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JOURNAL OF FUNCTIONAL ANALYSIS 69, 178-206 (1986)

Beurling-Lax Representations Using Classical Lie Groups with Many Applications. IV. GL(n, R), u*(2n), SL(n, C),

and a Solvable Group

JOSEPH A. BALL

Virxiniu Tech, Blacksburg. Virginia 2406 I

J. WILLIAM HELTOK

Unicersity of C’alijornio, Sun Diego. La Jolla, Culijornia 92093

Communicured by /he Edirors

Received March 13, 1985

This paper concerns the representation of an “invariant” subspace .I of L:(P) as

A = Qff*(cn), (*I

where 8 and 8 ’ are bounded matrix-valued functions on IL, the unit circle. We also consider more general representations .I = [OH” (C)l , where 8 and 8 ’ have square-integrable entries. Such representations are highly non-unique in that many different functions 8 produce the same space 4. In this paper we shall be interested in functions 8 whose boundary values lie in a particular Lie group I- of matrices, that is, Q(e”) E f for a.e. t. The class of all such 6 with L’” functions as entries (resp. with L* functions as entries or with rational entries) will be denoted L”f’ (resp. L*I’ or af). The central question of this paper is: Gioen a purticulur r what are all spaces .A? with u representation (*) whose 8 together with 63 ’ is in L”/; L*f, or WT! The classical theorem of Beurling-Lax-Halmos asserts that one may always take 0” to be in Lx-U(n), where U(n) is the group of unitary n x n matrices. Here we shall settle natural variants on this question when I- is one of the groups GL(n, R), U*(2n), SL(n, R) (Sects. 1.b and 2.a). Besides these simple groups we also do a basic solvable one (Sect. 2.b).

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BEURLINGLAX REPRESENTATIONS 179

Earlier works [B-HI, B-H41 treated the groups of matrices defined to be the isometries of a sesquilinear form on C” or bilinear form on C” or both, while [B-H21 presented a very general representation theorem corresponding to GL(n, C). Results here for SL(n, C) combine with these earlier ones to treat the classical groups with determinant one. Thus we have derived a Beurling-Lax theorem for each of the classical Lie groups.

The groups GL(n, I&!) and U*(2m) are defined as those elements of GL(n, C) which commute with a conjugation or skew-conjugation, respectively, on C”; in Section l.b we actually get more flexible theorems for- mulated in terms of a conjugation or skew-conjugation on L’(C) rather than strictly in terms of GL(n, R) or U*(Zm). Section la is a self-contained summary of the basic facts on conjugations and skew-conjugations which we shall need in the sequel.

The surprising feature of these Beurling-Lax theorems is that they typically have many applications to classical analysis; in particular to factorization of matrix-valued functions, to interpolation and to approximation. For the Lie groups Gl(n, R) and U*(2n) the main application is to factorization and this is described in Section l.c.

At this point our study of the classical groups is complete. We devote Section l.d to summarizing in a condensed form (namely, tables) all of our invariant subspace representation and factorization theorems for them. The classical groups can be characterized as being the commutant in GL(n, C) of one or of a commuting pair of involutions. This provides a unified setting for most of the results in the series of papers [B-H 1 -B-H41 together with sections lh and lc of this paper.

In Section 3 we give a Beurling-Lax theorem for lower rank invariant subspaces corresponding to the group SL(n, C). In Section 4 we describe an occurrence of this in engineering.

NOTATION GUIDE

Let L’(C”) be the usual Lebesgue space of norm square-integrable C”- valued functions on the unit circle, and H2(@“) the usual Hardy space equal to the L*(C”) closure of {eik’x) k 2 0, x E C”}. As usual, the overbar denotes complex conjugation; for Y a subspace of L’(C) we denote {J:jkY’} by 9. F or .!Y’ a linear manifold in L’(@“), [Y] denotes its closure in L’(V).

The m x n matrices are denoted by M,., and GL(n, C) denotes the invertible matrices in M, (=M,,). By L”(M,,,) we denote the matrix-valued functions on the unit circle, rr, with entries in L”. Likewise L*(M,,,) and H”(M,,,) are matrix-valued functions with entries in L2 and H”, respectively. H{ denotes all functions in HP whose zeroth Fourier coefficient

180 BALL AND HELTON

vanishes. The functions in L” U(n), those KE L”(M,) with unitary boundary values a.e., will be called phase functions.

A function F in L’(M,) is Outer if FH=(C”) is dense in H’(@“). Sometimes we use [H”(M,)] ’ to denote the set of functions whose reciprocal is in H”(M,). A subspace A of L’(V) is invariant if e”.X c .X, full range int;ariant if also UN z o {e IN’&} is dense in L*(@“), and simply invariant if also nN ao eiN’.A = { 0 ). We say .4? is *-invariant (resp. full range *-invariant or simply *-invariant) if .a is invariant (resp. full range invariant or simply invariant). Finally let P denote the orthogonal projec- tion of L*(C”) onto H*(@“). If X, Y, Z are Banach spaces with Y and Z subspaces of X, we write X= Y + Z for a direct sum decomposition.

We shall need to refer to a couple of general invariant subspace theorems from other work of the authors.

THEOREM A (Theorem 1.1 in [B-H3]). Suppose JY and ,V” are sub- spuces of L’(@“). Then ./X is simply invariant, ..X-‘ is simply *-invariant and L*(C”) = ./I’ t .I if and on& if there is u matrix function 8 E L’(M,) with imerse 0 ’ E L’(M,) such that

.l= [QH’,(@“)] , ..M“= [OH;(V)]

and QP0 ’ defines a bounded operator on L2(C”). Moreooer, if 0, is another such 8, then 8, = 8U where U E M, is an invertible constant matrix function.

The representer Q is in L”(M,) with inverse 8 ’ also in L “(M,) if and only if also the Schauder decomposition

L*(V)= + r”‘Y, .Y=e” 4i’n.A .I ,= - I

is similar to an orthogonal decomposition. Theorem A extends to a Fredholm setting as follows. Let us say that a

pair of closed subspaces A’ and .I forms a Fredholm pair of subspaces of L*(C) if the span A” + J is closed and if both dim(A.‘n .1) and codim(&“+ A) are finite. By a winding matrix D = D(e”) we mean a diagonal matrix function of the form

D(e”) = diag{ elk\‘,..., erkn’)

for some choice of integers k, G k, d . . . < k,.

THEOREM B (Theorem 1.2 in [ B-H3]). Suppose .,# and A” are suh- spaces of L’(Q)“). Then J? is simply invariant, .,+V” is simply *-invariant and { JI”, .,rY } form a Fredholm pair of subspaces of L*( C”) if and only if there is


a matrix function Q E L’(h4,) with inverse 8 -’ E L’(A4,) and a winding matrix D(e”) = diag{ eikl’,..., eikn’} such that

.I = [QDH”(@“)] , A”= [8H(y(@“)]

and 9P8 - ’ defines a bounded operator on L’(@“). The indices (k, ,..., k, } of the winding matrix D are determined by the subspaces .Xx and .I according to the formula

c (rc-k,)=dim[e’“‘.X’n.I], k, < K

K= “’ - I, 0, 1 )....

There is a useful characterization of when the representing function 8 in Theorems A and B may be taken to be rational or pseudomeromorphic. A matrix function Q on the unit circle is said to be rational (resp. pseudomeromorphic) if all its entries are rational (resp. pseudomeromorphic). A scalar function Q on the unit circle is pseudomeromorphic if it is the boundary value function of a meromorphic function of bounded type in the interior as well as the exterior of the unit disk.

THEOREM C. (Theorem 4.3 in [B-H3]). Let .I and ./I’ be subspaces of L’(C”) as in Theorem A or B.

(a) Then the representing function 8 is rational if und only if both dim[A/.X n H*(F)] and dim[M”/.M-‘n H2(@“)] are finite.

(b) The representing function 8 is pseudomeromorphic if and only if both A n H’(C”) and 2 n H*(F) are full range simply invariant subspaces of LZ(@“).

