spaces, characteristic functions and Štraus extensions
Transcript of spaces, characteristic functions and Štraus extensions
Pacific Journal ofMathematics
UNITARY COLLIGATIONS IN 5κ -SPACES, CHARACTERISTICFUNCTIONS AND STRAUS EXTENSIONS
AAD DIJKSMA, HEINZ K. LANGER AND HENDRIK S. V. DE SNOO
Vol. 125, No. 2 October 1986
PACIFIC JOURNAL OF MATHEMATICS
Vol. 125, No. 2,1986
UNITARY COLLIGATIONS IN ΠK-SPACES,
CHARACTERISTIC FUNCTIONS AND
STRAUS EXTENSIONS
A. DIJKSMA, H. LANGER AND H. S. V. DE SNOO
Dedicated to Earl A. Coddington on the occasion of his 65 th birthday.
The main result of this paper is the description of certain linearmanifolds Γ(λ), associated with a symmetric operator, in terms ofcertain boundary values of the characteristic function of a unitarycolligation.
1. Introduction. Let φ be a Hubert space and let φ be a ττrsρace,i.e., a Pontryagin space with K negative squares, such that φ contains φand the indefinite inner product on φ restricted to φ coincides with theHubert inner product on φ; we denote this situation by $ c s § . Let A bea selfadjoint subspace in |>2 with nonempty resolvent set p(A). With A weassociate as in [8] a family { Γ ( / ) | / G C U {OO}} of linear manifoldsΓ ( / ) c φ 2 defined by
Here P denotes the orthogonal projection from |> onto φ. We note thatA Π φ 2 is a symmetric subspace in φ 2 , with adjoint (P ( 2l4) c, i.e., theclosure in φ 2 of the set
= {{Pf,Pg}\{f,g}<EA}.
The following inclusions are obvious:
A Π § 2 c T(l) c P<2>Λ, l e C u f o o } ,
and also
Γ ( / ) c Γ ( / ) * , / e C u { o o } ,
with equality when / e p(^4).Now let S be a symmetric subspace in φ 2 . We consider the selfadjoint
extensions^ c |>2 of S, with nonempty resolvent set ρ(A), where § C 5 § .The corresponding families { Γ ( / ) | / G C U {oo}}, form the class of Strausextensions of S and T(l) for / e C \ R was characterized in [8], to which
347
348 A. DIJKSMA, H. LANGER AND H. S. V. DE SNOO
we refer for notations and definitions. An important tool in this note is
the characteristic function of a unitary colligation of the form
(l>, S, @; U), where the inner space § is a 77^-space, the outer spaces are
fixed and given by g = v(S* - μ) and ® = v(S* - μ), the correspond-
ing defect spaces of S, and where U is the restriction of Cμ(A)9 the Cayley
transform of A, where μ e p(A) \ R . The main result of this paper is the
description of T(λ) for real λ, i.e., λ e R U {oo}, in terms of certain
boundary values of this characteristic function. Straus [19], [21], [22]
investigated the case where fc = 0 and the manifolds involved are single-
valued, i.e., (graphs of) linear operators. His method we could not easily
extend to the case where K > 0 and the manifolds are multivalued.
However, by generalizing the theory of unitary colligations from the case
where the inner space is a Hubert space (see Brodskii [4]) to the case
where it is a πκ-space, we obtain a method which is simpler than the one
used by Straus (for instance, the two cases λ e R and λ = 00 need not be
treated separately) and, at the same time, works just as well in the more
general situation.
We outline the contents of this paper. In §2 we consider unitary
colligations and their characteristic functions and state results to be used
in the rest of this paper. The proofs will appear in [9]. Such characteristic
functions associated with 7rκ-spaces were also considered by Krein and
Langer [12], [13], [14]. The new ingredient in our treatment is the sys-
tematic use of unitary colligations. Closely related results are announced
by Arov and Grossman [2], Azizov [3] and Filimonov [10]. In §3 we give
the above mentioned characterization of Γ(λ), λ e R U {00}. We give a
sufficient condition for Γ(λ), λ e R u {00} to be selfadjoint. This in-
cludes a result of Stenger [17]. Also we will characterize the symmetric
linear manifold A Π φ 2 as the intersection of a finite number of mani-
folds Γ(/), thereby sharpening and generalizing a result of Brown [5]. In
[20] Straus presented a characterization of the subspaces {{/, g} ^ S*\g
= λ / } , λ e R, for the case of a densely defined symmetric operator. In
order to apply his previous theorems from [19], [21], he had to introduce
special Straus extensions in an auxiliary Hubert space. In §4 we show that
such results when S is a symmetric subspace follow directly by making use
of the theory of unitary colligations.
