It turns out that this dimension is independent of which particular .¢-maximal strictly negative subspace one chooses, and thus negative signature is well-defined; if .7 is nondegenerate
Trang 1A BEURLING-LAX THEOREM FOR THE LIE GROUP U(™, n)
WHICH CONTAINS MOST CLASSICAL
INTERPOLATION THEORY JOSEPH A BALL anc J WILLIAM HELTON
== U(n, 0)) has very strong consequences for Nevanlinna-Pick, Carathéodory-Fejér, etc interpolation theory We obtain directly from our theory a simple linear frac- tional parameterization of all solutions in @H™(M,,,,) or BH}°(M,,,,,,) of the most general interpolation problem for a finite number of points and strong results for infinitely many points Moreover we obtain a test to determine if any solution to a particular interpolation problem exists Finally in the last section we apply an ex- tended form of our Beurling-Lax theorem to the setting of the Sz.-Nagy—Foias com- mutant lifting theorem
Here @H?°(M,,,,,) denotes the closed unit ball of mxn matrix valued functions
on the unit circle with meromorphic continuations onto the unit disk with at most / poles there; multiplicity must be counted carefully — see [16] As usual U(m, 7) denotes the group of (m -E n) x (m -L n) matrices g which leave the form
x BY, X ® Vin = % om — Ws Mow
(x @yeC"", xeC", yeC", where <-,-) is the usual Euclidean inner product) invariant
The linear fractional parameterization and the test for existence was obtained for 7 =: 0 or for m =n = 1 by Adamjan, Arov and Krein [1], [2] The test for exis- tence was obtained in general by Ball [5] and very refined results due to Arsene, Ceausescu and Foias [3] when / = 0 are also available Also Nudelman has recently
Trang 2108 JOSEPH A BALL and J WILLIAM HELTON
obtained such results [27], see also (26) Fuither strong results aie also due to
T S Ivanchenko [19], [20] S V Kung obtained a set of solutions to the general
!,m:-: a problem in [30] However the theorem in this paper is appealing not only because of its generality but also because of the relative simplicity of the proof This simplicity permits many easy applications [7] and suggests many extensions [8], [9] The subsequent article [7] uses this method to obtain the Wiener-Hopi fac- torization of a (not positive) self-adjoint matrix function (due to Nikolaicuk and Spitkovskii), Potopov’s symplectic inner-outer factorization, and Darlington’s theo-
em While in this article and in [7] we have refrained from the great generality needed in our treatise [10] on the mathematics of amplifier design, these methods generalize trivially to that case and the authors think of the U(m, 1) Beurling-Lax theorem as a single result from which most of the tools developed in [10} follow
In a completely different vein the forthcoming articles [8], [9] deal with the clas- sical Lie groups (other than U(m,n)) We prove a Beurling-Lax theorem for them and give applications to mathematics and to theoretical engineering
The results of this paper were announced in [6] The authors are grateful to
P DeWilde for encouragement regarding the engineering value of a complete theory of shift invariant subspaces of L2(C”) with signed bilinear form Such spaces arose in his studies of Darlington synthesis for muitiports
1 PRELIMINARIES ON INDEFINITE INNER PRODUCT SPACES
We begin with some preliminaries on indefinite inner product spaces A com- prehensive reference for indefinite inner product spaces is Bognar’s book [11}, but
we shall depart slightly from his notation and terminology We shall be working with complex vector spaces # having a Hermitian bilinear form, denoted by [ ]
