Báo cáo toán học: "A Beurling-Lax theorem for the Lie group U(m,n) which contains most classical interpolation theory " pot

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A BEURLING-LAX THEOREM FOR THE LIE GROUP U(™, n) WHICH CONTAINS MOST CLASSICAL

INTERPOLATION THEORY JOSEPH A BALL anc J WILLIAM HELTON

INTRODUCTION

In this article we generalize the farnous theorem of Beurling, Lax, and Halmos from the Hilbert space H?(C") to a space with a signed Hermitian form Our proof is an adaptation of Halmos’ wandering subspace proof of the theorem [14] and of McEnnis’ analysis of shifts on a space with an indefinite metric [23] Our Beurling-Lax theorem for the Lie group U(m, n) (as opposed to the classical one where U(n) = == U(n, 0)) has very strong consequences for Nevanlinna-Pick, Carathéodory-Fejér, etc interpolation theory We obtain directly from our theory a simple linear fractional 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 extended form of our Beurling-Lax theorem to the setting of the Sz.-Nagy—Foias commutant 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

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108 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 factorization 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 classical 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_

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symmetry 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 complement #” 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

{x,} ¢ M such that

limsup 2 2 — ọ

n—-00 YE M ||xall lIzll

Equivalently # is nondegenerate if and only if„Z ' is dense in Z, and is regular 1f and only if in addition ⁄Z +7’ is closed (and thus # =.“ #/’) It is an easy corollary of [11], Theorem V.3.5 that @’ is regular (in our terminology) if and only 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 nondegenerate, 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

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= (.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 positive 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 subspaces 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 subspace 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 problems 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

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=.'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

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func-112 JOSEPH A BALL and J WILLLAM HELTON

tions on the unit circle {|z| =: L} square-integrable in norm, and let #*(CX) be its subspace 2z fe L2(CN) ‘| <f(e#), x) eltdt == 0 for xE CY, k= 1,2, 0 the usual Hardy space H* @ C¥% If 2 +m = N we can define a Hermitian form on L2{CX) and H*(C®) by Qn 1 Hà it (f, 8)2¢cm,ny = 2m \ [ fle )s gle Vem, ndt 0 in addition to the usual form 2z 1 : LB ikew = 5 \ Cle), g(e))gqdt 0 Frequently we suppress the subscripts 2(C”.") and /*(C*) on[,] and <,) The operator S: L°(C¥%) > L°(C¥) defined by [Sf] (e") = eX fie")

is called the forward shift operator, or sometimes multiplication by e (M,;,) Its restriction to H*(C¥) is also denoted by S In any case it is an isometry in both the indefinite Z°(C”-")-inner product and the Hilbert space Z?(C%) inner product 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 addition 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 functions & 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

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THEOREM 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- _ tion E such that 4 =[8 H®(C”+"9J' Moreover, (1) & is analytic - 4 < H”(C™") (2) Suppose / c H*(C”'") Then ¡is a full range invariant subspace << <> = mand n =n (3) Suppose < H*(C™") and is full range Then = is a rational function = <> H°(C™") 9’ is finite dimensional

Proof The idea of the proof is to adapt Halmos’ wandering subspace argument for the proof of the Beurling-Lax theorem for the definite case [14], an idea already used by McEnnis [23] for studying shifts in an indefinite metric Set P=.“ n

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> [> ]Â)-unitary = operator â: C"' + & We then can extend Zin a unique fashion to the po- My, thy, lynomials in H2(C ” *) with range equal to %, by demanding 8 M.,:= SE

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

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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 subspaces “& 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 where tự ® v ® we Cc" ® C1 @ Cc ecatytey (Necessarily m, +p, < m and n+ p, <n.) To set up notation we define a degenerate inner product [; Ty m on tsb bp

C44?) by [uO v @ 9, @ v @ w) Lm: my” cu, Hồ mm — C0, Doms and use this to define a degenerate inner product [,]}] „ „ „ on LA(C”1† 0“) by

LYC TV U71)

integration.)

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ii) 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 (2.1) L+ SM = MSM 1M)’ In general we wish to establish the identity: ⁄+SZ + S1 + S2 = (2.2); = M0 | V S⁄⁄n ay) k=0 To prove (2.2),, we note [ V 5n “|= | V Skt n (Skity |= k=0 k=0 =F) (SKM)! + (S0) k=0

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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 (2.5) (— {#Z + SZ + + 8~1Ø + S4} =.n.A” g»0 In order to establish (2.6) V SEP = MAN", k>o it suffices to establish (2.7) VSP = mhẤ(#+ S# + + §-LØ + SE Epo k>0

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 k>0 S' PY and Y S* Py Ifa vector xin nA’ were simultaneously [, +orthogonal k>@ k>o to :+ S*Y, and ¢, )-orthogonal to y S*¥,, then the decomposition (2.5) would k>0 k>o imply x 6ƒ S*.Z < = {0} (since 4 is simply invariant) This implies (2.7) and hence k>0 also (2.6)