We dedicate this paper to Earl A. Coddington, whose work in the
theory of subspaces and its applications to ordinary differential equations
has been very stimulating for us. The first and last author wish to express
their gratitude to Prof. Coddington for many inspiring ideas and many
years of cooperation.
UNITARY COLLIGATIONS IN ΠK-SPACES 349
2. Unitary colligations. In this section we will collect some state-ments about unitary colligations in τrκ-spaces, which will be proved in [9].Let g and @ be arbitrary Hubert spaces and let § be a τrκ-space. We shalluse [ , ] as the notation for the scalar or inner product for these and otherspaces; it should be clear from the context to which space it refers. BySκ(%> ®) w e denote the class of all functions Θ with the following twoproperties:
(a) Θ is defined and meromorphic o n D = { z e C | | z | < l ) , with0 e © θ , the domain of holomorphy of Θ in D, and has values in [g, ©].
(b) The kernel
has K negative squares, i.e., for arbitrary choices of n G N, z,- e ® θ andf. E g , i = l,,,,,«, the n X n hermitian matrix
has at most K and for at least one such choice exactly K negativeeigenvalues.
Let Δ be a unitary colligation, i.e., a quadruple of the form Δ =( $ , S , ® ; t f ) , where
is unitary, i.e., isometric and surjective. Here ( | ) (( | )) is the orthogonaldirect sum of $ and g (©, respectively), T e [|>, |>], F e [g, §], G G[ § , ®] and ffe[g, ©]. According to M. G. Krein the characteristicfunction Θ = ΘΔ of Δ is defined by
(2.2) Θ(z) = H + zG(l - zTyλF, z"1 e p(Γ),
see Brodskii [4]. As U is unitary, it is easy to see that T is a contractionand hence the spectrum σ(Γ) of T consists of points from Dc, the closureof D in C, and of at most K points from C \ D C , which are normaleigenvalues of T, cf. [11]. Thus Θ(z) is defined for z e D, with theexception of at most K points.
The unitary colligation Δ is called closely connected if the linear spanof all elements of the form Γ T / or (T*)nG*g with m, n e N U {0},/e gand g e ©, is dense in $, or, equivalently, if there exists no nontrivialsubspace S c | with Γ(®) = β and Γ| f t is isometric. In particular, if Δ isclosely connected, Γhas no eigenvalues on the unit circle 3D. Finally, two
350 A. DIJKSMA, H. LANGER AND H. S. V. DE SNOO
unitary colligations Δ = (φ, g, © U) and Δ' = ( $ ' , g, © W) are calledunitarily equivalent if there exists a unitary operator Z e [ § , §'] suchthat
THEOREM 2.1. (i) Le/ Δ = (§,g,@;t/)&efl unitary colligation whereI? w α πκ-space. Then ΘΔ e S ^ g , ©) /or some κf with 0 < κr < K. If Δ isclosely connected then κf = /c.
(ii) // Θ e 5κ(2ί5 ©) Λ « /Λ r ejcύ/j a unitary colligation Δ =(|>, gf, © £/), wΛer φ w β πκ-space, such that Θ = ΘΔ. ΓΛe colligation Δαz« Z>e chosen such that it is closely connected also, in which case it isuniquely determined up to unitary equivalence, and 2)Θ = { z e C | z ~ 1 £
It follows from the construction, that if the colligation Δ is closelyconnected, the dimension of the inner space |> is equal to the sum of thenumbers of negative and positive squares of the kernel SΘΔ.
The next theorem deals with the behaviour of Θ e S^g, ©) on theboundary 3D of D. By lim z^ ? we denote the nontangentiallimit a s z G Dtends to ξ e 3D.
THEOREM 2.2. Let Θ <Ξ S^g, ©), ξ e 3D, ψ e g. Then the followingstatements are equivalent:
(i) there exists an element φ e ©, such that \\φ\\ = ||ψ|| <
lim -—1 -
(ϋ)
lim1-1*1
(iii) there exists a sequence (zn) in © θ with zn
sup < 00.