or [, J,,, which induces an inner product on # which is not necessarily positive-deti- nite If in addition # can be written as a direct sum #:-.#, #_ where (Ha €;5 Sx.) and (#_,¢, yw) are Hilbert spaces, and the inner product on #” has the form
Ly, y] " éx¿ ’ 1+3, a éx_ ,› Yd
where xt: Ny 2 XLV 524 $V (Xn, E HM ,, XL € H_), then” is said to
be a Krein space Given a Krein space # =: #, +- #_, it is also a Hilbert space
in the inner product
(XV) = XE, Varn, + xL, y~-)x_
If P,:x x„ is the projection of ⁄ onto #'„ along./_ and P_ ::J P„: xox
is the projection onto # along #, , then J == P, —- P_ is said to be a fundamental
Trang 3symmetry for, 4, and the above Hilbert space inner product can also be written as
(x,y) =[Jx,J] (== éx, y}„) with norm
lIxIỦ = [J>, x]
(or simply ||x||Ê if the choice of / is understood) While the associated Hilbert space norm || ||, depends on the choice of fundamental symmetry J, the induced norm topology is independent of /, and thus intrinsic to (%, [, ])
Certain general geometrical properties which we now discuss arise in the context of any Krein space # If (, ) is a Hermitian form on %, two vectors x and y are said to be ( , )-orthogonal if (x, y) = 0 If @ and are two subspaces of # such that #@ 9 WY = {0}, 4+ is closed and [x, y] = 0 for all x in.# and y in
MN, we write “FW for “4 +N; if wand are closed subspaces with (x, y), =0 for xe and ye, we write 7 @, WV for @ +A Any subspace 4% has a closed [ , ]-orthogonal complement
AM’ = {x:[x, y] = 0 for all y in 4};
the , ),-orthogonal complement of # is denoted
MAT = {x: <x, »>, = 0 for all ye 4}
or sometimes simply “+ if the Jis understood Note that the [, ]-orthogonal comple- ment #” of a Krein space % is {0}; however, a subspace / and the restriction of { , ] toitneed not have this property The subspace / is called nondegenerate if no x
in @ is[ , -orthogonal to 4 (i.e.,.4@ nM’ = {0}), and regular if there is no sequence
if #@ is regular Also /@ is regular if and only if the restriction of the Hermitian form [ ; ] of Z#' to makes # a Krein space in its own right If # is merely nonde- generate, at best one can only decompose # as W4 =.Z, +.Z_ where the restriction of [,] to #, and of —[,] to “ _ respectively make “%, and Z_ pre-Hilbert spaces
We say that the subspace # of a Krein space % is pseudo-regular if 4 +
is closed For an arbitrary subspace 4%, it is always the case that -} Z' is dense in (4 4'Y; thus 4 is pseudo-regular if and only if we have the equality 4 + 4’ =
Trang 4110 JOSEPH A BALL and J WILLIAM HELTON
= (.4%.@')' Clearly @ is pseudo-regular if and only if @’ is pseudo-regular Equivalently @ is pseudo-regular if and only if @ is of the form Z =.Z'-'.Z⁄s where #, is a regular subspace of % and.&@, -=.@0.@#' is a null subspace ([x, 1}: : 0 for all x and yin.@,) Thus in this case the form[,] on #% induces a Krein space structure on the quotient space Z/(.Z n.⁄)
A subspace @ of an indefinite inner product space (%, [,}) is said to be posi- tive provided [x, x] 2 0 for each x in #, strictly positive if in addition [x, x] = 0 for some x in.@ implies x = 0; by the Cauchy-Schwarz inequality, for positive sub- spaces this is equivalent to the condition [x, y] =: 0 for all y in @ implies x: - 0