1. BEURLING-LAX REPRFSENTATIONS WITH GL(n,R) AND U*(2m) AND APPLICATIONS TO FACTORIZATION

a. (Skew)-conjugations on C”

In this section we review the basic facts concerning conjugation and skew-conjugation operators on C”. By a conjugation we mean an operator V on C” which is a conjugate linear involution, that is

and

qax + jly) = !s(x) + p-T!?(y)

W(%(x)) =x

182 BALL AND HELTON

for x, y E C” and a, /? E C. The canonical example is componentwise complex conjugation on C”: %,;I: x + .?. We can generate more examples by composing this with a linear change of basis of C”:

that is,

‘6: gx+gx for .Y E C”,

%:x-+gg ‘x, (1.1)

where g E Gf.(n, C) is fixed. If V? is any conjugation, let us say that x E C” is %-real if V.Y = x and g-imaginary if VX = --.Y. Then for any J EC”, X, = i(x + %x) is V-real and X, = 4(.x - %.x) is %-imaginary and x = x, + x,. Thus C” is the (real) direct sum of the subspace of q-real elements and the subspace of g-imaginary elements. From this it is easy to deduce that any conjugation operator % is similar to standard complex-conjugation %‘,, that is, any conjugation 59 has the form (1.1) for a g in GL(n, C). In particular, if WE GL(n, C) has the property w = 6 ’ then one easily checks that %\,.: x --) w - ’ .f is a conjugation. We conclude that any such w has a factorization of the form w = g g ’ for some g E GL(n, C). Conversely any M of the form w=Rg ’ is easily seen to satisfy the identity w = % ‘. The classical group GL(n, R)= { ge G(n, C)lg =i} arises as the subgroup of GL(n, C) consisting of all g which preserve the canonical conjugation %,:GL(n, R)= (g l GL(n,C)I%,g=g%,}. If w=E ’ and %w:x+w-‘.f is the associated conjugation, note that a g E GL(n, C) intertwines %, with %,,,. (g%,=9?wg) if and only if 2 ‘wg=I, or M.=& ‘.

Similar remarks hold for a skew-conjugation, that is a conjugate-linear skew-involution %:

%(ax + py, = Lwx + js?y,

W(Ux) = -x,

for x,y~C” and a/?EC. In this case the canonical example is %:x-a [ym 0 ‘m] -, Y w h ere n = 2m is even. We generate more examples by composing with a linear change of basis,

or

(1.2)


where gE GL(2m; C) is fixed. If g is any skew-conjugation on C” and 0 # x E C”, it is easy to see that {x, %‘.x} is linearly independent; indeed, %‘x = Ix implies that -x = %“x = Q?(J.x) = Xhpx = 1112x, a contradiction. Similarly, if {x, ,..., x~; %7x, ,..., Ux, } is a linearly independent set and xj + , is chosen to be linearly independent of the span of these vectors, then it follows that

is a linearly independent set. We conclude by induction that C” has a basis of the form {xl ,..., x, ; Vx, ,..., %?x,,, }; in particular n = 2m is even. With respect to this basis, w has the form of the canonical skew-conjugation. We conclude that any skew-conjugation is similar to the canonical one, and therefore has the form (1.2) for some ge GL(2m; C). In particular if w E GL(n, C) satisfies w = -w ~ ‘, then V,;, : x + w ‘X is a skew-conjugation and so has the form in (1.2); from this we see that n = 2m is even and ut has a factorization of the form ~‘=g[ t d-1 g-’ for a ge GL(2m; C). Con- versely, one can verify directly that any H’ of the form w = g[ L

_I ,‘-I g ’

has the property &l-w . The classical group U*(2m) = {ge GL(2m, C)) [i: -,‘-I g = g [E -,‘-I } is just the subgroup of all gE GL(2m, C) which preserve the canonical skew-conjugation: U*(2m) = {gEGL(2m;@)lwLg=flV}, where c=[:L ,‘-I. If H’= -* ’ and g :x--+H.-‘.f is the associated skew-conjugation, note g; GL(2m, C) intertwines the canonical skew-conjugation

that a 59,, with

~,J~~g=g%,.onC”)ifandonlyif~-‘rt’g=[L ,“],or~*=g[i -,‘-]g.-‘.

b. Invariant subspace representations using conjugation-preserving functions

Now we suppose that we have either a conjugation or a skew-conjugation V on the vector function space L’(C) which commutes with the bilateral shift U = multiplication by e”. Then V must be a direct integral of (skew)-conjugations g(t) on the scalars, which by the above discussion have the form %‘(t):x+ W(t)-‘x where W(t)= W(t)-’ (W(t)= -w(r) for the skew case). Thus %’ has the form %= +?‘,,:f-+ W-‘f forf’~ L2(C”) where WE L”(M,) satisfies W= + IV ‘. The type of generalization of the Beurling-Lax theorem which we wish to consider in this section is roughly, given a (skew)-conjugation ?Zw on L’(C”), characterize which invariant subspaces .A c L’(C”) have a representation .A = QH’(C”) where 8 EL ‘(M,) has the property that multiplication by 8 intertwines the canonical (skew)-conjugation & (%$ = $7,” for the conjugation case, $9” = P?~ for the skew-conjugation case) with the given (skew)-conjugation %?,+,; in short, 6 -’ W8 = I,, (resp. 6 ’ Wt3 = [t -,‘-I). In fact, insisting that Q E L”(M,) does not lead to a good result; instead we demand only that 8 E L’(M,). The precise statement is the following:

184 BALL AND HELTON

THEOREM 1.1. (a) Suppose WEL”(M,) is given with W= iii-‘. The simply invariant s&space .A? c L2(V) has a representation A= [@H-(Q)“)] for some QE L’(M,,), with @P@--’ bounded on L’(C”) and 6 ’ W@ = I, if and only if

L2(Cfl) = e “W ‘./a+&

If 8, is another such representer, then 8, = @lJ where U E GL(n, R) is u constant.

(b) Suppose WEL”(M,) isgiven with W= --w ‘, son=2m iseven. The simply invariant s&space .X c L2(@“) bus u representation A= [@HZ(C”)]- for some QE L’(M,) with @PC3 I bounded on L2(C”) and&‘W@=[P_ ,‘-I ifandonlyif

L2( C”) = e “W ‘.A! t M.

If 0, is another such representer, then 8, = 811 where U E GL(n, R) is u constant.

Proof: Suppose .X = [@H”,(C”)] where 8 is as in part (a) of the theorem. Then

=e “W ‘@H”(C”)

has closure equal to e -“W- ‘..a. Similarly, if ..K= [8H*,(C”)] -, where 8 is as in part (b), then

OH,” (c”) = W - ‘6 - [p, -oIqHI;(C”)

= e -“W- 1 QH”(@“)

again has closure equal to e -“W- ‘.I. By the necessity direction in Theorem A, we conclude that .X and .X’ x e ” W ’ .R are, respectively simply invariant and simply *-invariant and that L2(C”) = .Xx t 4.

Conversely, suppose .Af and P “W- ‘.A? induce a direct sum decomposition

L’(@“)=e “W ‘.& + .U.

Since .M is simply invariant, .R is simply *-invariant. By Theorem A (suf- ficiency direction) there is a function 8, E L2( M,) with inverse 8;’ E L’(M,) such that, (i) [@, H”(V)] =A, (ii) [@,Hz(C”)]- = e-“W-‘.,# and (iii) 8, P8;’ is bounded on L’IC”). Apply the transfor- mation N + e “W ‘..B to both sides of identities (i) and (ii) and use that

BEURLMGLAX REPRESENTATIONS 185

w.]= +W to deduce that [W’G, Hi]- =e.“W-‘d and [ W-‘8, H”(P)]. =JZ. We conclude that (i), (ii), and (iii) are satisfied with W-‘8, in place of 9,. By the uniqueness statement in Theorem A, W(e”)-‘B,(e”) = 61’(ei’) U a.e., where UC M, is a constant invertible matrix. From the form of V ’ as 8;’ Wt3, we see that li ’ = 0 in case (a) and U ’ = -0 in case (b). By the discussion in Section l.a, for (constant) matrices, we see that U-’ has a factorization U’=gg ’ in case (a) and CI ‘=g [I” glm]g- ’ in case (b). Now set @(e”)=Q,(e”)g. Then

@(e”)- ’ W(e”) @(e”) =g ‘[@,(e”) ’ W(e”) @,(e”)] g, - ‘IJ ‘g= I” in case (a) g

Ct PI in case (b);

and Theorem 1 .I follows.

