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INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES

WARSZAWA 1997

UNITARY EXTENSIONS OF ISOMETRIES, GENERALIZED INTERPOLATION AND BAND EXTENSIONS

R O D R I G O A R O C E N A

Centro de Matem´ aticas, Facultad de Ciencias Jos´ e M. Montero 3006, ap. 503, Montevideo, Uruguay

E-mail: rarocena@cmat.edu.uy

Abstract. The aim of this paper is to give a very brief account of some applications of the method of unitary extensions of isometries to interpolation and extension problems.

I. Unitary extensions of isometries. A general method for solving several moment and interpolation problems can be summarized as follows: the data of the problem define an isometry, with range and domain in the same Hilbert space, in such a way that each unitary extension of that isometry gives a solution of the problem.

In this review paper, the method and some of its applications are briefly described.

We now fix the notation and then specify the content of the following sections.

Unless otherwise specified, all spaces are assumed to be separable complex Hilbert spaces and all subspaces are closed; L(X, Y ) denotes the set of all bounded linear op- erators from a space X to a space Y ; L(X) is the same as L(X, X), and “W” means

“closed linear span”; P

EX

≡ P

E

denotes the orthogonal projection onto the subspace E of X and i

XE

≡ i

E

is the inclusion of E in X. L

p

(X) denotes the space of X-valued measurable functions on the unit disk T with finite p-norm. L

p

(X, Y ) denotes the space of L(X, Y )-valued measurable functions on T with finite p-norm.

The isometry V acts in the Hilbert space H if its domain D and range R are (closed) subspaces of H. We say that (U, F ) belongs to U , the set of equivalence classes of minimal unitary extensions of V , if U ∈ L(F ) is a unitary extension of V to a space F that contains H, such that F = W{U

n

H : n ∈ Z}; we consider two minimal unitary extensions to be equivalent, and write (U, F ) ≈ (U

0

, F

0

) in U , if there exists a unitary operator X ∈ L(F, F

0

) such that XU = U

0

X and that its restriction to H equals the identity I

H

in H. An element (U , F ) of U with special properties is given by the minimal unitary dilation U ∈ L(F ) of the contraction V P

D

∈ L(H).

In Section II an isometry V is associated with a generalized interpolation problem in such a way that there is a bijection between U and the set of all the solutions of

1991 Mathematics Subject Classification: 47A57, 47A20.

[17]

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the problem. A parametrization of that set by means of contractive analytic functions is described in Section III.

A general method for solving interpolation problems is given by the Nagy–Foia¸s com- mutant lifting theorem ([Sz.-NF], [FF]). Parrott ([P]) has shown that a special lifting yields interpolation results for analytic functions with values in a von Neumann algebra.

In Section IV each commutant is associated with an isometry V in such a way that there exists a bijection between the set of all the Nagy–Foia¸s liftings and U , and that a Parrott type lifting is given by (U , F ).

The band method is a general scheme for dealing with many extension problems. It has been developed in a series of papers including [DG.1], [DG.2], [GKW.1], [GKW.2]

and [GKW.3]. In Section V the method of unitary extensions of isometries is applied to deal with one of the problems that in [GKW.1] is solved by the band method.

In Section VI, Schur analysis of the set of unitary extensions of an isometry is related with previously considered subjects.

A basic example of how the method can be applied is given by the problem of extending functions of positive type. Its bidimensional case is related to the problem of finding two commutative unitary extensions of two given isometries ([AF]).

The author wants to thank the referee for his comments on a first version of this paper. An expanded version of it, as an introductory and essentially self-contained series of articles, is being published in “Publicaciones Matem´ aticas del Uruguay” ([A.5]).

II. Generalized interpolation. The method of unitary extensions of isometries gives a proof of the following

Theorem (1). For j = 1, 2 let E

j

be a Hilbert space, S

j

the shift in L

2

(E

j

) and B

j

a closed subspace of L

2

(E

j

) such that

E

1

⊂ B

1

⊂ S

1−1

B

1

and S

2−1

E

2

⊂ B

2

⊂ S

2

B

2

. Let A ∈ L(B

1

, B

2

) be such that AS

1|B1

= P

B2

S

2

A. Set

F

A

= {w ∈ L

(E

1

, E

2

) : P

B2

M

w|B1

= A, kwk

= kAk}, with M

w

the multiplication by w. Then F

A

is nonempty.