A positive subspace is said to be maximal positive (with respect to #) or #-maximal positive if it is not contained in any larger subspace of % which is also positive The term £-maximal strictly positive is defined similarly We define the conditions negative, strictly negative, @-maximal negative and @-maximal strictly negative for a subspace 7 analogously By the negative signature of a subspace @ of the Krein space # we mean the dimension / (0 < / < co) of any @-maximal strictly negative subspace of 7 It turns out that this dimension is independent of which particular ¢-maximal strictly negative subspace one chooses, and thus negative signature is well-defined; if 7 is nondegenerate, this quantity is also the dimension
of any @-maximal negative subspace The negative cosignature of the negative sub- space W is the codimension of WV as a subspace of some maximal negative subspace A’, of 2; this quantity also is well-defined, that is, is independent of the choice of maximal negative subspace , containing 1
The following general lemma is basic for our analysis of interpolation pro- blems to come in § 3
Lemma 1.1 Suppose “ is a pseudo-regular subspace of # Then each /-
«maximal negative subspace of @ has negative cosignature 1 equal to the negative signature of <Z'
Proof The space @ = ’' has a [, ]-orthogonal decomposition
%=Z.::2-mếo
info a striclY positive subspace #„, a strictly negative subspace #_ and a null S2ace 2¿ (=@n”), where dưn4_ is the negative signature of @ (If 2 is pseudo-regular, 2, and & _ are Hilbert spacesin[ , ] ahd —[, ] respectively; in gencral they are only pre-Hilbert spaces.) Suppose Y is a é/-maximal negative subspace (.Z: @'); we claim that 4 + @_ is #-maximal negative From this it follows that any such A? has codimension equal to dim2_ in a 5f/-maximal negative subspace, and the lemma follows
To prove that A” + @_ is maximal negative, we first observe that since WW and
7, ace cach negative and are [, -orthogonal, clearly.⁄ #_„ is negative To prove
A ' is maximal negative, we need only show that (7 + @_yY=W' nA ¥ =:
Trang 5=.'n(⁄ +.) is positive Since.#ˆ =.Z,.W' splits as ” =(W“n.) + + and thus
4n(⁄+-#„) =(W 7n.) #„
Since W is @/-maximal negative, W’ “7 is positive By orthogonality and the positivity of 2,., it next follows that (VW’ n.#) + # is positive, as claimed Finally, in the sequel we shall need the angle operator-graph correspondence for negative and positive subspaces of a Krein space # Suppose # = 4H, AIH —
is a [, ]-orthogonal decomposition of the Krein space # into a maximal positive subspace 4% ,, and a maximal negative subspace #_; then any maximal positive subspace “, is of the form
SF, = {xT x | xe XH 4}
for some operator T,: 4, > 24 _ which is a contraction when (%,.,[,]) and (#_, —[,]) are considered as Hilbert spaces The operator TJ, is said to be the angle operator for £4 (with respect to the decomposition #7, Fix _ for #) and Ff,
is said to be the graph of T, Similarly, a maximal negative subspace “_ is of the form
Z_={T ;y#qylyeZ-_}
for a contraction operator T.: 4% > +
2 REPRESENTATIONS OF SHIFT INVARIANT SUBSPACES
The most concrete instances of Krein spaces arise as follows The vector space C% naturally decomposes as
cy=c"™@c where N = m + n; define the Hermitian form [, ]om,, on it by
[u, Vom, nh” (Un , Đà cụ, — (Up > Unren
if u=u,, © u, and v = v,, ® v, where Up, Vm € C™, Uy, V, € C” Here ¢ , >, denotes the usual Euclidean inner product on C’ The set of ([, Jom, m>0> Jom,n)-isometric mappings will be denoted U(m,, m; n1,”) Note U(m,, n,; m, n) is empty unless
m, < mand mn, < n In all this discussion, we may take any of the integers m, n,
mm, m to be +co; one then interprets C® as the Hilbert space /? in the obvious way With this concrete class of Krein spaces we associate a class of functional Krein spaces as follows Let L2(C%) be the Hilbert space of measurable C¥-valued func-
