)

(Tj(a). 1 1 ZW*

=

1

1 }

a)*l_zw,,l_zv,

~I(BN)

1 -- ZW* =

1 - z w * ' 1 - za* 1 --'zv

,)"

1 -

wv* 1 wj

H(S~)

-

H(e~)

wa*

vw*)(1 - aw*)"

(I -

[] The main result of this section is the following. T h e o r e m 3.3 Let f E H(BN). Then there exist f u n c t i o n s g l ( z ) , . . . , g ~ ( z ) E H(BN) such that N

f(z) - f ( a ) = ~ ( z j - a j ) a ( z ) .

(3.2)

1

We define the functions gj(z) in a unique way by the formula gj(z) = T j ( a ) * f . The decomposition is not unique in general if one does not impose further restrictions on the gj(z), as is illustrated by the example

1 = I + zl(0) + z~(0) = 1 + zl(z2) + z2(-zl), which holds since the functions zl and z2 and the constant functions belong to H(BN).

Alpay, Kaptano~lu

Proof of Theorem gj = T j ( a ) * f . Then

9

3.3: We first consider f ( z ) to be a kernel, i.e., f ( z ) =

f(z)

-- f(a)

=

=

1

Let

1 1 - zw*

1 1 - aw* 1 . E . (1 - zw*)(1 - aw*) (zj - a j ) w j 3

=

~(z~

- ~)g~(z).

Thus the equality f ( z ) = f ( a ) + ~'~(zj - a j ) ( T j ( a ) * f ) ( z ) J

(3.3)

holds on the linear span of the kernel functions, and hence on all of H(]~N) by continuity. [] We can rewrite (3.3) as M=j_ajT~(a)* = • - c o

(3.4)

J where Ca is the evaluation at the point a. The problem of existence of such functions gj in a given domain and m a given class of functions is called Gleason's problem. See [29, p. 116] for a list of spaces where this problem has a solution. Rudin gives two formulas for the gj(z) in various spaces, and in particular in the Hardy space of the ball. The first formula is due to Ahem and Schneider (see [2]) and is valid in the space HI(~N) associated to the ball, and is equal to

gj(z) = f~

C(z, ~) - C(a,

U--~)-~

~)

~;f(~)~~

(3.5)

In this expression, C ( z , u) = (1 - ZU*) - N is the Cauchy kernel, S is the boundary of the ball and a is the Lebesgue measure on S such that ~r(S) -- 1; see [29, pp. 12-13]. The second formula holds for a = 0, is valid for functions analytic in the ball and continuous up to the boundary (i.e., in the ball algebra), and is due to Leibenson (see [23]). The gj(z) are now given (when a = 0) by

9j(z)= /t ~---~jf(tz)d~.

(3.6)

A long computation using Propositions 1.4.8 and 1.4.9 of [29] shows that the Ahem-Schneider solution coincides with our solution for any a E ] ~ N for N = 2. The computations for N > 2 and a ~ 0 seem similar but longer. On the other hand, for a --- 0 the situation is simpler and we have the following.

i0

Alpay, Kaptano~lu

L e m m a 3.4 Let N > 1 and a = O. The operator Tj(0)* coincides with the formulas of Ahem and Schneider and of Leibenson on H ( ~ . ) . P r o o f : We first consider an element of the form f(z) = 1--=7-~w1 9 Since it is in the ball algebra, it follows t h a t the formulas of A h e m and Schneider and of Leibenson coincide; see [29, w p. 117]. We now check that for such f

f(tz)dt =

w~ . 1 - zw*

But ~zj

1 - zw

(1 - t z w * ) 2 '

and so

fol O@jf(tz)dt = fo ~ ( 1 - W~w,)2d t = 1 - wj* t zw*" []

The result follows by continuity. We conclude with some further properties of the operators Tj(a)*. P r o p o s i t i o n 3.5 / t holds that Tj(a)*(1) = 0 and

r~(~)*z~ = 5~,

where 5ij denotes the Kronecker symbol. P r o o f : Let v and w in the ball. We have

(Tj(a)*l)(v)

=

zv* H ( ~ )

a)*l'l-

= (1,(I_za,)ZlI_zv,))H(~N) = Similarly,

(rj(a)* z~)(v)

=

/

(

zj (1 -

za*)(1

zv*) z:O

-

)"

=0.

1)

Tj(a)*ze, 1 - zv* H(~)

= {ze,(l_za,)Zil_zv,)}H(~N)

and the result follows.

=

~

1

=

~ze (1 - za*)(1 - zv*)

-

zw*

'(1

-

za*)(1

-

zv*)

.~

Alpay, Kaptano~lu

11

Proposition 3.6 The operators Tj(a)* commute. P r o o f : It is enough to prove that their adjoints commute on a dense set, but this is clear from Tj'(a)Tj2(a)f = (1 - za*) 2f(z " for f ( z ) =

1

I=1

1--zw*

P r o p o s i t i o n 3.7 The common eigenvectors of the resolvent operators Tj(a)* in exactly the kernels 1--Z~* 1 "

H(~N) ar e

P r o o f : Let ] 9 H(]BN) such that Tj(a)*f = Ajf for possibly different complex numbers hi. Then it follows from (3.3) that f(z)(1 - ~-~.jAj(zj - aj)) = f(a), that is,

f(a) As already remarked, f is analytic in the ball and in particular in a neighborhood of the c and origin, and so 1 + ~j Ajaj ~ O. Thus f(z) = 1-z~= with c -- I+Ejf(a)~ 1

w * = l + ~.~jAjaj( A1 )~2 "'"

AN ).

If [wl _> 1, the function f would not be in H(~N)- So ]w I < 1 and f is a kernel.

[]

We have already remarked that a similar result holds in the setting of the polydisk. It is of interest to note that it also holds in the setting of de Branges-Rovnyak spaces defined on compact real Riemann surfaces (see [13], [12]).

4

B l a s c h k e factors

It is well known that one can extract zeros of functions in H2(D) via Blaschke factors. The counterpart here is the following. P r o p o s i t i o n 4.1 Let a E ]BN. Then the cl• b~(z) = (1

aa*'l/2~ (z

-

1

-

-

function -

a)(Iiv

-

a'a) -1/2

(4.1)

za*

satisfies 1 - ba(z)ba(w)* 1 - aa* 1-zw* = (1-za*)(1-w*a)'

z, we]Blv.

(4.2)

In particular, bo(z)bo(z)" Lastly, b~ belongs

to

H(]~N)l•

< 1, 1,

[=

if zz* < 1, if z z * = l.

(4.3)

12

Alpay, Kaptano~lu

Formula (4.2) appears in Rudin's book on the ball [29, Theorem 2.2.2, p. 261 (with an apparently different choice of b,, but in fact, up to sign, the same; see L e m m a 4.2). It expresses the fact that the one-dimensional vector space spanned by the function z ~+ 1--Za* 1 endowed with the metric of H(BN) has reproducing kernel of the form 1-bo(z)b~(~)" That 1--Z~t]* formula in Rudin's book was the trigger that made us understand that part of the analysis of H2(D) could be extended to the space H(BN). We also note that (4.1) can be rewritten aS

bo(z)

=

bo(o) + b (z) - b (o)

=

- ( 1 - aa*)l/2a(I,~ - a'a)