Theorem 1.1 has the following extension to a Fredholm setting:

THEOREM 1.2. (a) Suppose WE L”(h4,) is given nith W= W ‘. The simply invariant subspace A c L2(@“) has a representation .A?= [SHz(C”)] for some @EL’(M,) with QP@-’ bounded on L*(C”) and 6 ’ WO = Dfor some winding matrix D = diag{ e”“,..., eiknr} if and only if {e i’W-‘X,.A} f arms a Fredholm pair of subspaces of L2(Cn). When this is the case, the indices {k, ,..., k, } of the winding matrix D are uniquely determined by the subspace .1 according to the formula

k;, (K-k,)=dim[e”” -“‘W ‘..#n.,K].

I

(b) Let WEL”(M,) be given with W= -W ‘, so n=2m is even. The simply invariant subspace .A c L’(V) has a representation .A? = [@Hx(@“)]- for some 63~ L2(Mn) with 8P8-’ bounded on L’(C”) and 6 ’ Wt3 = [ O, ‘0’1 for some m x m winding matri.x D = diag{ eik”,..., elk,’ } if and only if {e -i’ W ‘.A?, .I } forms a Fredholm pair of subspaces of L’(C). When this is the case, the indices {k ,,..., k, } of the winding matrix D are uniquely determined by the s&space A according to the formula

k~K(~-kj)=jdim[e”‘-“‘W~‘.#n.l]. I

Note that the condition 8W@ = D is equivalent to &I$?,, = VW8 and that VW8 = [ “,, t] is equivalent to

186 BALL AND HELTON

Thus one can view Theorem 1.2 also as representing invariant subspaces ..K with functions 8 mapping some canonical conjugation 9?,, or skew- conjugation

to a given (skew)-conjugation %,+,.

Proof. Suppose .x = [@H” (C”)] where 8 E Lz( M,) satisfies 8W19 = D for a winding matrix D, and @P@ --’ is bounded on L*(C”). Since D is a winding matrix, {e WL,(H2(C")), H*(C")) = {D -'H,+(C), II*( is a Fredholm pair of subspaces of L*(C”). As in the necessity part of Theorem B, { [@e “%II(H’ (C”))] , [8HX,(C”)] ) must also be a Fredholm pair of subspaces. But since S%?D = VwQ, we see that [ Qe “W,J H” (a=“))] = [e “V,(8H” (C”))] = e “Ww(..U) = e "W I.#. The argument is exactly the same when D is replaced by [ O,, $1 throughout. The necessity part of Theorem 1.2 follows.

Conversely, suppose that JZ is a simply invariant subspace of L’(C”) for which {e "W-'.d?-,.,M} IS a Fredholm pair. By Theorem A there is a 8, E L*(M,) with 8Pt9 -.’ bounded on L*(C”) and a winding matrix D, such that [S,D, 'H:(C")]- =e "W ‘.,# and [Q’H‘“(C”)] =,N. Since e “Wiw(e "W '.a)= .A+', the first identity implies W ' [e, D, Hz(Q)“)] = .,K, and similarly, an application of e “VW to the second identity gives W ' [ $, H; (C")] = e “W ‘.,R. In particular

[s,Hn'(cn)] = w '[O,D,H'(C")]

and [8,D, 'H;(V)]= W '[B,H,"o]

so the L’(M,) matrix function D, ' 8, ’ W@, is outer and the L’( M,) matrix function 8, ’ WQ, D; ’ is *-outer. If we set H = 8, ’ W@, , we thus have that

(i) H=l? ' in case (a); H= -n ' in case (b).

(ii) D, ’ H is outer. (iii) HD;' = +(D, ’ H) ’ is *-outer.

Thus we may apply the following lemma to H = 6, ’ We,. We defer its proof until after we have completed the proof of Theorem 1.2.

LEMMA 1.3. Suppose HE L’( M,,) satisfies

(i) either (a) H=A-‘, or (b) H= --R-l, (ii) D, ‘H is outer, (iii) HD, '= +(D, 'H) ' is *-oufer

BEURLINCSLAX REPRFSENTATIONS 187

for some winding matrix D, = diag{ciki’ ,..., eiknr}. Then, in case (a) there is outer polynomial matrix function A(e”) = x,?= , ale”’ with polynomial inverse such that H = A-’ DA where D = D, . In case (b), n = 2m is even, each index k, of D, occurs in the set {k, ,..., k, } an even number of times, and there is an outer polynomial matrix function A(e”) = J$VCO a,e@ with polynomial inverse such that H = A-‘[ E ; “1 A where D = diag{ e”“,..., erlmr) is the m x m winding matrix function, each of whose indices I, equals some k, but occurs in the set {I, ,..., I, } half‘ as many times as in the set {k , ,..., k,,, k,, + , ,..., k,, }.

If we assume the lemma, it is now easy to complete the proof of Theorem 1.2. In case (a) we see that

8,. ‘W@,=J- ‘DA

for an outer polynomial A such that A ’ is also a polynomial, where D=D,. Set @=@,A ‘. Since [e, H”(F)] =A and A-’ is outer, we see that [BHz(C”)] - =A, and clearly 8-l W8 = D as desired. In case (b), 8 = 9, A -’ is as desired by a similar argument. The formula for the indices follows from the indicial formula in TheoremB.

Proof of Lemma 1.3. Let N be the number of distinct indices {j, ,..., jh; ) among the indices {k, ,..., k, ) of D, . Note that we may write D, as a block- diagonal matrix D, = diag{ ei/l’I,,, ,..., e ‘j”‘IPy ), where pz is the multiplicity of the index j,. By considering e ““H in place of H and e-.““D, in place of D,, we may suppose that j, =0 and j,>O for 2<x< N.

We induct on the number N of distinct indices {j, = 0, jz,..., j,} of D,. If N= 1, D, = 1,. By (ii) and (iii), H is both outer and *-outer and hence is constant. Therefore, by the discussion in Section l.a, since H= 17. ‘, we have that H=A ‘A in case (a) and H=&‘[P_ 21 A in case (b) for an invertible constant matrix function A. This is the conclusion of the lemma for N= 1.

By induction assume that the lemma holds if the number of distinct indices of D, is at most N - 1, and we wish to prove it for N distinct indices. Write D, as a block 2 x 2 matrix D, = [L 21 where I is the j, x j, identity matrix and A is a (n-j,) x (n-j,) winding matrix all of whose N - 1 distinct indices are positive. If we set L = D, ‘H, by (ii) L is outer and from H= +R-’ we see that ,!- ‘D, = +D, L. Let L= [;;; ,;;I and -- L ’ = [!!! .?!?I be the block matrix decompositions for L and L- ’ com- .?I, 1’22 patible with D, = [A z]. S ince L is outer, x,, and y,, are matrix H’- functions. The equality L. ‘D, = D, L forces

188 BALL AND HELTON

We thus have matrix function identities

(i) j,, = fx,,, (ii) j,*A = *x,,,

(iii) s,, = *Ax,, 7 (iv) jzzA= fdx,,. -

From (i) we see that the entries of x,, are in H’ n 9, and hence x,, - must be a constant matrix. From (iii) we see that the entries of y,, are in zn C” H’ (since all indices of A are positive), and therefore yZ, and ,r2, must be zero. From (ii) and (iv) we see that x,? and xZ2 are at most analytic trigonometric polynomials. Since x2, = 0, and L is invertible, necessarily the constant x, , is invertible, the trigonometric polynomial .Y*? is outer, and the matrix form of L ’ can be calculated directly as

Identities (i)- (iv) now become

(i’) XI;‘= -+x,,,

(ii’) -.Y,,“.Y,2,t,,‘A= +x,,,

(iii’) x2, = 0,

(iv’) 2~’ A = + Ax,,,

where + signs occur in case (a), - signs occur in case (b). Now (i’) implies that there is an invertible matrix cl,, such that

(v) a ‘ea,,, -XII = II where e = I,, in case (a), e = [.t ,‘p] (p=g,)incase (b), by the N= 1 case of the lemma. By the induction assumption, (iv’) implies that there is an outer matrix trigonometric polynomial a,, with inverse also a trigonometric polynomial such that

(vi) Ax,,=G ’ Eu,,, where E = A in case (a), while in case (b) E= [do. [‘I where A’ is the winding matrix with the same indices as A but with half the multiplicities. Finally, if we set a,*=te ~‘~xIz, we may use (iv’), (v), (i’), and (ii’) to see that


from which we see that

(vii) -

ea12 - a,, Ax ?i- 22= II J-12.