Trang 6112 JOSEPH A BALL and J WILLLAM HELTON
tions on the unit circle {|z| =: L} square-integrable in norm, and let #*(CX) be its subspace
We next describe the general notion of “phase function’’ and “inner function” appropriate for our Beurling-Lax theorem First we call a measurable function =
on {|z|==1} with values in U(m, n,; m, n),a( my, nụ; m, n) phase function; if in addi- tion Ze) is the boundary value a.e of a function 3(z) analytic on the disk {izl < 1}, we say that © is an analytic (mạ, nm; m, n)-phase A full range subspace
of H*(C¥) is one with the property that at some z, in the disk {|z| < 1}, we have (fle) : fe M@}==C% It is easy to check that if this happens at one z, then it happens
at all but an isolated set of z, (see [15]) We shall only be concerned with phase func- tions & such that Z(e!)x is in L°(C%) for any x in C%1;thus any such © has
\'2Cllcen,om 4 < oo Finally a closed subspace Z of L2(CM) is said to be
simply invariant if it is invariant under S and 9 S% = {0} For example every
Trang 7THEOREM 2.1 If @ is a regular simply invariant subspace of L?(C™"), then there are nonnegative integers m, < mand n, <n and an (m,, n3m, n)-phase func-
n (M in HY Since # is regular, so is Y; furthermore, the spaces S*7 =M „2 are mutually [, ]-orthogonal, and by regularity, the [, ]-orthogonal decomposition
M= POSH [/15%*./m S2+1 holds for all g = 0,1, Thus any vector [,]-orthogonal to all the spaces Š$*Z(k =0, 1,2, ) is in ƒ\ S*.Z = {0}; hence
k>0
Mo = LHI SLU 11 S72}
g>
is dense in „#Z
Now the [, J-inner product restricted to & makes ¥ a Krein space (since Y
is regular), and so & has a fundamental decomposition
Lz LLL where #, is a positive subspace and £_ is negative If m, = dim.Z, and n, =:
= dimY_, we can use this decomposition to construct a ({, Jom, m> [> ]¢)-unitary
By orthogonality, this extended operator is ([, ]
If we deRne Z (e!) a.e by
E(elt)x = (Ex) (e#), xe C™",
acm, my» [+ ]ygocem, m)-isometric
then we see that & is the operator M, (ein Of multiplication by the matrix function E(e%)
8 — 2484 9
Trang 8114 JOSEPH A BALL and J WILLIAM HELTON
Since & is a ([, Te" mys My? L; Tae" m)" -isometry, it follows in a standard way [25] that the boundary values 3(e'') are (, J mys my [; Jom, n)-isometric a.e., that is Z is a (?m,, 1; m, n)-phase This then forces m, < mand n, < n Since = maps C”:'": onto 4c H%(C”*"), it follows that the matrix entries of © are square integrable, and thus 2 extends by continuity to H°(C™") Since ZH(C””"1) contains %,, we see that [ZHW*®(C”r 1)" == 4,
Clearly = is analytic if and only if < H?(C™") Suppose < HC") and is full range; that is {f(z))| fe} =C™" for some ze {|z} < 1} Thus Ran &(Z) =: C’"”" which forces m, + n, > m-+n Since it was previously shown that m,<m and vn, <n, we see that m,=m and ny=n Conversely if m,-:m and
n, =n, then there is a z)€ {jz| < 1} for which RanZ(z)) = C”" and thus Z is full range If 4 < H°(C™") is full range, then & is rational if and only if Puen, m 42* H*(C™") is finite-dimensional; it is not difficult to see that this is equivalent to H2(C"")n-@' being finite dimensional
Our next task is to obtain a useful representation for simply invariant sub- spaces “& of L°(C™") which are pseudo-regular By an (7, 11, Pi; m, n)-phase junction we shall mean a square-integrable ÄM„„.„, my +0, +P, matrix-valued func- tion &(e!) which is injective for a.e t and such that
[EZ(@”) (u @ 9 @ w), 5”) (U @ 0 ® Wlemn = (ts Um, — (% 0) on, for ae t
Trang 9ii) ÿ c H*(C"") and H*(C":"\n.' is finHe-dimensional, then 5 is uni-