//" one o/ ίΛese statements is valid, then φ in (i) w uniquely determined andψ = l im z J H θ(z)ψ.
If Θ = ΘΔ wAere Δ is β closely connected unitary colligation, thenequivalent to (i), (ii) or (iii) is
(iv)
UNITARY COLLIGATIONS IN ΠK-SPACES 351
and then (iv) implies
Let Θ e Sκ(g, @) and let Θ = ΘΔ where Δ is as in Theorem 2.2. Asζ ^ 3D is not an eigenvalue of T we may extend the definition of Θ to ζby putting
In general, 2)0(Θ(£)) need not be closed, but, of course, if we also havethat £-1 G p(Γ), then © 0 ( Θ (f)) = S and the definition of θ(f) coincideswith the one in (2.2). Theorem 2.2 shows that this extension of Θ to D c
can be described without making use of the fact that it is a characteristicfunction of a colligation: for f G 3D we have
®o(β(£)) = {ψ ^ g | lim m
1 i ^ ^ 1 1 existsj
and
(z)ψ, strongly, ψ e δ o ( θ ( f ) ) .
We note that for f G 3D, Θ(f) is an isometry on ©0(Θ(£)). W e remarkthat ®o(®(£)) i s i n general contained in the set
< ψ G g I lim Θ(z)ψ exists strongly!,
cf. [19]. Also we note that if ξ G 3D and 1/? G p(Γ), then Θ(f) is aunitary mapping from g onto ©. In particular, this happens whendim $ < oo, for then 3D c p(Γ).
THEOREM 2.3 (maximumprinciple). Letθ e 5K(g, ©), ψ G g, φ G ©αwJ assume that the relation φ = Θ(z)ψ holds for more than K pointsz G Φ θ . ΓΛen we Λαt e ίΛe inequality | |φ | | < ||ψ||. //"we have | |φ | | = ||ψ||, ίΛeπ
φ = Θ(z)ψforallz G Φ θ .
If Δ = (|>, g, © U) is a unitary colligation, where U is of the form(2.1), then it is clear that H maps the nullspace v(F) isometrically ontothe nullspace v(G*) with inverse H*. In terms of the characteristicfunction Θ = ΘΔ, we have that
352 A. DIJKSMA, H. LANGER AND H. S. V. DE SNOO
and hence with ψ e v{F) and φ = #ψ, that
(2.3) φ = θ(z)ψ, ||Φ|| = JIΨH, z e ® Θ .
Conversely, if we have (2.3) and Δ is closely connected if /c > 0, then
ψ e ι>(,F), φ ε *>(<?*) and φ = i/ψ.
3. Straus extensions. Let § be a Hubert space and let S c $ 2 be asymmetric subspace. It is well-known that S* can be written as
S* = S + M, + M;, direct sum in § 2 ,
where / e C \ R and Mι = {{/, g} e 5* | g = //}, is the defect subspaceof 5 at / G C. We fix μ e C \ R and consider a selfadjoint extension4 c φ2 of S with μ e p(A) where § is a Pontryagin space with K negative
squares such that φ D5 φ. The condition μ G ρ ( i ) is a restriction only ifK: > 0. For, if /c = 0 φ is a Hubert space and then C \ R c p(A). But ifK: > 0 then either p(A) = 0 or C \ R c ρ(A) with the exception of atmost 2κ points, which are normal eigenvalues of A, and one of these couldcoincide with μ. We denote the Cayley transform and its inverse at/ e C \ R by Cι and Fι respectively. Then
Cμ(A) = Cμ(S) + U, C-μ(A) = C-μ(S) + ί/*, direct sums in § 2 ,
and
(3.1) A = S 4- FM(l/) = 5 + Fp(ί/*), direct sums in | 2 ,
where ί/ is (the graph of) a unitary operator with a matrix representationof the form (2.1) in which | = § θ φ is a Pontryagin space with Knegative squares,
g = Φ(M~) = *>(£*- μ) and © = ©(ΛfJ = i>(S* - μ).
Writing the equalities in (3.1) in full detail, we obtain:
(3.2) φ —(T*(p 4- G*φ) \ I μφ — μ(T*φ + C/*φ)
φ ~ ( F * φ + H*φ)J\μφ - μ(F*φ + H*φ)
direct sums in φ 2 .