We now use the representation (2.6) for 2đ.#” to construct the desired (4,1, Dị: HỤ n)-phase function 5 We construct an (;, m)-phase Ẩunction ấy such that My = V SEL = [Ê: -H®(C”n "7 k>o just as in the proof of Theorem 2.1 Also one can show VW -= YY S*&@,, and from the k>0

đecomposition (2.6), Z; ˆ is dense in Z n.#” Since #ˆ 1s a closed simply invariant subspace, by the usual Beurling-Lax theorem there is a phase function YEH? (Minsn, ry) (isometric in the usual Euclidean metrics) such that = « W#H*(C"1) 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

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E(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

LoL =M iM + SMM’)

Thus/ = S(t’) This allows us to take any Ơ which maps H(Cđ k>o

onto / 9’ (not just the Beurling-Lax ¥') Here p = dim #4’ While generi- cally det ¥(z) 40 it may happen that det ¥(z) = 0 for all z In this case one modi- fies ¥(z) to ¥,(z) by throwing out just enough columns of ¥ to make the remain- ing ones linearly independent

3 APPLICATIONS TO INTERPOLATION

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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 continuation 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]

1 — 2,2, KEL aN

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

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invariant 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) đ #L”(M„.,)

where Fy is any function in H?®(Äƒ„ „) which satisfies the interpolation conditions F,(z;)"p; = 9; for j= 1, ., N and F(z))p; = qj for j = 1, ., N’ The inner functions 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 multiplication operators with multiplier in K -+ 0N?(My,„)@ :

M = Mx 4,9 = the closure of {[ 7 Jerre ẩ J#âđl

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

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120 JOSEPH A BALL and J WILLIAM HELION

Proof Suppose F is in Cx,o, o(/) Then F has a representation

Fe K+ 0GY-19

where Ge H*(M,,,,) and ¥ € ZH™(M,) is a matrix Blaschke product of degree at most / Since G maps H™(C’) into H?(C*) and gg* = J, this representation implies

; Jovarey c.Z By continuity,

9= H eo PAC’) c Z

Sinee || #||„<1, the subspace Y is a negative subspace of L°(C”) @ o*H*(C*) Since WY is a matrix Blaschke product of degree at most /, p*YH°(C") has codimension at most / as a subspace of g“H?(C") Therefore Y has codimension at most / 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 codimension Jin a L°(C”) @ o*H7(C")-maximal negative subspace That Y is invariant and

‘as codimension / in a maximal negative subspace means #= ; Jove

where Fis in ZL™(M,,,) and ¥ ¢ H™(M,) is a matrix Blaschke product of degree / Since also Y c “@, we have for any he HC’) F “Wh = lim K1 SA@ 0 js HỆ tell? Loy" for some 4 € H°(C") and 4@ € H°(C*) From this we get g"Wh =- lim0AM and qo then (F — K)o*Vh = lim 04 Thus (F — K)p"¥ = 0G where Ge H(M,,.), so gơo

Fe K+ 0H*(M,,,)Ơ1 Since ¥ has degree /, we conclude Fe K + 0H7(M, n)o Since from the above we also have j|Fi{.< 1, we conclude that F is in Cy ;,„() as desired The uniqueness in the correspondence can be arranged by demanding that G and ¥ are right coprime in the representation F = K 4- 0G¥-19

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LEMMA 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 invariant 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 negative 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 F 0} a pw my Py G = the closure of Ƒƒ 0| (H°(C )@ H°®(C *) x YY @ 0 7

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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 subspace 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?

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L 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 xX y @ i: H(C") > HC") @ {0} c H{C) @ HAC?) = HXC") and j: HC") {0} @ H*%C”) c H*(C*) @H*(C”) = HC’) be the natural inclusion maps Define a mapping Â, from BLđ(M,,,.n,) into %L(M„,„) by

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 BHđ(M,,,,) and 06Â@2H(M,,,) and gy € BH™(M,) are phase functions In addition suppose the associated invariant subspace 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 = x yy @ = p, =n, such that Cx 6, of + h) = 9 ABHP(Mn + ))

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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 negative 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

a se = a

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 F 0 7 ÿ n g:- Ề B l 7 0l[Zn(H“(C®9) @H=“(C)]L, ⁄“.} @ 0 1 7 ?

where #% is an invariant subspace of H?(C )@ H°(C '?) of codimension at most I,, since P chow therefore has codimension at most /, in H*(C") {0}, we must have that F is in BHM m,.n,): Using the inclusion maps i and j defined - ip _ above and identifying # as a subspace of H*(C * “*), we can rewrite the form for G above as FT” + yi® + wj* G xx l4 -ˆ Bi? + | Zn HC) |”

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 cosignature 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

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graph theorem) even if K is unbounded Then one easily checks that

, == f &

(Mx, 0,9)' = Lr, cay From this the next lemma follows almost immediately

LEMMA 3.5 The negative signature 1 of a subspace (Mx, 9,9)’ is the dimension of the negative eigenspace of the self-adjoint operator I —Ig, 4(K)V9, (K)* on # = L^(C") Q 0H 1c"), Moreover, M(x,6,9 is regular if and only if L—P6, (K)Fo,9( K)*is invertible ; and pseudo-regular if and only if I—T9, g( K)I 4, 9 K)* has closed range