formly bounded and A = SH*(C”+ "+ 73),
Proof Since # is pseudo-regular and S is a bounded isometry, S.# is also pseudo-regular, and thus S@ + (SAY =(SAn(SM'Y = (Sin) Thus,
if we set # = “1 (SM), then from this it follows that
Trang 10116 JOSEPH A BALL and J WILLIAM HELTON
This in turn holds if
(2.4 a 41 SIP =: SIM 0 [M1 (S9*1.0)']
Using the definition of ¥ (=: @ N (S.#)') and that S is a [, ]-isometry, one can easily check directly that (2.4),4, is true This establishes (2.2)g+1, and hence (2.2), for all g Letting q tend to oo in (2.2),, one gets
First note that the containment ¢ in (2.7) is obvious For the reverse containment,
we note that & is pseudo-regular and thus has a decomposition Y =: &,':: Ly where £, is a regular subspace and #, is Yn #’ Thus Y S*£ is spanned by
To see that = == [5,7] is the desired (m,, ™,, p,; m,n)-phase function, it remains only to show that = is one-to-one To see this, note that since #(e”) is an isometry
from C”""" "1 into C”™" in their respective indefinite metrics, the columns
Trang 11E(e") G = 1, ., my + + py) of 5(€) satisfy
1, l<j=k MN mM, [é(e**), é,(e")] myn c —1, m< j = Kk S My + ny
0, otherwise
Suppose Ee) f(e*) = 0 ae for some f in HC’? ”) Then
0 = [6(), 5) SEM nin =
= {5% Ge nn HE where we have written Zf = )) f,¢, Therefore f,(e")=0 a.e for 1<j <m +m,
Ẽ
so ƒ has the form 0 @ 0 @ f for an in H°(C” and #ƒ = Vf Now, since ¥ was the traditional Beurling-Lax representor for W (and hence in particular is one- -to-one), it follows that ? , and thus also f, is 0
Clearly, © is analytic if and only if 7@n.W' ¢ H°(C™") or @ c HA(C™"),
If @ < HC") and H2(C™") 7’ is finite dimensional, then @ = H?(C™") n
n @’, being finite-dimensional invariant subspace for the backward shift operator, consists of uniformly bounded functions ‘Then since Y < ¥ + SX, so also does L From this one can deduce that = is uniformly bounded in norm
Some final remarks might help with the computation of & First we thank Bruce Francis for pointing out that
3 APPLICATIONS TO INTERPOLATION
In this section we describe the connection of Lemma 1.1 and Theorem 2.1
to generalized matrix interpolation problems We first describe a specific class of interpolation problems
Trang 12118 JOSEPH A BALL and J WILLIAM HELTON
a THE GENERAL CONSTRUCTION
Let z := {Z/}JˆoU {2/}/=¡ be points in the unit disk and p == {p)}¥., u {pj}, and q = {9,}9.1 U {qj}, be vectors in C™ and C” respectively The problem of
interest is to describe the set
N-P(z, p,q) = {te 2H°®(M„,„)Ì F(2)ŸP; = đ;, j = 1, , N and
F(2}) pj =7 Qj, J= 1, ., N’}
Here ZH™(M,,,,,) is the set of all (mX2)-matrix valued functions with analytic con- tinuation to the unit disk {]z| < 1} with ||Fi|.<1 Similarly @L™(M,,, ,) will denote the set of (7 x< n)-matrix valued L©-functions F with |jF[,, < 1 We include the pos- sioility of m or # = co; then C” stands for a separable Hilbert space of dimension
m, M,,,_, stands for the set of bounded linear operators from C" into C” The classical solution to the problem of determining if N-P(z, p,q) is non-empty for the case N’ -:0 is: There exists a function F in N-P(z, p, q) if and only if the matrix
A 2p, a = | —————————————- _ (Pi> Pry — (Gis 2]
iS positive-definite
More generally, consider the class @H7°(M,,,,) of functions F which have a representation of the form F = G@-1 where Ge ZH™(M,,,,) and @ is a matrix Blaschke product of degree at most / Consider the problem of describing the larger set
N-P,(z, p,q) ={Fé BHP(M,,,,) | F(z;)"pj = 9;, 7=1, ,N and
F(z) pj = GY, = 1, ., Nh