If we now set A = [“A’ ;;;I, we see that A and A -- ’ are analytic trigonometric polynomials, and from (v), (vi), and (vii) that

AD,L=D,A,

that is,

where D, = [;; ;.I. In case (a) D, = D as desired and the proof is complete. In case (b), we may write D, = 77- ’ Dn where D is as in the statement of the lemma and IC is a permutation matrix, then the proof of the lemma is complete upon substituting nA for A.

When we combine Theorems 1 .I and 1.2 with Theorem C from the Introduction, we obtain the following:

THEOREM 1.4. Let .X be a simply invariant subspace of L’(C”) as in Theorem 1.1 or 1.2.

(a) Then the representing function 8 as in Theorem I. I or 1.2 is rational ~fand only Q-also dim[.A/.,K n H’(C”)] < CX.

(b) Then the representing function 0 as in Theorem 1.1 or 1.2 is pseudomeromorphic if and only lf also .X n H’(@“) is a full range simpl) invariant subspace of L’(C).

Proof If dim[.&/.An H’(C”)] < x, then also dim M”/[.,tVn

H2(Cn)] < cc where A” = e “X. Similarly, if A n H’(C”) is full range simply invariant, then .X” n H’(@“) is full range simply *-invariant as well. Theorem 1.4 now follows from Theorem C and the construction procedure for the representing function 8 in Theorem 1.1 or 1.2.

c. Applications to Factorization

Much as in the spirit of [B-H3], invariant subspace representation theorems such as Theorems 1.1 and 1.2 above have immediate applications to factorization. For KE L”(M,) we define the Toeplitz operator T,: H’(@“) -+ H’(V) by TK(f)= PH~(c:nj(Kf). Note that TK(f)=O if and only if KfEH’(C”)‘; if also K-‘EL=(M,,), we see that dim Ker T,=dim[H2(@“)‘nKHZ(C”)]. Note also that T,(f)=g (where J ge H’(V)) if and only if Kf - ge H2(Cn)‘. From this we see that

dim[H2(Q)“)/T,H2(Q=“)J

= dim[L’(Q=“)/[H’(@“)’ + KH2(@“)]].

190 BALL AND HELTON

In particular, Y, is invertible if and only if we have the direct sum decomposition L*(C”) = H*(C”)’ i KH*(C”), and T, is Fredholm if and only if {H2(cn)‘, KH*(u?)} . IS a Fredholm pair of subspaces of L’(C”) (see the Introduction for the definition). These observations will be useful to us below.

We first present a very general type of factorization theorem corresponding to Theorem 1.1.

THEOREM 1.5. LPI W and F he t~lo given jitnctions in Lx (M,) klh inverses W ’ and F ’ also in L ’ (M,,).

(a) Suppose W= WI-‘. In order that F have a factorization F= QG where 63” E L’(M,), @PC!- dgfines a bounded operator on L’(C”). 6-l WQ = I, and G” E H*(M,,), it is necessary and sufficient that the Toeplirz operator RF I wc’ on H*(C”) he invertible.

(b) Suppose n = 2m is even and W = - p ‘. In order that F have a factorization F = 8G nlhere 8 * ’ E L2( M,), QP8 ’ defines u bounded operator on L*(C”), 63 ’ W@ = [P, i-1 and G f ’ E H*( M,), it is necessary. and sufjicienr that rhe Toepplitz operator TF 1 Ic’b on H’(C”) he invertible.

(c) If W and F are rational (resp. pseudomeromorphic) in (a) or (b) [hen so also are 8 and G.

Proof: Theorem 1.5 is equivalent to Theorem 1 .I (including its refinement in Theorem 1.4) for the special choice of the subspace .M as FH*(C”). Indeed, if . K = FH’(C”), then e “W ‘.@=

e’ “W ‘FH*(C”) = W- ‘FH’(C”)‘. Upon multiplying by P ’ W, we see that L*(C”)=e “W I.2 t .K if and only if L2(Cn)= H’(C”)l t F-’ WFH*(C”); by remarks above this is equivalent to the inver- tibility of the Toeplitz operator TF ’ WF. On the other hand, if there is matrix function 8 E L’(M,) with inverse 8‘ ’ E L’(M,) and 8P8 ’ bounded on L*(C”) such that FH’(C”)= [@Hx(Cn)] - and 8 ’ W8 = I,, in case (a) (&‘Wf3= [L -p] in case (b)), then F=@G where G=@-‘F has G” E L*(M,) and [G -‘H”(C”)] = H*(C”). This last condition is equivalent to G * ’ E H*( M,).

Finally, if W and F are rational, then dim[ H’(V)/

(FH*(C”)n H’(C”))] < cx: as well as dim[H*(C”)/(e “W-‘FH*(C”)n H*(V))] < 3c;, and hence 8, and therefore also G = @-IF, is rational. Similar arguments hold if both W and F are pseudomeromorphic. This completes our sketch of the proof of Theorem 1.5.

Theorem 1.2, in a completely analogous way, leads to the following non- canonical factorization theorem.


THEOREM 1.6. Let W and F be two giuen functions in L”(M,) wifh inverses W ’ and F- ’ also in L”(M,).

(a) Suppose W= W’- ‘. In order that F have a factorization F= 0G where 0*’ E L*(M,), 0P0’ defines a bounded operator on L*(C”), 0- ’ W0 = D for some winding matrix D, and G *’ E H*( M,), it is necessary and sufficient that the Toeplitz operator Tr I wr on H*(@“) he Fredholm.

(b) Suppose n = 2m is even and W = - W ‘. In order that F have a factorization F= @G where 0 * ’ E L*( M,), 0P0 ’ defines a bounded operator on L2(Cn), 0 ’ WO = [i .;n] for some winding m x m matrix function D and G * ’ E H*( M,), it is necessary and sufficient that the Toeplitz operator Tr I ‘,,r be Fredholm.

(c) If W and F are also rational (resp. pseudomeromorphic) in (b) or (c), then so also are 0 and G.

For the reader’s convenience we offer some special cases of Theorems 1.5 and 1.6. The next theorem ( 1.7) is an analogue of “spectral factorization” for a self-(skew)-conjugate-valued function W rather than a self-adjoint valued function W.

THEOREM 1.7. Let W and its inverse W- ’ be gisen matrix functions in L * (M,). Then there is a matrix function G with G * ’ E H*( M,) and G - ‘PC a bounded operator on L*(C”) such that

w= If and only if

(I) G ‘G W=@ ’ and Tw is invertible

(2) G-’ ; [ 1 m -2 G W = - w-’ and T,,, is invertible

(3) G ’ DG for some winding matrix D

w=w ’ and T,. is Fredholm

G for some W= - w-’ and T, is Fredholm

winding matrix I)

Our next theorem amounts to innerouter (or phase-outer) factorization with the unitary group U(n) replaced by either GL(n; R) or U*(2n).

THEOREM 1.8. LRt F be a given matrix function in L”(M,) with inverse F ’ also in L”(M,). Then F has a factorization F=QG with G” E H*(M,), G ‘PC bounded onL*(C”) and with

192 BALL AND HELTON

If and only if

(I ) e(P) E GL(n, R)

(2) fqe”)E u*(2m)

(3) 6 ‘@=D,Dsome winding matrix

TF ‘b is invertible

T. F ‘[,2 ,:-I’ is invertible

T&.1, is Fredholm

for some winding matrix n

Remark. All the results of this and the next sections have extensions to L”(F) for 1 <p < x. Indeed [B-H3 J contains the appropriate LP-version of Theorems A, B, and C, and all the other techniques have easy Lp- extensions. We leave the statements and proofs to the interested reader.

d. Groups D+ned by! lncolu~ions

We indicate in this section how many of the invariant subspacc representation and factorization theorems of our earlier work can be unified by introducing the concept of an involution. Let g +g’ be an involutive automorphism on GL(n; C), that is

(i) (gIgr)‘=g;g; and (ii) (g;)‘I=g, for all g,,g: in GL(n,C). Let us say that the

involutive automorphism ’ is Type I if

(iii, I) F‘ E H” (M,,) whenever FEH’(M,,) and F‘EH’(M,,) whenever FE H” (M,,),

and is Type III if (iii, II) F’E H”(M,,) whenever FE H’(M,) and F’E H”(M,,)

whenever FE H” (M,,). The main examples of Type I involutive automorphisms arc

the main examples of Type II involutive automorphisms are g -+ g’ ‘,


where also 0 <p < n and m = n/2 where n is even. It can be shown that a biholomorphic involution of Type I or II on GL(n, C) is similar to one of the above examples.