Thus Theorem 3.4 combined with Lemma 3.5 not only proves that Cx 9, ,(I') is nonempty if J— I, ,(K)I's, o(K)* has closed range and negative eigenspace of dimension at most /’, but also effectively parameterizes the set Cx ,(/’), once

_ - [a

we have a procedure for computing the associated phase function 5 = [ B ° x y @ Later we shall illustrate how one can compute & for a simple example First we obtain the result on existence for the general case where “Wx, , may not be pseudo-regular

THEOREM 3.6 Suppose K is in H*(M,,,,) and 0€ L(M,, ,) and pe L°(M,, ,) are phase functions Then Cx 9,g(1) is nonempty if and only if the negative eigenspace of T—V 6, (KP 9, (K)* has dimension at most 1

Proof Lemmas 1.1 and 3.1 give necessity of the dimension / condition immediately If J—Iy, ,(K)I'o, (K)* is invertible (or even if only it has closed range), sufficiency follows from Theorem 3.4 and Lemma 3.5 To remove the invertibility assumption, we use an approximation a-gument of the type used by Adamjan, Arov and Krein [1], [2] Suppose J— Ig (K)F'o, (K)* has negative eigenspace of dimension at most / and is not invertible Replace the L°(C”.") inner product by

12(C””?), where for f @ ge L(C") ® LAC) = L(C™"), [fs f@8) mn =f) ) LC LYC™) —_ éq + 88 8) 20%" I : With this new inner product, for ¢ > 0 sufficiently small, ¿| # is regular 0, “ with negative signature /, and its [,] vạch ng -orthogonal complement Z£ ¿ „ 1C ` °®)

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In applications it is sometimes useful to know that there is an element F in Cx,,6,9(/) with boundary values F(e"’) which are isometries The following is a fairly general sufficient condition

COROLLARY 3.7 Suppose K is in H*(M,,,,),0€ L™(M,,,,) and g & L™(M,) are phases, and that I — Ig, (K)P'o, 9 K)* has closed range Set | equal to the dimension of the negative eigenspace of I — Tg, (K)I9, (K)* and assume 1 < 00 Then if m,>n,, Cx, 6, pf) contains an element with isometric boundary values

Proof By Lemma 3.1, elements of Cx,»,,(/) correspond to invariant x, 6, o- -maximal negative subspaces of “x, 9,4; in this correspondence, isometric subspaces of Cx, 9, 9(/) correspond to invariant (x, ¢, N’)-maximal negative subspaces (V=:Y S*Mx, 690M, 0,9)) Which are null spaces Lemma 3.3 establishes a

kp

correspondence between invariant (./x, 9,4 ’)-maximal negative subspaces and

F739 Ft

° s ˆ

invariant maximal negative subspaces of H?(C * * °, Moreover, there exist invariant maximal negative subspaces of HC 55 which ‘are null spaces if and only if m, 2 n, This establishes the corollary

As an instructive special case, consider the scalar case (m = n = 1), and suppose k == 1 Then if Cz, 4 ,(/) is nonempty, we must have either m, =: ny» 1, p,: - 0, Or m+ my == 0, py = 1 In either case, if I — Fạ ;(K)I¿ ;(K)? has closed range, then Cx, 9,:(/) contains a function of modulus 1 on the boundary if it contains any junction at all Without the special topological assumption on the operator

f-— Po (K)P9,.(K)*, it is known that this is no longer the case (see eg [13]) b CONCRETE EXAMPLES

As an illustrative example of the above theory, consider the interpolation problem N-P,(z, p,q) mentioned at the beginning of this section for the case N’ 0 Set Cx, 6, o(/) equal to Cx, (1) if ọ = 1, and similarly for Zy „ and Pạ(K) While N-P,(z, p,q) corresponds to Cy ,(/) for a certain choice of K and @ we need not actually compute them to see that /, , is the space / defined by

J

M {re wor [ [2], re) | = 0 for j=1,2, , wt

L cm,n

Since the function (1 — zw)-} is a reproducing kernel for #2, we have

Lo} Jen [2— 3 [FO Jy

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by the functions (s) {a — 2z,)-} l7] j=), oN} qj A typical element k(z) of ’ is x - P kŒ) = Š c( — 28) [ | j=l qj from which we compute aa = Pi) FP [k, KÌ taycmom = > x ec —Z;Zj) 1 l7} [ || = ; qi cen =>»? I=l jo {“ NT — qj , đicn } - fC i om ZjZy _ = CFA, pg

when c = [c, .,€y]? In other words, the Pick matrix is the Grammian of the basis («) for @ Thus the negative signature of the space #’ is precisely the number of negative eigenvalues of the Hermitian matrix Aj, p,q) Therefore the matrix test for when N-P,(z, p, q) isnonempty mentioned atthe beginning of this section is an immediate consequence of Theorem 3.6 and Lemma 3.5 Alternatively, one can check that the space #” = H* © 6H" is the span of