(If F happens to have a pole at z;, interpret the condition F(z,)*p; = 9; as G(z;)*p; ==
sz O(z;)"q where F == GO@-1 is the representation for F mentioned above Similarly one handles the condition F(zj)pj == qj if F has a pole at z}.) The solution of the existence problem (for the case N’ = 0), given by Ball [5] in this generality, is N-P,(z, p, q) is nonempty if and only if the associated Pick matrix Ag, p, g has at most J negative eigenvalues
To begin our analysis, we write the set N-P,(z, p,q) in a different form, which
in turn will suggest a more general problem (that of “generalized interpolation”
in the sense of Sarason) For the sake of simplicity in the present discussion, we assume that no point z, is the same as some point zj in the disk Let H°(M,,,,,) be the class of (#x~»)-matrix functions K analytic on the disk with boundary value function K(e') square-integrable in matrix operator norm; H?(M„,„) then consists
of matrix functions K such that K¥e H(M,, ,) for some inner V in GH™(M,,, ,) of
degree at most / Note that {fe H{C”)| (p;, f(z))) =0 for j=1, ,.N} is an
Trang 13invariant subspace for the shift operator S on H*(C”), and hence by the classical
Beurling-Lax theorem is of the form @H*(C”) for some matrix inner function
de @H™(M,,) Similarly there is a ma:rix inner function @ € @H*(M,) such that
pH(C’) = {fe HC’) | épj ƒj)) =0 for j=1, , N},
where @(€!) —= ø(e~)*, Then it is not đifficult to see that
N-P,(z, Pp, q) = (Fo + 6H?(M„,n)0) ñ #L”(M„.,)
where Fy is any function in H?®(ă„ „) which satisfies the interpolation conditions
F,(z;)"p; = 9; for j= 1, ., N and F(z))p; = qj for j = 1, ., N’ The inner func-
tions 0 and @ arising from a set N-P,(z, p, q) in this way are very special; they are
rational and have only simple zeros Allowing F,, @ and @ to be I?-functions gives
a more general problem without an interpretation as an interpolation problem as
above We say a function Q in ZL™(M’,, ,) is a phase function if its values Q(e')
are isometries a.e The general problem to be analyzed in this section is the following
GENERALIZED INTERPOLATION PROBLEM Describe the set
Cy,s,) = (K + 0Hƒ(M,,„) 0) n 2ØL® (Mm,n)
for any given Ke L*(M,,,,) and phase functions @ and go in @L™(M,,,) and
BL~(M,) respectively
Our approach is to make use of the angle operator-graph correspondence
between contraction operators and maximal negative subspaces of a Krein space
described in §1 to obtain an equivalent more geometric version of the problem
Thus we consider the space L?(C™) @ *i4?(C*’) with the Krein space inner product
inherited from 2°(C™*) Form the span of all subspaces which are graphs of mul-
tiplication operators with multiplier in K -+ 0N?(My,„)@ :
M = Mx 4,9 = the closure of {[ 7 Jerre È J#©®l
Since the angle operators defining the spaces are multiplication operators, it is clear
that # is invariant under the shift operator S The following is basic to our analysis
LEMMA 3.1 Let K be an element of L?(M,,,), let 0€ BL(M,,,) and
epeEBL”(M,) be phase functions, and sei dd equal to Mg 4,9 as above The angle
operator-graph correspondence induces a one-to-one correspondence between Cx 9o(1)
and shift-invariant negative suspaces of u< which have codimension of at most 1 as
a subspace of some L?(C™) ® o* H2(C")-maximal negative subspace In particular,
when " = [LA(C") @ o*(A2(C"))] 4-4 has negative signature I, these are the shift
invariant @-maximal negative subspaces of
ke /
Trang 14120 JOSEPH A BALL and J WILLIAM HELION
Proof Suppose F is in Cx,o, o(/) Then F has a representation