Groups of matrices arise in the following three ways:

(I) g(x) = { gE GL(n, C)lg=g”} where -’ is a Type I involutive automorphism.

(II) g( + ) = {g E GL(n, C) 1 g = g+ } where + is a Type II involutive automorphism.

(III) 9(x, +)= {gEGL(n, C)Ig=gX=g’} where x and + are commuting involutive automorphisms such that x is Type I and + is Type II.

By checking the list of classical simple Lie groups (see [Hlg] ) one sees that they are all of one of the above types. The group g(x) with g”= c5 -:-,I g*-‘[t -‘J is the classical group U(p, n-p), the main concern of [B-Hl, B-H2]; if g” = g, B(x) = GL(n, R) while if g”= c-p, 51 &T; -,lm] then 5’(x) = U*(2m), the main concerns of Section 1 of this paper. Also invariant subspace representations using groups of the form Y(x, + ) and c??( + ) were the main concern of [B-H4]. Thus general theorems on groups of the form F?(x), g( + ), and 9(x, + ) serve to unify all this work.

The next theorem uses the following notation. If &? is a full range simply invariant subspace, J% has a Beurling-Lax representation Jlil = ryH2(Cn) for a !P with Y * ’ E L”(M,). If g + g” is an involutive automorphism of Type I, let A” denote the subspace Ax= !PH’(@“)‘; similarly, if N = YH*(@y is full range simply *-invariant, we define JV-’ by JV” = !PH*(@“). By the definition of Type I, J# + Jz” is a well-defined involution on the collection of full range simply invariant or *-invariant subspaces. If g + g+ is a Type II involutive automorphism, we define [ !PH2(Cn)] + = F’H*(@“) and [ !P~z(@“)~]+ = !P+H’(@“)‘; by the defining property of Type II, this also is a well-defined involution of invariant subspaces.

The following theorem summarizes the invariant subspace representations from [B-Hl, B-H2, B-H41 and Section 1.b of this paper in the unifying language of involutions. Direct proofs of this theorem and of the next are straightforward adaptations of the techniques from Sections 1.b and c and [B-H4], and so will be omitted here.

THEOREM 1.9. Let JZ be a given full range simply invariant subspace of L2(@“) and let g+g” and g-g+ be given commuting involutive automorphisms of Types I and II, respectively. Then J.& has the representation A = [@Hm(@*)] ~ for some matrix function 0 with 0” E L*(M,), such that @PO-’ defines a bounded operator on L’(C) and such that

194 BALL AND HELTON

If and only if

(I) t3(e”)EqX)

(2) tqP”)E%( +)

(3) 8(r“)EqX, + )

(4) 8’-’ WC3 = constant where W= Wvm’ is given

(5) 8 - ’ V8 = constant where V = V ’ is given

(6) 8’ ’ W@and8+~‘Warc constant, where W= W‘-’ and V = V ’ ’ are given

(7) 8’ ‘We=D where W=W‘ ’ is given, D = D r ’ 0 LI winding matrix in a canonical form appropriate for .Y

.K‘ t .u = L’(C”)

U=.K’

..U‘ t .H = Lz(C”) and ..K = .U ’

w ‘.U^ -f .v = L2(C”)

.u= v. ‘.U‘

lv’.Af” -f .u = L?(F), .x = v 1.u - and(V’WV ‘W’-‘).U’* =.U”

( W ‘.X ‘, .X } is a Fredholm pair of subspaces of L2(C”)

The reader should note that statements (1) (2), and (3) in the theorem follow from (4), (5), and (6), respectively, but with the extra content in (I), (2), and (3) that the constants can be chosen to be the identity.

Finally we give a summary of the general factorization results arising immediately from these invariant subspace representations.

THEOREM 1.10. Let W, V, and F he given functions in L fi (M,,) with inverses W- ‘, V- ‘, and F- ’ also in 1~ ’ ( M,, ), g + g ‘. und g 4 g + he given commuting involutive uutomorphisms of‘ Type I and Tripe II, respectively, and suppose that W= W’ ’ and V = V’ ‘. In order thut F has u factorization F= 8G with G’ ’ E H2( M,) such thut G ’ PG dsfines u bounded operator on L’(C”) and such thut

It is necessary and sullkient that

(I) @km’VVeisaconstant T,. ‘WF is invertible

(2) 8’ ’ ve is a constant (F’ ‘VF)“E H’(M,)

(3) both 8” ’ IV8 and TF, 1 ,,isinvertible,(F’.‘VF)f’EH’(M,) 8 - ’ ve arc constant and [P ‘V-‘~--‘V~WF]“EH”(M,)

(4) 8’ ‘We=DwhereD=D’ ’ T,, ‘,,isFredholm is a winding matrix in an appropriate canonical form


TABLE I

Type I involution x Canonical form for winding matrix D = D”-’

(I) g”=g,--’

D(e") =

I where D,(P)=

diag{&“,..., &‘}

In connection with table line 7 in Theorem 1.9 and table line 4 of Theorem 1.10, Table 1 gives the appropriate canonical form for D.

2. BEURLINC-&LAX REPRESENTATIONS WITH SL(n, C) AND

A SOLVABLE GROUP

a. SL(n,@)={~:~GL(n,@);detg=l1

For this group we consider only rational full range simply invariant subspaces ./K because of technical difficulties for the general case. We begin our treatment informally and just give the basic idea. Suppose .I = QH2(C”) where 8 is a rational function in LX (M,) with det 0 = I. If z0 is a pole of 8 inside the unit disk, then z0 also is a zero of the same order. This induces a constraint on .X; the “order of z0 as a pole of the subspace .X” must equal the “order of z0 as a zero of the subspace .X.” All that remains is to make the expressions in quotes precise.

So suppose M is a rational full range simply invariant subspace. Adapting the definitions of [B-G-K] from a single matrix function to a subspace of vector functions, we say that a chain (x0, x, ,..., x- , } of vectors in @” with x0 # 0 is a zero chain for the suhspace .A? at the point z. in the disk if, for all f~ .&‘, xl,f(z) is analytic at z. for k = O...., I- 1 and

196 BALL AND HELTON

xi-0 [xif]“.-“(;,,)=O forj=O, I,..., I- 1. At any given point z0 in the disk, choose a zero chain {-\-A, .Y: ,..., ,~j, 1 j of maximal length. Then choose any subspace Y, of C” complementary to the span of the vector x,‘). Choose a maxima1 zero chain (possibly vacuous) {x:, .~f,..., .v;, , ) of maxima1 length such that .Y: is in the subspace .$d. Continue in this way. If .K is a rational full range simply invariant subspace, this process must stop, say with k nonvacuous chains. One also proves that the lengths of the various chains is independent of the choice of maximal chain and of complementary subspaces at each stage. We define the order of‘q, us a zero of the suhspace .X to be the nonnegative integer ZV(=,,) 2 C”,-, I,.

Similarly a chain of vectors { yo, y, ,..., y7 ,I in C” with y, # 0 is said to be a pole chain for the subspace .I ut the point z0 if there is some function g in H2(Cn) such that cj:tyj(z - z,j)r ’ + g is in , K. Given a point zo, choose a (possibly vacuous) pole chain { yh, yt ,..., yf, : of maxima1 length 7,. Then choose a subspace .q, complementary to the span of the vector yd. Then choose a pole chain { y;, yi,..., J$ , ) of maxima1 length with yi in .Y, . Continue in this way. The process stops with finitely many (say k) nonvacuous chains if .I is rational. We define the order of 2” as II poft~ of‘the subspace 4 to be the nonnegative integer P,,(z,) & x5-, 7,. Moreover, if J? is rational, then there can be at most finitely many points z0 in the disk D where Z.,(z,,) or P.,(zO) is nonzero.

THEOREM 2.1. Suppose .A is a rational fill range simply invariant subspace. Then A = @H*(C”) with QE L”SL(n, C) if and onlJ1 [/ Z,,(z,) = P.,(z,) fiw all points z. m the unit disk. Moreover in this case 8 can he taken to he a scalar outer multiple of a rational matrix function.