A direct consequence of these formulas is a description of A Π φ 2 :
A Π ίp == *S-l-ί{ψ — /f ψ, μ ψ — μi/ψ } |ψ ^ ϊ'(i^))
= 5 4- {{<p - H*φ,μφ - μi7*φ} | φ G *>(G*)}, direct sums in $ 2 .
UNITARY COLLIGATIONS IN Π-SPACES 353
Using definition (1.1) we obtain from (3.2)
S +{{ψ -(Gψ
(3.3)
= S + { { φ -(F*φ + ff*φ), μ<p - μ{F*ψ + H*φ)} \
φ e © , φ <Ξ § , (/ - μ) φ = (/ - μ)(Γ*φ + G*<p) } ,
direct sums in § 2 ,
Γ(oo) = S +{{ψ -(Gψ + tfψ),μψ - μ(Gψ +
/ e C,
+ H*φ)} |
φ e G , φ e § , φ = F*φ + G*φ}, direct sums in φ 2 .
Let CM = { / € C I Im /Im μ > 0}, let z: Cμ-+Ό be the fractional lineartransformation z(l) = (/ — μ)/(/ — μ) and put z(oo) = 1. Then (3.3) im-plies that the Straus extension of S associated with A via formula (1.1) canbe written in the following way: for all / e Cμ with z{l) e ® θ
(3.4) T(l) = S + {{ψ - Θ(z)ψ, μψ - μΘ(z)ψ} | ψ e v{S* - ji)},
and
(3.5) = S +{{φ - Θ(z)*φ, juφ - φ
direct sums in § 2 , where Θ = ΘΔ is the characteristic function of theunitary colligation Δ = ( $ , g, © t/) and belongs to class S^g, ©) forsome ic' with 0 < κf < K. We note that Δ is closely connected if and onlyif A is closely connected, i.e., the linear span of all elements of the form(A — iγιhy A e $ , / G p(^4), together with the elements of § is dense in| ) . It follows from Theorem 2.1 that iiA is closely connected, then K' = K.Theorem 2.1 and (3.3) imply the following result
THEOREM 3.1. Let S be a symmetricsubspace in φ 2 andfieC\R.(i) Let { Γ ( / ) | / G C U {OO}} be a Straus extension of S associated
with a self adjoint extension A of S in | ) 2 with μ e p(A), where § ^>s& is aπκ-space. Then there exist uniquely a κf with 0 < κf < K and a functionΘ e Sκ.(v(S* ~ μ), v(S* - μ)) such that, for all I e Cμ with z(l) e ©Θ,T(l) is given by (3.4). Furthermore, for these values of 1(3.5) is valid. If A isclosely connected then K' = K.
(ϋ) If for some Θ e Sκ(v(S* - μ), ^(S* - μ)) and all I e Cμ
z(/) e 35θ, Γ(/) c $ 2 ώ gwe/i ί/ G C U ( O O ) SWCA ίΛαί {Γ(/) | /
(3.4), /Λ « Γ(/) cα« Z? extended to all
C U {OO}} is a Straus extension of S
354 A. DIJKSMA, H. LANGER AND H. S. V. DE SNOO
associated with a self adjoint extension A of S in | ) 2 with μ ^ p(A), whereφ D 5 § is a πκ-space and then T(ϊ)9 z(l) e ©Θ, w gwe/i by (3.5). Λ α«d $can be chosen such that A is closely connected, in which case they areuniquely determined up to isomorphisms which, when restricted to φ, areequal to the identity on φ.
This theorem is actually another formulation of Theorem 5.1 of [8]and shows that the description of Straus extensions given there is one interms of characteristic functions of unitary colligations. In [18] Strausidentified the mapping Θ in (3.4), restricted to the operator case withK = 0, with a characteristic function (in his sense) of some operator in $.It can be shown that this notion is equivalent to that of a characteristicfunction of a colligation associated with this operator.
For λ E R U { o o } w e reformulate (3.3) as follows
(3.6) Γ(λ) = S + {{ψ -(if + ζG(I -
ψ e g, Fψ €= 3t(/ - fΓ)}, direct sum in § 2 ,
where f = (λ - μ)/(λ - μ) if λ e R and f = 1 if λ = oo. Theorem 2.2now implies that T(λ) for these values of λ can be characterized as aboundary value of Γ(/), / e Cμ.