(x*) {q — Zzj)~}p, j= 1, , N}

and P;(K)*:(1 — zz;)—1p; = (1 — 2z;)-47; Thus Ajz, p,q; can also be viewed as the matrix representation for J—I,(K)I',(K)* with respect to the basis (*«) for #

While the very geometric approach we take appears abstract, actually it serves very well to organize and simplify concrete computations We now analyze the scalar case (n = m = k = 1) of N-P,(z, p, q) thoroughly In this case the space At has the form

„— Í[¿Ì* HM(C11) | w; g(j) = ƒŒ) for j=1, so] &

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-: & (SM) One easily checks (S.Y = (SH*(C119)' + SM and thus (S.)” consists of the functions /⁄.(60= [5 ]+ ¥ atl — ergy 2 Ì j Ww; for all a, b, a; in C The functions in & satisfy in addition 1 l 0 [reum ee] = 1 —e Zy Wy HỀ%(Cnb N 1 — www =a— wyb + ` cjg, i) jel 1 — 2;2, for all w:- 1, , N So the prescription for arriving at a basis for Y is to solve z1 : ¬ wạZ¡T Aa = : : Ae = : > zy? waznt where A is the Pick matrix A = [(1— w,w,)/(1 —Z,z,)] (invertible, since # is regular) Define o by oS(eit) == S2 | } | (s =1, 2) j=l ] — ze" W;

‘Then the functions

faery -=[ )]-ore and fey = [1] + 2%

are a basis for &

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The case not covered is when / 11.’ # {0}, or A is not invertible To settle N it, note that ‹Zn.⁄Z' consists of all vectors g,(e*) = W œ(1 — e“z,)~1 |; | J j= 1 1 0 =} 2a; I = (1 —€! 2y) Wy H*(CbD N ]—wyy ——————— ) such that ‘IS — j=l 1— 2,2;

that is o is in the kernel of the Pick matrix A For this special case (m =n = 1), P= 1 >m =n, =0 and thus “nV’ =W where

N = NV {elt g,| œe kerA}, J>0

and the solution F of (N-P), (where /is the number of (strictly) negative eigenvalues of A) is unique We obtain this unique inner function F as the quotient ¥, Yz}

% | (C1) For example, take

of the components of a representation W == |

2

N N - -1

F= ya — ez) | ¥ al — ez,;)~ %,]

jel jot

where a = [0, , @y]¥ is any vector in the kernel of A Thus computing the unique solution of (N-P), is reduced to an eigenvalue problem, much in the spirit of Hintzman’s work [18] on Hankel matrices Higher order interpolation problems can be handled in much the same way by using the Pick matrix as in [28] or [5]

c L™® APPROXIMATION FROM H@

A second special case of interest is when both phase functions @ and @ are the identity This corresponds to supremum norm approximation of a given function by H®™ functions We are given an L7(M,,,,) function Kand wishto know when there exists a function F in the set

4g) = {K + HỆ(M„,„)} n 2L (M„„,u) (= Cr, 1,10)

and if possible, to parameterize A,(J) when it is nonempty The set 4„() can be regarded as the set of all error functions of Z® norm less than one obtainable by approximating K with the error functions in H7(M,,,,) In this case the appropriate operator I'x,,,, is the Hankel operator

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defined by

Hg: ke > Pg on FR) ac")

for ke /I°(C") The following is simply a restatement of Theorems 3.4 and 3.6 for the special case 0=:9 =I

THEOREM 3.8 Suppose K ts in L?(M,,,,) Then the set Ag(l) is non-empty if and only if the self-adjoint operator Ì — 2,32% has negative eigenspace of dimension ai most l Moreover, if I — 3, H¢% has closed range and | is the dimension of its nega- Bow 3 tive eigenspace, then there is an (nu, tị, py: m, n-phase function Š : | % ¢ with my -+ py»: nsuch that Axl + h):= #z(2H/ (Mu): d BOUNDARY INTERPOLATION

Another application of invariant subspace techniques here are to boundary interpolation This concerns the extension of a function fin LZ (where A is a subset of the unit circle of positive Lebesgue measure) to one in @H® Our treatment isin the spirit of a proof given by Rosenblum and Rovnyak [28] adapted to our setting We actually give a result considerably more general than the classical one or the one ¡n [28] Suppose p and g are matrix-valued functions on A, namely, pe L7ƒ(M, ;) and øc 12M „)

THEOREM 3.9 There is a function F in BH;7?(M,,,,) in the set

Kp, q, A) = {Fe L°(M,,,,) : F(e)*p(e")= ge) for almost all e* in A}

if and only if the dimension 1 of the negative spectral space of the self-adjoint operator

= = — *

A,, IA Moy, P er Mex, May, P sqm, Moy,

acting on L*(C*) is at most I’ If p(e**) is rank n on a set of positive measure, then the solution F with Il’ =1 is unique (whenever it exists) by analytic continuation Here x4 is the characteristic function of the set A and My is the operator of multiplication by the function F