Proof: Bart, Gohberg and Kaashoek [B-G-K, Chap. II] define “the zero multiplicity Z(e, q,) of Q at .q)” and “the pole multiplicity P(@, zo) of Q at zc for a matrix-valued function which is meromorphic at zo. If .X is a rational full range simply invariant subspace of L’(C) such that ~2 = @H’(C”), it is a simple matter to check that

and

Z,,(z,) = the zero multiplicity of Q at z0

P.,(z,) = the pole multiplicity of 8 at z0

Also the difference Z(Q, .zO) - P(@, zO) is the order of q as a zero of det 8 (if positive), and the negative of the order of z0 as a pole of det Q (if negative). We conclude: det 63 has no zeros or poles in the unit disk if and onfy $ Z.,(z,) = P.,(z,) for all z0 in D. Thus if det 8 = I, then Z.,(q,) = P,,(z,). Conversely, suppose Z.,(z,) = P,,(z,,) for all z. in D. Then det Q has no zeroes or poles in D. Since 4 is a rational subspace, any representing 8 is (rational). (outer), so det 8 having no zeros or poles


on the disk is equivalent to det t3 being outer. Then det 63 has an outer nth root (det @)I”‘, and if we set 6 = (det @) -““6, then A = &H*(C”) and det P(z) = 1.

Note that if A is a subspace as in the theorem, we can always represent A as 8HZ(C”) where 8 E WM, and det Q is outer. Normalizing 8 to have determinant 1 involves multiplying by a scalar algebraic (in general non- rational) outer function.

If we had used the representing 8 in @U(n) guaranteed by the classical Beurling-Lax theorem, then we would actually have proved that there is a 8 in L”SU(n) representing A. Clearly this could be generalized to give results for L”SU(p, q), L”Sp(n, R), and the other classical Lie groups. The point is that if A= @If’(V) with a Q in 3ier then the condition “Z,(z,) = P,(z,) for all zO in D” gives that det 0 is outer. Thus (det 63)“” is outer and 8, & @/(det 8) Iin has determinant one and is still in Lx I: The function 9, is in L”ST and .A = [@(I-I”)] . b. f={g~GL(m+n,C):g=[; f]}

This is the group I- of matrices g with block-triangular decomposition g= c;i fi] induced by the direct sum decomposition Cm +n 2 C” @ 43” of Cm+“. lb the classification scheme for Lie groups [Hlg], this is one of the more common types of solt~hle Lie groups (as opposed to the simple Lie groups which we have been studying up to now). Formally clny full range simply invariant subspace A of L’(@‘“+“) such that An [L2(Cm)@ {0}] is a full range subspace of [ L’(C”‘) @ {0}] h as a Beurling-Lax representer 8 with the values @(e”) in the group f, but Q may be badly unbounded. Instead of making this precise, we give necessary and sufficient conditions for the representer and its inverse to be bounded and a sufficient condition for the representer to be in L2.

We shall write elements of L2(Cnrc “) as [L] where .f~ L’(@“) and g E Ll(F). Let P,: LZ(Cm+” )-{[~]EL*(C~+~)~~~L~(C~)} and P, : L2( @” + ” )- ([,“]EL2(Cm+” ) 1 go t*(C”)} be the orthogonal projec- tors.

THEOREM 2.2. Let .M he a full range simply invariant s&space of L2(Cm+n). Then

(a) .M = 8H2(@” + “) with f3 and Q ’ in L”T $and only if P,.M is u closed subspace and A n P, L’(C”+ “) is a fill range subspace of P, LZ(Cm+n).

(b) If P,.lnL”(C”‘“) = (P,.M) nLz(C”‘“) and .AnP,L’ (Cm+” ) is a fulf range subspace of P, L’(C” +“), rhen ,ff = [QH” (C” ’ “)I with 8 and 6? ’ in L’T.

Prooj: If A = @H’(C”+“) where both 8= [;; F] and 8-l = C’“’ a,!4y-‘] arein L”(M,+,), then P2.A= { [,“I 1 gEyH*(C”)} isclosed

198 BALL AND HELTON

by the boundedness of 7 - ’ and .A n P, L*(C” + “) = [ XH*hCm)] is a full range invariant subspace of P, L*(C” + *).

Conversely, suppose to start with that P,.d n Lx(Cm +“) = (P,.X) n L-*(v+“) and An P, L2(Cm+m ) is a full range invariant subspace of P, L*(C” +.). Since .I is full range and simply invariant, {gEL*(c”): [;]E(P*J?-} is a full range simply invariant subspace of L2(Cn). By the classical Beurling- Lax theorem, (P,Jl) = [,Jc”,] for a phase function o E L”(M,); in fact we are willing to settle here for any Q with o and o -’ in L”(M,). By assumption .Af n P, L*(C’“+“) is a full range invariant subspace, so has the form .A n P, L2(Cm+“) = [+“zo(c”‘] for a phase function + in L”(M,). Let x be the subspace L*(V) 0 $H*(C”). For each f such that [,,‘$I E P,.M, a dense subspace of H*(C”), there is a unique 2” in x such that [Xi] E 4. Moreover the mapping X so defined is a closed operator since .X is a closed subspace. The assumption P2.XnL”(C”‘“)=(P2d~) nL‘(C”‘+“‘) implies that the domain of X includes the constant functions in HL(V). Deline an operator-valued function FE L*(M,,,,) by F(e”) c = (Xc)(e”) for c~ 43”. Then it is easy to check that

={[:I H’(C”)) +[‘H;c”‘]

= (H*(C”)OH’(C”))}

={[; ;]Hx(Cm+,i)} .

Moreover, if 8 = [$ t;], then 8 and Q -’ are in L*( M, +,) since I+?, II/ ~ ‘, u and W-’ are uniformly bounded on the unit circle and F is in L*(M,,,,). This gives the desired Beurling-Lax representation for A.

If in fact P2.X is closed, then Dam(X) is closed, so X is bounded by the closed graph theorem. In this case it still might happen that the matrix function F as defined above is unbounded. However by the lifting theorem of Sz.-Nagy and Foias [N-Fl], one can show that there is a KE W*Mn,,) a~ ’ such that )I F+ K(I oc, = IIXlI < ,x. If we set F, equal to F+K and 8, equal to [ f L;], then 8, is a bounded Beurling-Lax representer with bounded inverse.

3. ORBITZ AND ANALYTICITY

The main mathematical issue of this section is orbits of linear fractional map representations of some of the groups we have been discussing where

BEURLING-LAX REPRESENTATIONS 199

analyticity of certain functions is crucial. We concentrate on SL(n, C) since the corresponding problem for U(m, n) is the classical ‘Dartlington embed- ding’ of circuit theory (cf. [Ar; D-H]) which is treated in the subspace setting [B-H 11.

We shall begin with a representation theorem for degenerate (nonfull range) invariant subspaces for the group SL(n, C), and then show how this is an “orbit” theorem and in the next section sketch an occurrence in system theory.

Recall that L”SL(n, C) is the set of matrix functions g in L^(M,) with g(e”)ESL(n, C) for a.e. t. By Cramer’s rule, g-’ E L”SL(n, C) whenever geL”SL(n, C). A more natural object than .%‘SL(n, C) (rational matrix functions in L”SL(n, C)) is the set dSL(n, C) of g in L”SL(n, C) of the form sR, where s is a scalar outer function and R is a rational matrix function with no zeros or poles on the unit circle. Note that a rational matrix function R in L”(M,) n (L” (M,)) ’ can be normalized to a function K= sR in WSL(n, C) by multiplication by a scalar outer s if and only if det R is outer. For our engineering application we shall be interested in the subgroups H”SL(n, C) A L%SL(n, C) n H’(M,) and .$H”SL(n, C) P 8’SL(n, C)n H’(M,). Again by Cramer’s rule, g ‘EHrSL(n,@)(KH”SL(n,C)) whenever gEHnSL(n, C)(.%H”SL(n,C)). Finally we define the runk of a simply invariant subspace dK of L’(C”) to be the (uniform) spectral multiplicity of the unitary operator M,, 1 .,@, where J?= [UNr,,e jN’.M] is the doubly invariant subspace generated by &I. The following is a Beurling--Lax theorem for degenerate invariant subspaces corresponding to the group H”SL(n, C). For r <n, If*( {0, ,} $ C’) denotes the set of H’(V) functions which take values in the subspace {0}@@‘c@“~‘@@‘~C@“.