THEOREM 3.2. Let S be a symmetric subspace / n § 2 , μ e C \ R and letT(l) c φ 2 te gjircn ty (3.4), / e Cμ, z(/) e ®θ, wterβ θ e 5,(^(5* - μ),
= (λ -we have
T(λ) = S + {{ψ - θ(f )ψ, μψ - μθ(f )Ψ} I Ψ
= lim θ(z)ψ, ψ e ϊ
Theorem 3.1 restricted to the operator case with K = 0 coincides withthe main results of Straus in [19], [21] and [22].
UNITARY COLLIGATIONS IN ΠK-SPACES 355
As Γ(/)* = T(ϊ)9 1 e p(A\ T(l) is a subspace for / e p(Λ). How-ever, for those λ e R u { o o ) which do not belong to ρ(A) T(λ) need notbe closed in general. Sufficient conditions for T(λ) to be closed for allλ e R u {00} are that dim v(S* - μ) < 00 and dim v(S* - μ) < 00.Another sufficient condition is that dim φ < 00, as follows from thefollowing theorem.
THEOREM 3.3. Let A be a selfadjoint subspace in |>2 with ρ(A) Φ 0,where φ is a πκ-space, and let φ c ^ be a Hubert space such thatdim | θ φ < oo. Let P be the orthogonal projection from φ onto φ. Then
T(λ)={{Pf,Pg)\{f,g}eA,g-\feξ>}, λ e R,
and
Γ(oo) = {{/, Pg) I {/, g } e Λ / e i } = PA\9,
are selfadjoint in φ 2.
f. Let μ e p(A) n (C\R) and let Δ = (φ, φ, φ; ί/) where § =§ θ φ and U = Cμ(Λl). Then Δ is a unitary colligation and without loss ofgenerality we may assume that it is closely connected. Write
T F
and let Θ = ΘΔ. Then, as dim φ < oo, for all ξ e 3D we have thatl/ζ e ρ(T) and therefore
is unitary for all ξ G 3D. Formula (3.6) with S = {{0,0}} c φ 2 andS = φ now implies that Γ(λ) is selfadjoint for all λ e R U { oo}.
The statement in Theorem 3.3 about Γ(oo) coincides with a result ofStenger [17] in case fc = 0 and A is a selfadjoint operator. The theorem isstill valid when φ is a 77 -subspace of φ, 0 < /c' < /c.
THEOREM 3.4. Lei § be a Hubert space and ^ be a mκ-space withφ C5 φ. Lei A be a selfadjoint subspace in φ 2 w/iΛ p( 4) # 0, wΛ/cΛ wclosely connectedifκ>0. We assume μ e p(^4)\R, 5 iΛαi /Ae correspond-ing Straus family is represented for all I e Cμ w/YΛ z(/) e ΦΘ Ay
Γ(/) = {{ψ - θ(z(/))ψ,μψ - |»θ(z(/))ψ} |ψ e $}/or 5ome Θ e Sκ(φ, φ). // SΊ c φ 2 is α symmetric subspace in φ 2,
356 A. DIJKSMA, H. LANGER AND H. S. V. DE SNOO
for more than Kpoints I e Cμ with z(l) e φ θ , then we have
Proof. We assume S1 c Γ(/), for / = ll9... ,/ κ + 1 in Cμ w i t h z ^ ) G © Θ ,
i = 1,... ,κ + 1. Let (α, /?} G Sv then
for some ψ e φ , The symmetry of SΊ implies \\β - μa\\ = \\β - μα||, so
that we have
for more than K points /z such that z(lt) e © Θ . Using Theorem 2.3 and the
formula following (3.2) with S = {{0,0}}, we obtain the desired result.
This result, stated in a slightly different way, can be found in [8]. The
present proof is similar to the one given by McKelvey [16] who showed
this result for the case of a Hubert space |> and operators S and A. A
direct consequence of Theorem 3.4 is
() < K + l ,/κ + 1 \
A n £ 2 = Π HIM n r(/, ), l <
where /,., i = 1,... ,/c + 1, are distinct points in Cμ with z(/z) e ® θ . This
can be seen by checking that the set on the right-hand side is symmetric.A more general result is contained in the following corollary.