Proof Set

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Then # is an invariant subspace of H®(C”") and if Fe BH“(M,, ,) satisfies F(e1)* p(e”) :: 4(©') a.e on A, then Bie is an invariant H2(C”™")-maximal negative subspace of /@ Conversely, if W is an invariant H?(C”-")-maximal negative subspace of @, then its angle operator gives rise to an Fe BH™(M,,, ,) satisfying F(e")* p(©!) =: q (e) ae on A Analogous statements apply for solutions F in ØH?*(M„,„) and invariant negative subspaces of ” having codimension at most /’ in a H2(C™")- HN subspace By Lemma 1.1, the existence of a solution F of I(p, q, A) in BHP(M,,,) then implies that the negative signature/ of # is at most 1’ Conversely, if hộ = /1'—1/>0, then, again by Lemma 1.1, invariant negative subspaces of /@ with /-negative cosignature /, are exactly the invariant negative subspaces with negative cosignature J’ which are contained in ⁄ If is pseudo-regular, such subspaces exist in abundance by Lemma 3.3; otherwise, we can perturb the metric and use an approximation argument as in the proof of Theorem 3.6 to still get the existence assertion Thus Theorem 3.8 follows once we establish that the negative signature of the space @’ is equal to the dimension of the negative eigenspace of A, a4 To see this, observe that an equivalent way to define / is Ms =: | ƒ6 H1") ,P „ p =0 for all pe LCL | HC” | |e | cm or all ( | From this representation, it is clear that ye P AM! = {Ƒz.e- | q le

Thus we see that @’ is parameterized by 17(C*) and the self-adjoint operator A, 4, 4 induces the inner product on L*(C*) equivalent to the H?(C™")-inner product restricted to Z“ Therefore the negative signature of @’ is the dimension of the negative eigenspace of A, 4,4 as desired

If p is full rank ona set of positive measure, then Page LPX @|l@c1?(C*)} i dense in ae This implies that if f, and f, are two elements of / with nh= P enc? , f2, then in fact f, = f This implies that /% is a graph space (⁄ = l?] Dom(M) where H: Dom(H) c H*(C') ¬ Hˆ(C") ¡is a

QE #9]

P oren%c’)

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We conclude this section with a converse to Theorem 3.4 Theorem 3.4 cha- acterized a generalized interpolation set Cx, »,,(/ -:-1,) (for the pseudo-regular case) as the image under a certain linear fractional map ¥- of a function disk BAP Mm, mi): The phase function 5 and the index / in principle can be computed divectly from K, 9 and ø Conversely, in the next theorem we start with the phase function 2 and indicate how one can compute K, 0, @ and / directly from £ This result thus computes the image of BHP(Ma, m) unđer a linear fractional map #z and also gives a cleaner less computational proof of a result of Helton [17]

a pw THEOREM 3.10 Let © -

⁄.? 0 |e a (Mm, my py; my n)-phase function with n,-' py đ Suppose also that

i) the closure of xi®(C”) <i yH®(C”) + œH1*®(C”) is a full range simply invariant subspace of L?(C"),

ii) the closure of yH %(C”®) + wH(C’?) has finite codimension in the closure

sƒ zH®(C”) +: yH®(C®) +- òH®(C”), and m+n, +p asthe iii) @ ==(2-H°(C * +) is @ simply invariant subspace of LC } Then G(BH(Mm, n,)) = Cx, a) where a) 9E BL>(M,) is a phase function such that [P oyaryen “1 = GHC); b) d€ BL™(M,,,,) is a phase function such that ⁄⁄ n [H*(C”) @ {0H = 0H*(C*) @ {0} ; c) K is any L*(M,,, ,)-function such that ý | ọ*H®(C") c / and đ) / % the codimension of {yH®(C”) + œW*®(C”)}” as a subspace of {xH°®(C”®) + yH®(C”) + œH®(C”)}”

Proof By the classical Beurling-Lax theorem there exist phase functions Ø and ¢ as prescribed in a) and b) Thus

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Consider the space /' = L2(C”) @ @*H?{C”) as a Kreĩn space with inner product inherited from /2(C""), By Lemma 3.3, invariant maximal negative subspaces of Al 14C”? " "5 match up with.Z-maximzl negative subspaces in Zˆ via multiplication

by & A particular maximal negative sutspace of

ACY" P9 ~ A200") @ HC") @ HC”) is

Ny = {01@H*(C”) @ H2(C”)