When B

1

= H

2

(E

1

) and B

2

= H

2

(E

2

) := L

2

(E

2

) H

2

(E

2

), the above is Page’s extension of Nehari’s theorem (see [N]). When E

1

= E

2

= E, B

1

= H

2

(E) and B

2

= H

2

(E)⊕K, with K a closed subspace of H

2

(E) such that S[H

2

(E) K] ⊂ H

2

(E) K, we have Sarason’s general interpolation theorem [S]. For convenient choices of the data, F

A

is the set of all the solutions of the Nevanlinna–Pick problem or of the Carath´ eodory–Fej´ er problem.

Lemma (2). Let A ∈ L(B

1

, B

2

) be a contraction between Hilbert spaces. There exist a Hilbert space F and isometries r

j

∈ L(B

j

, F ), j = 1, 2, which are essentially unique, such that F = (r

1

B

1

) ∨ (r

2

B

2

) and A = r

2

r

1

. Moreover , if U

j

∈ L(B

j

) is a unitary operator , j = 1, 2, and U

2

A = AU

1

, there exists a unique unitary operator W ∈ L(F ) such that

W r

j

= r

j

U

j

, j = 1, 2.

(3)

S k e t c h o f p r o o f o f (2). Let F be the Hilbert space generated by the linear space B

1

× B

2

and the sesquilinear positive semidefinite form

h(b

1

, b

2

), (b

01

, b

02

)i ≡ hb

1

, b

01

i

B1

+ hAb

1

, b

02

i

B2

+ hb

2

, Ab

01

i

B2

+ hb

2

, b

02

i

B2

; define r

1

, r

2

by b

1

→ (b

1

, 0) and b

2

→ (0, b

2

), respectively; set W r

j

b

j

≡ r

j

U

j

b

j

, etc.

S k e t c h o f p r o o f o f T h e o r e m (1). We may assume that kAk = 1. There exist H and two isometries u

j

∈ L(B

j

, H), j = 1, 2, such that A = u

2

u

1

and H = (u

1

B

1

) ∨ (u

2

B

2

); an isometry V acting in H with domain D = (u

1

S

1

B

1

) ∨ (u

2

B

2

) is defined, with obvious notation, by V (u

1

S

1

b

1

+ u

2

b

2

) ≡ u

1

b

1

+ u

2

S

2−1

b

2

.

If (U, F ) ∈ U , an isometric extension r

j

∈ L[L

2

(E

j

), F ] of u

j

such that r

j

S

j

= U

r

j

is well defined; the following equalities hold: r

1

S

1−n

b

1

= U

n

u

1

b

1

, n ≥ 0, b

1

∈ B

1

, and r

2

S

k2

b

2

= U

∗k

u

2

b

2

, k ≥ 0, b

2

∈ B

2

. Since S

2

r

2

r

1

= r

2

r

1

S

1

, there exists w ∈ L

(E

1

, E

2

) such that M

w

= r

2

r

1

; then w ∈ F

A

. Moreover:

Theorem (3). In the same hypothesis of Theorem (1) assume kAk = 1. Set w

(z) = zP

E2

S

2

A(I − zS

1

)

−1

i

E1

. There exist an isometry V acting in a Hilbert space H and two isometries π

j

∈ L(E

j

, H), j = 1, 2, such that a bijection from U onto F

A

is defined by associating with each (U, F ) ∈ U the function w ∈ F

A

given by w(z) = w

(z) + π

2

P

H

U (I − zU )

−1

i

H

π

1

.

Concerning this section, details can be seen in [A.2].

III. Parametrization formulas. A set δ = {E

1

, E

2

, X; A}, where E

1

, E

2

, X are Hilbert spaces and A = [A

jk

]

j,k=1,2

is a bounded operator from the space X ⊕ E

1

to the space E

2

⊕ X, is called an operator colligation; it is unitary if A is a unitary operator;

a unitary colligation δ is called simple if the contraction A

21

= P

X

A

|X

is completely nonunitary (c.n.u.), i.e., no nontrivial restriction of A

21

to an invariant subspace is uni- tary. The colligation δ

0

= {E

1

, E

2

, X

0

; A

0

} is equivalent to δ iff there exists a unitary operator λ ∈ L(X, X

0

) such that A

0

(λ ⊕ I

E1

) = (I

E2

⊕ λ)A.

A colligation can be seen as a discrete linear system with response function Ψ ≡ Ψ

δ

given by Ψ (z) = A

12

+ zA

11

(I − zA

21

)

−1

A

22

, which is also called the characteristic function of the colligation. Two simple unitary colligations are equivalent iff they have the same characteristic function.