COROLLARY. Let $ be a Hubert space and |> be a πκ-space with
§ c s | , Let A be a self adjoint subspace in | ) 2 with p(A) Φ 0 , which is
closely connected if K > 0. We assume μ ^ ρ(^4)\R, so that the correspond-
ing Straus family is represented by Θ e Sκ(φ9 § ) . Let ll9... Jκ+1 e C^ be
K + 1 distinct points such that z(lt) e ® θ , i = l , . , . ,κ + l, and let
mv.. .,mκ+ι G Cμ be K + 1 distinct points such that z{mt) ^ Φ θ , i =
l,...,fc + 1. Then
κ + 1
( ( / ) n r(m,)).
Proof. It is sufficient to show that the manifold on the right-hand is
symmetric, because then we may apply Theorem 3.4. So we assume
Π
UNITARY COLLIGATIONS IN ΠK-SPACES 357
which implies by (3.4) and (3.5) that
{β - μα, jβ - μa) = {ψ,θ(z(/,))ψ} = {Θ(z(mz))*φ, φ}?
for ψ , φ e § , i = l,...,/c + 1. This representation yields ||ψ|| = ||φ||, seeTheorem 2.3, or, equivalently, Im(α, β) = 0, and the proof is complete.
This corollary contains a result of Brown [5]. He considered a denselydefined symmetric operator S in φ, and assumed that A is a self adjointoperator extension of S in the Hubert space φ. In that case he proved
/eC\R
or, strictly speaking, the equivalent result
ίeC\R
4. A special extension of a symmetric subspace. Let φ be a Hubertspace and let S c § 2 be a symmetric subspace. We define the linearmanifolds S(l), I e C U {oo}, by
fs(/) S + , /eC,
In this section we will study the boundary behaviour of S(l) as / tends toλ ^ R U {oo}, analogous to the results in Theorem 3.2. First we notesome obvious consequences of the definition. We have S c S(/) c S*,/ G C U {oo}; S(/) is maximal dissipative for I G C + , S(l) is maximaldissipative for / e C", S(/)* = S(ϊ) for / G C \ R and S(λ) is a (notnecessarily closed) symmetric linear manifold for λ G R U {OO}. Notethat for λ e R S(λ) is selfadjoint if and only if
dl(S - λ) = 9i(S* - λ) n($l(S - λ)) c ,
while (see [7]) S^oo) is selfadjoint if and only if
The manifold *S(oo) plays a role in determining whether an extension of Sis an operator or not, cf. [6] and [22].
THEOREM 4.1. Let S be a symmetric subspace in φ 2 and let μ e C \ R,
C . Then we have
358 A. DIJKSMA, H. LANGER AND H. S. V. DE SNOO
where Θ e S0(v(S* — μ),v(S* — μ)) is the characteristic function of the
unitary operator colligation
,(s*-\
where Pμ denotes the orthogonal projection from φ onto v(S* — μ).
. For a fixed μ e C \ R we have for / e C^
φ = 9ΐ(5 - /) + P(S* - μ), direct sum.
This decomposition defines a projection of φ onto y(S* - μ), parallel to9l(S - /), which we denote by P / μ . Completely analogous to Straus [20]we obtain
ϊ+-£*;*>(* Hi-fi)(s(μ)-1)'1),
where Pμ denotes the orthogonal projection from § onto v(S* - μ).Using the identity
we obtain
Note that the identity
CjtiSiμ)) = aμ(S)\m(S_-μ)® 0\v(S*_μ),
shows that CμiSiμ)) is a partial isometry on φ. We have for all ψ e^(S* - μ) that Cμ(S(/))ψ e ^(S* - μ) and
(/ - μ)ψ - ( / - μ)Cμ(5(/))ψ e SR(S - /).
Using the notion of parallel projection we obtain the desired result.
Applying Schwarz' lemma to this characteristic function Θ, noteΘ(0) = 0, we obtain the following corollary.
COROLLARY. For μ G C \ R and I e C^ we have
l-μ
UNITARY COLLIGATIONS IN ΠK-SPACES 359
If for some ψ e v(S* — μ) and some I e C^
I I Ψ I I ,l-μ
l-μ
then ψ G v(S*) Π Sf*(0) *md, consequently,
I — μ μ
Let λ G R U {oo}, then it is not difficult to show that Cμ(S(λ))\HS*-μ)is the boundary value of Cμ(S(l))\v(S^_]i) as / -> λ, / e C . In order toapply Theorem 2.2 we remark that the symmetric subspace S c § 2 can bewritten as an orthogonal sum Sx Θ if, where Sτ is a simple closedsymmetric operator in φj, if is a selfadjoint subspace in §\ and φ y ,y = 1,2, are subspaces of φ with φ = φ x Θ φ 2 , cf. [15]. This shows thatf o r / G C U {oo}
where
{ } e 5f| g = //}, / e C,
In terms of the colligation the above splitting implies
i.e., a splitting in a unitary operator and a partial isometry, which does nothave a non-trivial unitary part. Hence without loss of generality weassume S to be simple, which is equivalent to the corresponding unitarycolligation being closely connected.