The Z-maximal negative subspace in K corresponding to as above 1s the subspace {yHe(C”® + œH*®(C”)}” By đ) and the deBnition ofZ⁄ and @, the codimension of this space in a #-maximal negative subspace is /, which is finite by condition ii)- By Lemma 1.1 any /-maximal negative subspace has negative cosignature / with

respect to #, or equivalently, “#’ = & =!.@ has negative signature equal to / We are now ready to show that a function K as prescribed in c) exists Let # equal to L^(C") Q0H?(C“) and ;et 2 c H?(C") be the linear manifold 2= Ptayen*(c) ⁄Z Thus 2Ø ¡is dense in @*H?(C") by definition Then for each ƒ in & there is unique Xfin # such that X7 @ fis in.@ This defines a closed operator

x:2 ¬# such that

AM = l; | 2 + [0H*(C*) @ {0}

From this representation it is easy to check that MS = X“ , |e

where & is the domain of X* and is dense :n H We saw above that.#’ has negative signature | < co; it follows that the self-adjoint operator J — YX* has a negative eigenspace of dimension at most /, and hence must be bounded Therefore X 1s bounded and the linear manifold @ is in fact all of p* H?(C")

Let X, be the restriction X | p*C" of the operator X to the subspace @*C" of @*H*(C"), and defđne an Ä„ „-valued function K by

it it it

K(e °)x = Xo(p* -(e °) x9) (€ °)

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K

for x in C’, and thus | I | @°C" c / By the invariance of the space 7/,

[; | p*H®%(C") c

This establishes the existence of a function K as in c)

Now suppose X is any function as in c) Then if F is any function in H°(M,, ,)

such that ance c.Z, then

r 9 | @°H*(CŒ') c li Jone +k [; | @*H*(C") c Z,

and therefore F Ke 0H*(M,,,,)o One can further check that / is spanned by such subspaces and therefore / == Mg 9,4 That G(BH(M ny , n)) » 2 Ceo, ell) now follows exactly as in the proof of Theorem 3.4

4, THE COMMUTANT LIFTING THEOREM

in this section we indicate how the work of Arsene, Ceausescu and Foias [4] on parametrizing the set of all contractive intertwining dilations of a given contraction intertwining two contractions can be put in the framework of this paper Furthermore we provide a more general lifting theorem which is the abstract analogue of Wƒ?-interpolation The fact that commutant lifting is intimately connected to interpolation is due to Sarason [29] Thus the reader who is already familiar with this connection probably already seen how our development will unfold For this reason we shall be brief

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TA = AT; we say that A intertwines T and 7) An intertwining dilation of A is an operator Aw € (2#, 2) such that A, € IU, U) and PA„=ÁP, where P:' =2 and P: 2ˆ — 3Z are the orthogonal projections When ||4|| < 1, it is the content of the commutant lifting theorem that there is a contractive intertwining dilation Aco of A (i.e [lAcol] <1, Uc == Aco U and P Ago =: AP) The goal of the work of Arsene, Ceausescu and Foias is to describe via an explicit parameterization the set CID(A) of all contractive intertwining dilations of A

Such a parameterization can be obtained via a slightly more abstract version of Theorems 2.1 and 2.2 as follows Let # = % 1# be the Krein space with Inner product

(kK, REA, =k, A, — Uk, Ay

and define a subspace (== (T, T; U U; A)) by

M = lƒ pre [z2] I M.(ø)Ì

If we let U be the operator lý | on x, then U is isometric in the [, ]2 -inner product, and since TE ICE IHS? lt =[4]7+ (4, I _U-T we sce that is invariant for U Furthermore, A, € &(K, K) intertwines U and ~ A A ~ U (UA = AwU) = its graph [ li is invariant under U, PA = APs the A graph | Z is contained in , and |Ì4œ|Ì < 1 <> the graph | 2Ì HX is a negative A

subspace of K As we have seen in the previous sections, a ZŸ-maximal negative subspace is always the graph of a contraction operator We have obtained

THEOREM 4.1 The angle operator-graph correspondence establishes a one-to- -one correspondence between the contractive intertwining dilations Aw of A (i.e the

A A ~ ~

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We next obtain the abstract analogue of H/?°-interpolation We say Awé @ CID,(A) if there is some U-invariant subspace W# of codimension at most / as a subspace of % such that Ax: W# > © satisfies All < 1, AolU, We) : UAcw and PAs : AP! WH (Thus PAco ‘3 agrees with A on a T-invariant subspace of codimension at most /.) Using the same analysis as above, we obtain the following refinement of Theorem 4.]

THEOREM 4.la The angle operator-graph correspondence establishes a one-to- ~0ne correspondence between the set CIDA) and O-invariant negative subspaces of

ia MT, T: Ũ, U: A) of codimension at most | ina 2È-maximal negative subspace of 2

Thus by Lemma I.1, a necessary condition that the set CID,(A) be nonempty is that the negative signature of ”’ be at most /, and conversely, once we establish that there exist @-maximal negative subspaces of @ which are also U-invariant, we shall see that this condition is sufficient as well Now it is easily checked that

Ml | y |? A®

and thus the negative signature of /#”’ is the dimension of the negative spectral subspace for the self-adjoint operator / — AA* on # Thus, modulo the gap mentioned above, we obtain the following generalization of the existence part of the Sz.-Nagy- -

Foias lifting theorem (which corresponds to the / =: 0 case of the following) THEOREM 4.2 The set CID,(A) is nonempty if and only if the negative spectral subspace of the self-adjoint operator I — AA* on # has dimension of at most i