The space H

(E

1

, E

2

) is the set of analytic functions Ψ : D → L(E

1

, E

2

) on the unit disk such that kΨ k

:= sup{kΨ (z)k : z ∈ D} < ∞. The characteristic function of a unitary colligation belongs to the set B(E

1

, E

2

) := {Ψ ∈ H

(E

1

, E

2

) : kΨ k

≤ 1} of contractive analytic functions. The converse holds: if Ψ ∈ B(E

1

, E

2

), by applying Lemma (2) to the contraction M

Ψ

, it can be proved that it is the characteristic function of a simple unitary colligation.

Let V be any isometry with domain D, range R, and defect subspaces N and M ;

that is, N and M are the orthogonal complements in H of D and R, respectively. To

describe the set U of equivalence classes of minimal unitary extensions of V is equivalent

to describing the set of all (nonequivalent) simple unitary colligations {N, M, X; A} with

given N and M . Thus, there exists a bijection between U and the set B(N, M ) of

contractive analytic functions:

(4)

Theorem (4). Let V be an isometry acting in a Hilbert space H with defect sub- spaces N and M . A bijection between the set U of equivalence classes of minimal unitary extensions of V and the set B(N, M ) of contractive analytic functions is obtained by asso- ciating with each (U, F ) ∈ U the characteristic function of the simple unitary colligation {N, M, X; U

|X⊕N

}, with X = F H:

Ψ (z) = P

M

U

|N

+ zP

M

U

|X

(I − zP

X

U

|X

)

−1

P

X

U

|N

.

If V is as in (4) and D is its domain, a unitary extension B ∈ U (H ⊕ M, N ⊕ H) of V is given by B(h, m) = (P

N

h, m + V P

D

h), ∀h ∈ H, m ∈ M . If L is a closed subspace of H and L

= H L, set δ

(V,L)

= {L⊕M, N ⊕L, L

; B} and let S

(V,L)

= [S

jk

]

j,k=1,2

∈ B(L⊕

M, N ⊕ L) be the characteristic function of the unitary colligation δ

(V,L)

. If U ∈ L(F ) is a unitary operator, then δ

(U,L)

= {L, L, F L; U } and its characteristic function is S

(U,L)

(z) = P

L

U (I − zP

F L

U )

−1|L

. Then:

Theorem (5). If (U, F ) ∈ U corresponds to Ψ ∈ B(N, M ) in the bijection given by Theorem (4), then, for every z ∈ D,

S

(U,L)

(z) = S

21

(z) + S

22

(z)Ψ (z)[I − S

12

(z)Ψ (z)]

−1

S

11

(z).

This formula was stated by Arov and Grossman ([AG]). As a consequence we obtain a parametrization of all solutions of the interpolation problems that can be solved by means of Theorem (1).

Theorem (6). In the same hypothesis and with the notation of Theorem (3), set L = (π

1

E

1

) ∨ (π

2

E

2

) and [S

jk

]

j,k=1,2

= S

(V,L)

. A bijection from B(N, M ) onto F

A

is given by associating with each Ψ ∈ B(N, M ) the function w ∈ F

A

defined by

w(z) = w

(z) + π

2

S(z)[I

L

− zS(z)]

−1

π

1

,

S(z) = S

21

(z) + S

22

(z)Ψ (z)[I − S

12

(z)Ψ (z)]

−1

S

11

(z).

Proofs of the Arov–Grossman formula and of the other statements in this section are given in [A.3].

IV. A lifting theorem. Parrott’s extension of the Nagy–Foia¸s theorem follows from:

Theorem (7). Let T

j

∈ L(E

j

) be a contraction with minimal unitary dilation U

j

∈ L(F j), j = 1, 2, and X ∈ L(E

1

, E

2

) such that XT

1

= T

2

X. Set (A

1

, A

2

) ∈ A if A

j

∈ L(E

j

) bicommutes with T

j

, j = 1, 2, and XA

1

= A

2

X, XA

1

= A

2

X; let b A

j

∈ L(F

j

) be the extension of A

j

that commutes with U

j

and is such that k b A

j

k = kA

j

k, j = 1, 2. There exists τ ∈ L(F

1

, F

2

) such that τ U

1

= U

2

τ , P

E2

τ

|E1

= X, kτ k = kXk and τ b A

1

= b A

2

τ ,

∀(A

1

, A

2

) ∈ A.