ia some L°(C”) © o*H*(C")-maximal negative subspace Clearly also GY is shift invariant
Conversely, suppose Y is a shift invariant negative subspace of # of codimen- sion Jin a L°(C”) @ o*H7(C")-maximal negative subspace That Y is invariant and
‘as codimension / in a maximal negative subspace means
Trang 15LEMMA 3.2 Suppose is a pseudo-regular invariant subspace of L*(C™") Set V equal to VV {S*(H# 1 M')|k = 0,1, .} Then a subspace G of is inva- riant and M-maximal negative if and only if Gad MN"' and G is invariant and (4 9 W"’)-maximal negative
Proof Suppose G is invariant and #-maximal negative Any /#/-maximal nega- tive subspace must contain /# 0.4’; since G is also invariant, it follows that A” c & Since G is a negative space and / is a null space ({x, x] = 0 for xe), this in turn forces § < WV’; hence ¥ c HNN" Since Y is /-maximal negative, a for- tiori # is (@ nN W’)-maximal negative
The converse direction does not require that Y be invariant Thus, suppose only that G is (4 W')-maximal negative Since (Zn.°)n(Zn.V'Y = 4W, then Y > /% Therefore if Y, is a negative subspace of / containing ¢ then 9, > V;
as in the first part of the proof, this forces 9, <¢ #0’, and thus 9, = ¢ by the maximality of Y in #04’ Therefore & is #/-maximal negative
We are now ready to use our symplectic Beurling-Lax theorem to parameterize the set of invariant /-maximal negative subspaces for an invariant pseudo-regular subspace @ of H?(C’™")
‘ LEMMA 3.3 Suppose @ is a simply invariant pseudo-regular subspace of L(C""), Then there is an (my, ny, py; m, n)-phase function 5S = b B 5] such
x y (@ that the invariant -maximal negative subspaces GY of 4 are precisely those of the form
Trang 16122 JOSEPH A BALL and J WILLIAM HELTON
Proof We do only the case /, = 0; the general case easily reduces to this By Theorem 2.2, there is an (7, 7, 21; m, )-phase function 2 = L7 B l such
* y @œ
that n.#”‹:{5.H®(C 13'')1~ where W= Vy S*M nM’) Suppose for the
kp moment that 2 is bounded, so #NW'’ = 5-H(C™ “ mì, Then multiplication
by Sis a metric-preserving isomorphism from xc" “1 *1y, with é,> om Cn
1 ci
to ⁄Z n 4W”; so a (⁄Z n 4ƒ”)-maximal negative subspace # has the form 2-%, where
#4 is amaximal negative subspace of H*C”ẻ “hy, Also since & is a multiplication operator, invariant subspaces of 4 MW’ correspond with invariant subspaces of H(C'? “ ’2) in this way Thus & is invariant and (.4 1 A’)-maximal negative
if and only if 9 = 2% for some invariant H*(C”ủ m ')_maximal negative sub- space Z But one easily checks that the invariant maximal negative subspaces of
F 0
HC" "'?ty are those of the form |Z 0|(HXC") @ H°(C")) for some F in
0 1 2H®(C”¿ T, By Lemma 3.2, invariant #-maximal negative subspaces are (4 1 4’)-maximal negative This proves the Lemma for the case where 5 is bounded
The proof for the general case involves the same ideas, but must be done with more care Given any invariant subspaces Y contained in “”=[2-H oe kh “To › one can argue that #; = #n z.H*(c> Tiến is dense in ¥ Then #,=2719,
is a negative submanifold of #'*%C”*”””*); denote its closurein W*(C””””*'”*)_ by
‹⁄ We claim that # is an invariant maximal negative subspace of Hic" Indeed, if # is not maximal negative, then there is a strictly larger negative
‘subspace @ which is also invariant By the classical Beurling-Lax theorem, we can produce a bounded F whichis in @ but notin # Then the closure of {Z-(2n H®(C”+ 5h in H*(C™") is a negative subspace of H°(C™") which is strictly larger than Y, a contradiction By a similar argument, one can show conversely that a subspace of the form
1’ ?)