In contradistinction to the previous sections of this paper, it may happen here that U |.” is not simply invariant (i.e it may happen that () U4 #35 !0})

>0 *

For example one may simply take 4 =: 0 and let 7 be any completely nonunitary contraction which is not C,» (see [25]) For such an isometry we must generalize Theorems |.1 and 1.2 and develop a Wold decomposition for the [ , ],-geometry

7

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j$ an isometry in both the Krẹn space [, ]g and the Hilbert space < -, -È, inner pro- ducts The next result is an extension of our Beurling-Lax theorem which handles isometries which are not necessarily shifts

THEOREM 4.3 Suppose U -l ni] is a fundamentally reducible isometry on the Krein space K = K Ft K and M is a pseudo-regular U-invariant subspace of K SetN=: V Ơ⁄M n M) Tihen there are a densely defined ([, lạ kee [, ]a)-Isometry Li i gj20 : G(E) (cK, (11 Ky) > K, and a fundamentally reducible [, In i -isometry 1 ụ by R °| on K,® K, such that 2([% N 2) 8 0 Gli 0 Ủy and MnN =(z 2(2))” A A

Here K, is a Kretnspace, K, is a Hilbert svace and

[ki ®kạ, k@ Kole cng, =: [ky, Kile and

Proof As in the proof of Theorem 2.2, we get a generalized Wold decomposition for M n N’: MnN ={VŨ'L+m0"M)" n>0 neo where L-=: Mn (My (Note that Y/ Ê"L and mM Ơ"M are [› orthogonal, but n>od nai)

may have nontrivial intersection.) Since M is pseudo-regular, L is also, and hence L as a [, ]-orthogonal direct sum decompesition L =- L, f;:!L, where L, is regular and Ly is a null space The space MN’ 1as the space N as its isotropic subspace; N in turn has a Hilbert space Wold decomposition N == (@ "(N â "N)] @đ n>0 en) Ị"N =N, @N, Note N, is contained in (7) U"M and is reducing for U and n?>Ũ n>o thus M, =: \JU"MON, is also reducing for U It is not difficult now to see that "n0

an equivalent form for the representation of Mn N’ above is

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aA

where each summand is invariant for U The theorem follows once we show that each summand has a representation {2-2(&)}- for some phase function 2 The summand VY Ù"L, is handled as in Theorem 2.1; N, and N, are handled easily in a

n>0

manner similar to that in the proof of Theorem 2.2, since they are null spaces It remains only to consider M,

The basic structural property of M, is that it is a reducing (in the Hilbert space sense) subspace for the Hilbert space unitary operator U which is also nondegenerate in the indefinite metric By the spectral theorem we can represent U as the mul-

2x

tiplication operator M_,;, on a direct integral Hilbert space # =: \ ® KX (t)dm(t),

0

where mis a scalar spectral measure for U and dim (t) is a multiplicity function for U (see [12]) By assumption there is a signature operator J which commutes with U and induces the Krein space inner product on & Sucha J must be represented as multiplication by a measureable field {J(t)}, <teoq Of Signature operators, where each J(ft) is a signature operator on #(t), which thus makes each fiber space -#(t) also a Krein space Now since the subspace M, is reducing for U, it must be decom-

2z

posable, that is, M, has the form M, = \ @® ‹ZŒ)dm(£) where each #(t) is some

0

subspace of #(rt) Since M, is nondegenerate as a subspace of #, a.e .//(t) must be nondegenerate as a subspace of the Krein space #(t) We can thus produce a densely defined Krein space isometry Y(t): X(t) > #(t) with range dense in (1) If we do this in a measurable way, then the operator Y of multiplication by ¥(1) gives a .x densely defined isometry from the Krein space \ @® X(t)din(t) onto a dense subset 0 22 of \ ® #@(t)dm(t) «= M, which intertwines M7 eit’ The assertion of the theorem 0 follows

As an aside, it is interesting to note that the Wold decomposition in the proof of Theorem 4.3 can be refined if the invariant subspace M is regular Indeed we get the following:

Ũ 0]; OU is a fundamentally decomposable '

^ ~ A

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9ƒ Đ Define subspaces L, M,(L) and M,, by L-=MF] UM, M,(L) == V 4U, k>0 M_.:- (ÊƠ*M Then M +(L) and M., are regular U-invariant subspaces of K, k>o Ồ | M,(L) is a_ shift, Ờ | M,, is unitary and the invariant subspace M has the Wold decomposition M = M,(L) fA M,,

Proof Since M is regular, the restriction of [, ly to M makes Ma Krein space Since U | M is also an isometry in the compatible Hilbert space topology on M, the norms I(Ð | M)*|| (A>0) are uniformly bounded The Wold decomposition of the proposition now follows immediately from a general result of McEnnis [24] The next Theorem extends Theorem 3.8 to a more general setting, which for 7:: 0 should be compared to the results of [4] To state it, we note that © is as in Theorem 4.4, and we decompose K into the maximal positive and negative subspaces