Assume kXk = 1. Set M

1

= W{U

1n

E

1

: n ≥ 0} and M

20

= W{U

2n

E

2

: n ≤ 0}. Let H be a Hilbert space such that H = M

1

∨ M

20

and P

MH0

2|M1

= X

0

:= XP

M1

E

1

. Every

(A

1

, A

2

) ∈ A defines an operator A ∈ L(H) by A(g

20

+ g

1

) = b A

2

g

20

+ b A

1

g

1

, ∀g

20

∈ M

20

and g

1

∈ M

1

. Set D = U

2

M

20

∨ M

1

; define the isometry V by V (U

2

g

20

+ g

1

) = g

02

+ U

1

g

1

.

Let U ∈ L(F ) be the minimal unitary dilation of the contraction V P

D

∈ L(H). We

may assume that F = F

1

∨ F

2

and that U

|Fj

= U

j

. Then A extends to b A ∈ L(F )

(5)

such that b AU = U b A, so b A

|Fj

= b A

j

. Setting τ = P

FF

2|F1

the result follows. Proofs and two-dimensional generalizations can be seen in [A.4].

V. A band extension problem. We are given the integers N and p such that 0 ≤ p < N −1, the Hilbert spaces G

j

, 1 ≤ j ≤ N , and the operators A

ij

∈ L(G

j

, G

i

), 1 ≤ i, j ≤ N, |i − j| ≤ p. The band A

(p)

:= {A

ij

: |i − j| ≤ p} is positive if the operators [A

kj

]

i≤k,j≤i+p

∈ L[L(G

j

: i ≤ j ≤ i + p)] are positive for 1 ≤ i ≤ N − p; A

(p)

is positive definite (p.d.) if [A

kj

]

i≤k,j≤i+p

is positive definite for 1 ≤ i ≤ N − p. Recall that an operator in a Hilbert space is positive definite if it is positive and boundedly invertible.

Set G = L(G

j

: i ≤ j ≤ N ). A positive operator B = [B

kj

]

1≤k,j≤N

∈ L(G) such that B

ij

= A

ij

whenever |i − j| ≤ p is called a positive extension of the given band. The following statement is related to one of the problems that are solved in [GKW1].

Theorem (8). Every positive band A

(p)

has positive extensions. If A

(p)

is positive definite, it has positive definite extensions and there exists one of them, A, such that [A

−1

]

rs

= 0 if |s − r| > p.

Assume p ≥ 1. If r ∧ s denotes the minimum of r and s, set C = {(i, j) ∈ Z

2

: 1 ≤ i ≤ N, i ≤ j ≤ (i + p) ∧ N }, G

ij

= G

j

for every (i, j) ∈ C and e G = L{G

ij

: (i, j) ∈ C}; thus, every f ∈ e G is given by [f

ij

]

(i,j)∈C

, f

ij

∈ G

j

; its support is the set supp f := {(i, j) ∈ C : f

ij

6= 0}.

Let H be the Hilbert space generated by the vector space e G and the sesquilinear hermitian positive semidefinite form in e G given by

[f, f

0

] ≡ X

{hA

jk

f

ik

, f

ij0

i

Gj

: (i, j), (i, k) ∈ C}.

For any f ∈ e G such that f

ii

= 0, 1 ≤ i ≤ N , let g = τ f ∈ e G be given by g

ij

= f

i−1,j

if (i, j), (i − 1, j) ∈ C and g

ij

= 0 if (i, j) ∈ C but (i − 1, j) 6∈ C. In a natural way, τ defines an isometry V acting in H.

For any v ∈ G

t

, 1 ≤ t ≤ N , let λ

t

v ∈ H be given by v

0

∈ e G such that supp v

0

= {(t, t)}

and v

tt0

= v. Then λ

i

V

i−j

λ

j

= A

ij

, ∀(i, j) ∈ C.

For any (U, F ) ∈ U , a positive extension A = A(U, F ) of the band A

(p)

is given by A

ij

= (i

FH

λ

j

)

U

i−j

(i

FH

λ

i

), 1 ≤ i, j ≤ N , and every positive extension of the band A

(p)

is obtained in this way.

Assume that A

(p)

is p.d.; then A := A(U , F ) is a positive extension of A

(p)

and [A

−1

]

rs

= 0 if |s − r| > p.

Proofs of the above assertions can be seen in [A.5].

VI. Schur analysis. Let V be any isometry acting in H, with domain D and defect subspaces N and M . If (U, F ) ∈ U set H

1

= H ∨ U H and V

1

= U

|H1

, let N

1

and M

1

be the defect subspaces of the isometry V

1

that acts in H

1

, and set ν

1

= P

M1

U

|N 1

. By

iteration a sequence of contractions {ν

k

: k > 0} is associated with each (U, F ) ∈ U , and a

Schur type analysis of the unitary extensions of an isometry is established ([A.1]). In fact,

when the method of unitary extensions of isometries is applied to the Carath´ eodory–Fej´ er

problem, those {ν

k

: k > 0} are the classical sequences of Schur parameters.