THEOREM 4.2. Let S be a simple symmetric closed operator in § 2 and letS{\\ λ e R U {00}, be given by (4.1). Let μ e C \ R am/ let Θ e^(^(S 1 * - μ), KS* - M)) 6* as g ^ « w Theorem 4.1. ΓΛew w/YΛ f =(λ — /x)/(λ - μ) //λ e R α«Jf = 1 ifλ = 00 we
S(λ) = S
®o(Θ(n)= ψ e r(S - p) I lim Ji^L^viivαi ^ lI - f x ~ lzl
Θ(z)ψ, ψ e S
360 A. DIJKSMA, H. LANGER AND H. S. V. DE SNOO
Straus [19], [21] obtained this result by extending the operator to aself adjoint operator in a larger Hubert space and then used his previousresults about Straus extensions [18]. We prove this result by directlyrelying on Theorem 2.2. In [20] Straus gives necessary and sufficientconditions for the operator S in Theorem 4.2 to be densely defined. Amore general version is given in the following theorem.
THEOREM 4.3. Let g and © be Hilbert spaces and let Θ <Ξ SΌ(g, @)with Θ(0) = 0. Then there exists a Hilbert space ίρ, a simple symmetricclosed operator S c § 2 , μ G C \ R , α n isometry F from g onto v(S* — μ)and an isometry Gfrom v(S* — μ) onto ©, such that
The operator S is densely defined if and only if for α,
ti?} 1 _ ι - f = °°'z->l 1 |Z|
or, equivalently, if for allψ e g and for all φ e © wzϊλ ||ψ|| = | |φ| |
lim1 - z
= 0 0 .
Proof. According to Theorem 2.1 there exists a closely connectedunitary colligation Δ = ( § , g, © [/), where § is a Hilbert space and Uhas the form
IT
\G oj \%! ' \%rwhere T is a completely non-unitary partial isometry. If we denote theisometric part of Γby F, we have
and we have
Θ(z) = G[zP*iV),(I- z Γ ) " 1 / ^ ^ ] ^ z e D,
where i 7 and G are isometries from g onto 9i(Y)± , and from S)(V) x onto '© respectively. For μ e C \ R w e define S = /^(K), so that S is a simplesymmetric closed operator in φ. Note that
UNITARY COLLIGATIONS IN Π.-SPACES 361
so that
Φ(F) X = p(S* - μ), ?H(V)±=v(S*-μ).
Hence we obtain
T
pμ
Cn(S(μ))
P , 0
and hence
Next we observe that S is densely defined if and only if S*(0) = {0}. IfS*(0) = {0}, then 5(oo) = S, which by Theorem 4.2 implies that©O(Θ(1)) = {0}. Conversely, if <£)0(θ(l)) = {0}, then we have by Theo-rem 4.2 that ^(oo) = S, which implies S*(0) = {0}. The final statement inour theorem follows from Theorem 2.2.
If we have © = g> then we may write G = F*W, where if is aunitary extension of V in $, see [12]. Our result resembles a similarstatement in [15]. The special case g = @ = C goes back to Livsic, andcan be found in [1].
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Received July 1, 1985 and in revised form October 10, 1985. This work was supported bythe Netherlands Organization for the Advancement of Pure Research (Z.W.O.).
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Pacific Journal of MathematicsVol. 125, No. 2 October, 1986
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for singular solutions of the capillarity equation . . . . . . . . . . . . . . . . . . . . . . . 317Donald M. Davis and Mark Mahowald, Classification of the stable
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colligations in 5κ -spaces, characteristic functions and Štrausextensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
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Palle E. T. Jorgensen, Analytic continuation of local representations of Liegroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397
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measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459Luen-Fai Tam, On existence criteria for capillary free surfaces without
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