K = K{QK which diagonalize 0 ( ?—|U 0 U= Lo i) and similarly, write K,=K,mK, Ũ ee A aA aw °|, then these decompositions of K and K, 1K, induce an opera- 0 U, 2-[* # Ý x Y @ tor matrix representation

for = In principle, all the objects in the following theorem are computable via our techniques The proof is completely analogous to that of Theorem 3.4 and will be omitted

with U; =F [

THEOREM 4.5 Suppose T, T, U, Ũ, on 2, 2, KH, A in L(K, H) and A = MT, T; U, U ; A) are as in Theorem 4.3, and suppose that Md is pseudo- -regular Then there isa Krein space 2 = KH 1/7! #, and a Hilbert space #4, a u% 0 0 fundamentally decomposable isometry | 6 U, O | L2 194L,= KH 111i A1@B 4%, 0 0 Ứ, a densely defined ({, l5 „Ï, ]„ )-isometry a= § B |: (5) > 2, such 1 tH› x ¥ y @

that S((Ủ, @ U, @ U,) | 2(8)) = (Ù @ U)S and

CID,(A):-={(aHi*-+ Bi* + j*)\(«Hi* + yi* +0j") 9H: 0, 2v,||MỊI<1, HU= Ủ,H}

where

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and

fH > 10} OH oH, OM are the natural injections

The same conventions for [, ], and [, ]a hold asin Theorem 4.4

ey 1 Hy OX,

In the case where the invariant subspace / «= /#(T, T; U, U; A) is regular, the degenerate subspace #, can be dispensed with, and Theorem 4.3 guarantees the existence of many elements in CID,(A), where / is the negative signature of /’ If @ is not regular (or even pseudo-regular) but #’ has finite negative signature, @ becomes regular after a slight perturbation in A; one can then use Theorem 4.1 and a continuity argument to complete the proof of Theorem 4.2

Finally we consider the question of uniqueness The first part of the following corollary refines a result of [3]

COROLLARY 4.6 Suppose A: Jé ~» 4 intertwines contractions Te (3#) and Te LH ) as above and that 1 AA*® > 0

(i) Kker2„ (2¿-:(1— AA*)¥2) is cyclic for the minimal isometric dilation U of T, then there is a unique A,, in CED(A) aid furthermore this unique A,, is an isometry

(ii) Assume 0 is an isolated point of the spectrum of Z@, Then there is a unique 44 in CID(A) if and only if the subspace & -'- i, is either a positive or a negative subspace of de Here L - » Mt U ẤN cị ĐOơ», where .4(T, T; U, U: A)

ADO

as above

This unique A,, in CID(A) is isometric if and only if & -:-.@,, is positive Proof (i) Let A,, be any element of CID(A) Since PA,, :: APand ‘4, <1, A,, has to be isometric on ker@,; from the intertwining condition it follows that A,, is uniquely determined as an isometry on (JU"ker @,; the U cyclicityof ker“,

neo

implies that A,, then is unique We thank the referee for pointing out that this result is elementary and does not depend on the machinery of our Beurling-Lax theorem (ii) If 0 is an isolated point of the spectrum of J—AA*, then the space -/ is a regular subspace of td, and a Wold decomposition as in Proposition 4.4 holds for.4# By Theorem 4.1, we know that operators in CID(A) are in one-to-one correspondence with U-invariant -maximal negative subspaces of #7 Since 2È ¬—

I 2- " a

= F | 2# is a positive subspace, by Lemma 1.1 &-maximal negative subspaces of dé are exactly the #/-maximal negative subspaces of #/ Asin Lemma 3.2, invariant /- -maximal negative subspaces must be subspaces of / ) A”’ where WV: = V Ù*( n.Z)

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By the decomposition (4.1) for “n.Wv' we see that if Mn’ has a unique maximal negative subspace, itis automaically U-invariant Now it is easy to see that this uniqueness occurs if and only if either.@ 9 4’ is positive or #1 AW’ is negative If.Zn.W” 1s positive, the unique maximal negative subspace of W@W" is the isotropic subspace of 4n.W’ and the corresponding element A,, of CID(A) is isometric If “9 MW’ is negative, taen the maximal negative subspace for 4n.” is the whole space Zn.#ˆ” By the decomposition (4.1) for #nWN’, we see that ⁄Z n.4” 1s positive (negative) if and only if Z + is positive (negative) and the desired conclusion follows

The main result of [3] is that CID(A) has a unique element if and only if at least one of the factorizations T-A = 4-T is a regular factorization It would be interesting to have a direct proof for the equivalence of this condition and that of Corollary 4.6 (ii) for the case where /—AA* has closed range

Both authors are partially supported bv National Science Foundation grants

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JOSEPH A, BALL J WILLIAM HELTON

Department of Mathematics, Department of Mathematics,

Virginia Polytechnic Institute University of California at

and State University, San Diego

Blacksburg, Virginia 24061, La Jolla, California 92093,

U.S.A U.S.A

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