G = closure of {8-(Ýn H®(C 7?
where % is invariant and maximal negative in HC, is invariant and (4 1) N')-maximal negative Finally it is not difficult to see that a linear manifold
Trang 17L is of the form % 1 H°(C'?'"”"") for some invariant maximal negative sub-
space X of H “cnr "1'?1y if and only if
F 0
#=|1 0|-°(C»@m*(C°)
0 7
for some F in BH (Mn, ,n,)- This completes the proof of the lemma
To parameterize a set Cx,9,,(/), by Lemma 3.1 it remains only to obtain a
parameterization of the angle operators corresponding to negative subspaces of
Mx,9,¢ having a prescribed “, 4 -negative cosignature To do this we need to in-
troduce a certain linear fractional transformation associated with a (m, , ny, Pi; mM, n)- -phase function = = * Boy Ì Assume m -+- p¡ = ứ and let
Go(F) = (aFi* + Bi* + Yi") (4 Fi* + yi* + @j*)-*
(We shall see below that (xFi* + yi* -+ wj*)-1 always exists if He @L°(M, m,.n,) and & = Ệ B 5] is a (m,,71, Pi; 1, n)-phase function.) The maps 9, can
“un y @
be used to parameterize the sets Cg ¿ ,(/) as we now see
THEOREM 3.4 Suppose K is in BH®(M,,,,) and 06¢@2H™(M,,,) and
gy € BH™(M,) are phase functions In addition suppose the associated invariant sub- space Mx 4 ,is pseudo-regular, and let I be the negative signature of (Mx, 9,)’ Then
a there is an analytic (m,,n,,p1; m,n)-phase function & = B l with ny =
Trang 18124 JOSEPH A BALL and J WILLIAM HELTON
Proof By Lemma 3.1 we know that the angle operator-graph correspondence sets up a one-to-one pairing between elements of Cy 4 ,(/’) and shift-invariant nega- tive subspaces of x, 9, of codimension of most /’ ina maximal negative subspace
of L(C") @g~1H°(C") By Lemma 1.1, such subspaces of “, 4, cannot exist if i’< 1, and for Il’ 3/1 coincide with invariant negative subspaces of #/ x 4, of -@y, 9, p-hegative cosignature at most /, = /' —/ By Lemma 3.3, such subspaces
of Mx,9, exist in abundance for J’ >/; indeed if 5 = B l is the
x » @ (mz, 1, P;; m, 2)-phase function associated with the invariant pseudo-regular subspace /, 9, aS in Theorem 2.2, then such subspaces are those of the form
For this negative subspace to have finite codimension in a maximal negative subspace
of H2(C”"), we necessarily have 7, + py=n and (%Fi* + yi* + @j*) (e*) invertible for a.e t The angle operator associated with this subspace clearly is
H= (aFi* + pir + cj) (“Fi* + yi? + œ/*)—1 = G (F),
and is in ZL°(M,,,,,) since Gis a negative subspace Putting all the pieces together,
we conclude that Cy ¿„( + /,) is the set #;(2HNP(M„ n,)) as claimed
1
For this result to be useful it is crucial to be able to compute the negative co- signature of a space (.@x, g, 9)’ directly in terms of the given Ke H*(M,,,,,) and phases
8, To do this let # be the subspace L*°(C”) Q6H(C™); if K is bounded we
simply deñne ỨƑ¿ „(K)*: + H*(C") by Po, (K)*: f> Poa") K*f If & is merely in L°(M,,,,,), use the same formula but insist as well that K*fe L°(C*) Note that I°,(K) may be defined on all of # (and thus be bounded by the closed