(6)

In general, a bijection between U and the set of (N, M )-choice sequences (see [FF]) is established. If (U , F ) ∈ U is given by the minimal unitary dilation of the contraction V P

D

∈ L(H), then the corresponding sequence of “Schur parameters” is such that ν

k

= 0 for every k > 0, and the corresponding solution of a generalized interpolation problem can be considered as the maximum entropy solution.

There exists a bijection between the set of all the Nagy–Foia¸s intertwining liftings of a commutant and a set of choice sequences (see [FF] and references therein); that result can be proved by means of the above sketched Schur analysis of the unitary extensions of an isometry. In [FFG], a “central intertwining lifting” is studied; it may be conjectured that it corresponds to the Parrott type lifting we considered in Section IV.

Concerning the band extension problem considered in Section V, this kind of Schur analysis of unitary extensions of isometries gives another proof of the follow- ing facts ([GKW1]): each positive extension A of A

(p)

is bijectively associated with an (N − p)-tuple of contractions {T

(k)

: p ≤ k ≤ N − 1}; when A

(p)

is p.d., A is p.d. iff kT

(k)

k < 1, p ≤ k ≤ N − 1, and A corresponds to T

(k)

= 0, p ≤ k ≤ N − 1.

References

[AF] R. A r o c e n a and F. M o n t a n s, On a general bidimensional extrapolation problem, Colloq. Math. 64 (1993), 3–12.

[A.1] R. A r o c e n a, Schur analysis of a class of translation invariant forms, in: Analysis and Partial Differential Equations: A Collection of Papers dedicated to Mischa Cotlar, C. Sadosky (ed.), Marcel Dekker, New York and Basel, 1990.

[A.2] —, Some remarks on lifting and interpolation problems, Rev. Un. Mat. Argentina 37 (1991), 200–211.

[A.3] —, Unitary colligations and parametrization formulas, Ukrainian Math. J. 46 (1994), 147–154.

[A.4] —, On the Nagy–Foia¸ s–Parrott commutant lifting theorem, in: Contemp. Math.

189, Amer. Math. Soc., 1995, 55–64.

[A.5] —, Unitary extensions of isometries and interpolation problems: (1) dilation and lifting theorems, Publ. Mat. Uruguay 6 (1995), 137–158.

[AG] D. Z. A r o v and L. Z. G r o s s m a n, Scattering matrices in the theory of unitary extensions of isometric operators, Soviet Math. Dokl. 27 (1983), 518–522.

[DG.1] H. D y m and I. G o h b e r g, Extensions of kernels of Fredholm operators, J. Analyse Math. 42 (1982/83), 51–97.

[DG.2] —, —, A new class of contractive interpolants and maximun entropy principles, in:

Oper. Theory Adv. Appl. 29, Birkh¨ auser, 1988, 117–150.

[FF] C. F o i a¸s and A. E. F r a z h o, The Commutant Lifting Approach to Interpolation Problems, Birkh¨ auser, Basel, 1990.

[FFG] C. F o i a¸s, A. E. F r a z h o and I. G o h b e r g, Central intertwining lifting , maximum entropy and their permanence, Integral Equations Operator Theory 18 (1994), 166–

201.

[GKW.1] I. G o h b e r g, M. A. K a a s h o e k and H. J. W o e r d e m a n, The band method for positive and contractive extension problems, J. Operator Theory 22 (1989), 109–

115.

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[GKW.2] I. G o h b e r g, M. A. K a a s h o e k and H. J. W o e r d e m a n, The band method for positive and contractive extension problems. An alternative version and new appli- cations, Integral Equations Operator Theory 12 (1989), 343–382.

[GKW.3] —, —, —, A maximum entropy principle in the general framework of the band method , J. Funct. Anal. 95 (1991), 231–254.

[N] N. K. N i k o l ’ s k i˘ı, Treatise on the Shift Operator , Springer, New York, 1986.

[P] S. P a r r o t t, On a quotient norm and the Sz.-Nagy–Foia¸ s lifting theorem, J. Funct.

Anal. 30 (1978), 311–328.

[S] D. S a r a s o n, Generalized interpolation in H

, Trans. Amer. Math. Soc. 127 (1967), 179–203.

[Sz.-NF] B. S z. - N a g y and C. F o i a¸s, Harmonic Analysis of Operators on Hilbert Space,

North-Holland, Amsterdam, 1970.

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