THEOREM 3.1. Let A c H’(@“) be on invariant .&space of rank r <n. Then .I=gH’({O, .,}OC’)/ or some g in H”SL(n, C) {f and only if .I bus an invariant direct sum complement ~ I‘ in H’(C”), that is H’(C”) = A + .N for some .N with ei’M c .A“.

Moreover, if .I is a rational subspace (i.e., ..X = 6)H2(@‘) where OE H”(M,.,) is rational), then g may be taken to be in .$H”‘SL(n, C).

Proc$ If g E H”SL(n, C) then g and g -’ are in H”‘(M,), so H’(F) = gH’(C”). Thus the direct sum decomposition H2(Cn) = H2( {0,, ,} @C’) + H’(C”-‘@ {0,}) for H’(V) implies

H2(C”)=./Z j- .N,

where .M=gH’({O,, ,}@C’) and N=gH’(C”-‘@{O,}). Conversely suppose J? c H’(V) is invariant of rank r with invariant

direct sum complement ./lr. Represent & as OH2(6’) and JV as $H’(@” ‘)

200 BALL AND HELTDN

via the classical Beurling-Lax theorem. Then set 8 4 [$0] E H”(M,); clearly @H’( { 0, ,) $ C’) :: 6)H2( C’) = .X. Moreover, the computation

@H21@“) = l//H2(C” ‘) + f9H2(C)

= ,t- + .H = fP(C”)

shows that 6J maps H’(V) onto itself. Note next that 0 = 8 [ /] = Iclf+ Oh for SE H’(@“- ‘) and h E H’(C’) implies that 1+5/= 0 and 0g = 0 since ..+’ n Jf = { 0). Since Ic, and 6’ are one-to-one mappings, this forces f= 0 and h = 0 and we see that 8 also is one-to-one as a mapping on H’(C” ). This implies that both 63 and Q ’ are in H”(M,,), and hence det 8 is outer. If we let s be an outer nth root of det 8, then g e s@ gives the desired representation of -4. The asertion for the case of a rational subspace J4 will follow from the proof of Corollary 2.3 to come.

There is another way to view this theorem which we need later. First, write g as a block matrix [: P J, where the blocks arise from thinking of g as acting on C”-‘$43’zC@“, and express each rank r invariant subspace JX as the graph of a multiplication operator

for some SEH’(M, ,,,I whenever possible. Then, if .H, and J& arc graphs of S, and S2, we have g.k(, = .M2 if and only if

s2 = (as, + /l)(KS, +-j) ‘.

Define $ to equal this linear fractional map whih sends S, to S2. Clearly, ..ai; = I &,) J is the graph of 0 and the .d’s described in Theorem 3.1 which are graphs actually correspond to the orbit

O. ii {S = $JO): g E H” SL(n, a3 ) )

of 0 under linear fractional maps with coefficient matrix in H”SL(n, C). In the future we shall often identify H”SL(n, C) with the associated linear fractional maps (4 / gc H”St(n, C>). Also of interesf is the orbit 0, of any ME M, r.,. In subspace terms this merely equals all 4 of the form

for some g E H x SL(n, C 1. Cfearly, the class of .I’s which arise in this way equals precisely those .1 in Theorem 3.1.

To convert this to algebra we need a definition. Two matrix functions N and P are called strictly left coprime provided there exist H”’ matrix

BEURLINGLAXREPRESENTATIONS 201

functions X, Y so that XN+ YD = Z. The Corona theorem as generalized in Fuhrman’s thesis says that such N and P are those for which

N(z) N(z)* + P(z) P(z)* 2 E z

for some E > 0 and all IzI Q 1 (cf. [Fl]). An easy consequence of Theorem 3.1 is

COROLLARY 3.2. (a) The orbit 0, of any A4 in M, ~ I r under H”SL(n, C) consists of all functions P of the form P= ND-” where NE H”(M,- ,,,) and D E H”(M,) are strictly left coprime. In particular, this includes a dense subset L “(M,).

(b) The orbit 590, of any M in M,-,, under &HmSL(M,) includes all of WM, - T,T.

Proof: A function P is in 0, if and only if P = ND-’ where d = [K] H*(C) is as in Theorem 3.1. From M c H*(C) we conclude that NE Hm(M,-T,r) and DE H”(M,). We also have that there exist A, B such that g 4 [4 z]~H~(iV,)n [H”(M,)]-‘. Write gP1 as [f 71. Then XEH~(M,,,_~), Y~H~(ikZ,)and [XY][g]=XN+YD=Z,.HenceNand D are strictly right coprime.

Conversely, suppose N and D are matrix H” functions such that there exist Hm-matrix solutions X, Y of XN + YD = Z,. To show that P= ND-’ is in OM, we must show that & = [,“I H*(@‘) is as in Theorem 3.1, that is, has an invariant direct sum complement M. Set JV equal to Ker[X, Y] in H*(C). Then JV is invariant, 4? n J = (0) and J+! + JV = H*(@“), since any f in H*(C) can be decomposed as

f=[J wlf+jf-[;] Wlf.),

where [g][xY]f~A and f- [g][xY]f~N. Finally suppose P is any rational function. Then P has a rational

coprime factorization ND-’ (see, e.g., [F2]), so there exist rational H”- solutions X, Y of XN+ YD = Z,. From this one concludes that both 4 = [,“I H*(C) and JV” = Ker[XY] are rational invariant subspaces, so the function 8 in the proof of Theorem 3.1 is rational with both 8 and 8-l in H”(M,). This implies that g =sO is in &!H”SL(M,) and d=g c &‘c,J as desired.

A problem complementary to describing the orbit 0, = (5&(M): ge H”SL(n, C)} is to fix a g in H”SL(n, C) and ask for a description of the range of 4 as a map on H”(M,-,,), that is, of

202 BALLANDHELTON

We give a solution in the following corollary; the solution is less explicit than that for the group U(m, n) (see [B-Hl]), but is adequate for an application which follows. The second statement in the corollary is given as a contrast to the situation for the group U(m, n).

COROLLARY 3.3. Suppose g = [: f] E H”SL(n, C) where the block decomposition is induced by raking C” = C” ‘0 C’. Then

(i) a function K is of the form gR( H) for some HE H”(M, ,.,) if and only if K has right coprime factorization N,D, ‘( NKe H”(M,. ,,,), D, E H”( M,)) such that

H2( C’).

(ii) If 3g(H”(M, ,.,))c H”(M, ..,), hen %“(ff” (M, .,.,))= H”(Mn. ,.r).

ProoJ Since g and g -’ are in H”( M,), multiplication by g or by g ’ is a linear isomorphism on H’(C). Also note that an invariant subspace ..9; c H2(C”) has the property H’(C”) = [6] H*(C” ‘) -f- .A: if and only if -4; = [‘j’] H’(C’) for an HE H”(M, ,,,). Since g[h] H’(C” ‘)= [ :] H’(@“- ‘) and g is an H’(V)-isomorphism, it follows that an invariant subspace ..JV c H’(C”) has the property

H’(C) = -LY 11 s

H2(C” ‘) t .t‘

if and only if .,Y‘=g[‘:] H’(@‘)= [:“,:!I H’(C’) for some H in H”(M,-,,,). Since g. Co , ’ H] has inverse in H’ (M,), it is not difficult to see that {rH + /?, KH + 7) is right coprime for all such H. We conclude that the set K= {(TH+P)(KH+~) ‘: HEH=(M,. ..,)} is characterized as claimed.

Now suppose that sR( H’“(M, _ ,.,)) c H” (M, .-,. ,). If K # 0, we can choose HE H”( M,,. ,.,) such that

(KH+.I’)(zJx=O

for some Z,,E D and x E c’, x #O. We claim that then (CXH + p)(z”) #O; otherwise, we could have g(z,)[“(,?‘)] x = 0, a contradiction to

g ’ E H”(M,). Conclude that (ci(H) must have a pole at zO. Thus, if ~g(H”(M, -,., ))c H”(M, ,.,), then K=O and g= [; &I. From gEH”Sf.(n,C) we see that x,z-‘EH~(M, ,), K,K- ‘EH~(M,,), and BEH=(M, ,.,). But then it is clear that

{%“(H)=(EH+/?)K-~: HeH%(M,, ,,,)}


4. AN ENGINEERING OCCURRENCE

We next describe an occurrence of these ideas in system theory; for a different approach more in the spirit of classical network theory, see the engineering article [B-H5]. The system z, described by the diagram in Fig. 1 mathematically corresponds to the system of equations

a, = Ca,,

a,=a, +d,

a,= Pa,,

a4= -a,+r.

Here C and P are matrix functions in L&(M”) and a, (i= 1. 2, 3,4), d and r are vector functions in L’(@“). The system is said to be internally stable if all internal signals a,, a,, a3, a4 are in Hz(Cn) whenever the inputs d and r are in H2(C”). Given a fixed rational P, the objective is to parametrize all C which produce an intrnally stable system. This was done by Youla, Jabr, and Bongiorno [Y-J-B], then simplified by Desoer et al. [D-L-M-S], and further simplified by Francis and Vidyasager [F-V]. The parametrization turns out to be very useful in solving various systems problems [Z-F; D-L-M-S; Y-J-B]. Our first observation is that their answer can be phrased as follows in terms of orbits.

THEOREM 4.1 [Y-J-B]. Let P be a rational matrix function with P and P ’ in L”(M,). Let g be anyfunction in .G%?H*SL(M,,) such that

(Fc:, ;/q(: $O)= -p.

Then the et ofC in L”(M,)f or which the system in Fig. 1 is internally stable equals the set {S”(W): WE H”(M,,)}.

Note that the existence of a function g as in the statement of the theorem is guaranteed by our Corollary 3.2. Also the existence of a g in 9H”SL(M,,) with

4: (l/q rJO)= -p

d

FIGURE I

204 BALL AND HELTON

is equivalent to the existence of a bounded uniformly invertible rational outer matrix function h with g,,(O) = -P.

The following criterion for stability together with the invariant subspace ideas presented above will be seen to lead to a new proof of Theorem 4.1.

THEOREM 4.2. The system L,, (Fig. I ) is interna1l.v stable if‘ and only if the mutrix ,function g = [ ‘!hp z:] has inverse g ’ in H”( M,,). Here P = N, D; ’ and C = N, D; ’ are right coprime .factorizations.

Proof. Consider the system Z, in Fig. 1. Key internal variables are determined from inputs via the equations:

Stability means (“,:) E HZ whenever (p) E H’. Thus we see that the system is stable if and ony if [ IP y] - ’ E H’. Let [ -‘P y] = ND --’ be a right coprime factorization of [ IP : 1, so [ .Ip ‘;I -’ = DN ’ (if it exists). By the coprimeness of N and D, this is in H” if and only if N ’ E H”, i.e., if and only if N is outer.

To relate this criterion to the theorems above concerning outer linear fractional maps $, we construct a specific useful coprime factorization of [ m’P 71 as follows. Let P = N,D, ’ and C = N, D, ’ be right coprime factorizations of P and C, respectively. Thus there exist Hm-matrix functions X,, Y,,, X,, and Y,. such that

X,N,+ Y,D,=I,

X, N, + Y,.D,.=l.

From P= N,D;’ and C= N, D, ‘, we see that

Since the first factor and the inverse of the second factor are in H”, this is a candidate for a right coprime factorization of [ ‘,, ;]. To see that this is indeed the case, observe that

We now can conclude that stability holds for our system if and only if [ -“Go 21 is outer (in the strong sense that its inverse is in H”). This concludes the proof of Theorem 4.2.

BEURLING-LAX REPRESENTATIONS 205

Since the first column of our test matrix [ !$ $1 involves only the given plant matrix P while the second column involves only the feedback matrix C, the stability criterion that this matrix function be outer is perfectly suited for application of Theorem 3.1 describing the range of an outer linear fractional map.

To obtain the result of [Y-J-B] (Theorem 4.1) is now just a matter of putting together the pieces. By Theorem 4.2 the problem of characterizing all compensators C which stabilize the system Z, is equvalent to the problem of parametrizing all coprime pairs {N,., DC} in H”(M,) such that h = [ --)~b ,“:I has inverse in H”‘(M,). By Theorem 3.1 this in turn is equivalent to parametrizing all invariant subspaces .&” = [z< J H2(Cfl) such that H2(C2”) has a direct sum decomposition

H2(C2”).A -f *,c-,

where A is the fixed invariant subspace [ ..$$ H’(F). Since D, and N, are right coprime, Corollary 3.2 also implies that there is at least one such X, say & = [ 21 H2(C2”), and hence also one such k, namely

Now by Corollary 3.3, the set of all invariant subspaces A” as above is just

D, H + Nc, -NpH+D,, 1

H2(C"):H~H"(M,),

so the set of all stabilizing compensators C is ((D,H+N,,) ( -NpH+D,,,)-’ =SJH): HE H"(M,)}. This gives Theorem 4.1 as claimed.

REFERENCES

[B-C-K] H. BART, I. GOHBERG, AND M. A. KAASHOEK, “Minimal Factorization of Matrix and Operator Functions,” Birkhauser, Base], 1979.

[B-HI] J. A. BALL AND J. W. HELTON, A Beurling-Lax theorem for the Lie group U(m, n) which contains most classical interpolation theory, J. Opera/or Theory 9 (1983). 107-142.

[ B-H21 J. A. BALL AND J. W. HELTON, Factorization results related to shifts in an indefinite metric, Integral Equations Operator Theory 5 (1982). 632458.

C&H31 J. A. BALL AND J. W. HELTON, Beurling-Lax representations using classical Lie groups with many applications II: GL(n, C) and Wiener-Hopf factorization, Integral Equarions Operator Theory 5 (1982) 632658.

F-41 J. A. BALL AND J. W. HELTON, Beurling-Lax representations using classical Lie groups with many applications. III. Groups preserving two bilinear forms, Amer. J. Math. 10s (1986) 95174.

206 BALL AND HELTON

[B-H51 J. A. BALL AND J. W. HELTON, Linear fractional parametrization of matrix function spaces and a new proof of the Youla-Jabr-Bongiomo parametrization for stabilizing compensations, in “Math. Theory of Networks and Systems,” Lecture Notes in Control and Information Sciences, Vol. 58, pp. 1623, Springer-Verlag, Berlin. 1984.

[DL-M-S1 C. DESOEK, R. W. LX. J. MURRAY, R. SAEKS, Feedback System design: the

[Fll

WI

[F-VI

[Wxl

Wsl

[N-Fl]

[N-F21

CR-RI

[Y-J-B]

[Z-F1

fractional representation approach to analysis and synthesis, IEEE Trans. Auromuf. Control, 25 (3) (1980). 399412. P. A. FUHRMANN, On the Corona theorem and its applications to spectral problems in Hilbert space, Trans. Amer. Math. Sot. 132 (1968). 5546. P. A. FUHRMANN. “Linear Systems and Operators in Hilbert Space.” McGraw-Hill, New York, 1981. B. A. FRANCIS AND M. VIDYASAGAR, Algebraic and topological aspects of the regulator problem for lumped linear systems, Automafica 19 (1983). 87-90. S. HEI.GASON, “Differential Geometry, Lie Crops, and Symmetric Spaces.” Academic Press, New York. 1978. H. HFLSON. “Lectures on Invariant Subspaces.” Academic Press. New York, I964 B. SZ.-NAGY ANO C. FLEAS, Dilations des cornmutants d’op&ateurs, C. R. Acad. Sci. Paris Ser. A 266 (1968). 493. 495. B. SZ.-NAGY ASD C. FOIAS, “Harmonic Analysis of Operators on Hilbcrt Space,” North-Holland-American Elsevier. Amsterdam, 1970. M. ROSEN~LUM ANU J. ROVNYAK. The factorization problem for nonnegative operator valued functions, Bull. Amer. Math. Sot. 77 (1971), 287 318. D. C. YOULA, H. JAHR. ANI) J. J. BONGIORSO. Modern Wiener-Hopf design of optimal controllers. I and II. IEEE Trans. Auromar. Conrrol 21 (February 1977). 3 13; (June 1977). 319-338. G. ZAMFS AND B. FRANCIS. Feedback, minimax sensitivity, and optimal robustness, IEEE Trans. Aufomat. Comrol AC-28 ( 1983), 585-601.

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