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ISSN 2074-1863 Уфимский математический журнал. Том 7. т 2 (2015). С. 123-144.
ON SPECTRAL AND PSEUDOSPECTRAL FUNCTIONS OF FIRST-ORDER SYMMETRIC SYSTEMS
V.I. MOGILEVSKII
Abstract. We consider first-order symmetric system Jy'-B(t)y = A(t)f (t) on an interval i = [a, b) with the regular endpoint a. A distribution matrix-valued function X(s), s £ R, is called a pseudospectral function of such a system if the corresponding Fourier transform is a partial isometry with the minimally possible kernel. The main result is a parametrization of all pseudospectral functions of a given system by means of a Nevanlinna boundary parameter t. Similar parameterizations for regular systems have earlier been obtained by Arov and Dym, Langer and Textorius, A. Sakhnovich.
Keywords: First-order symmetric system, spectral function, pseudospectral function, Fourier transform, characteristic matrix
Mathematics Subject Classification: 34B08,34B40,34L10,47A06,47B25
1. Introduction
Let H and H be finite dimensional Hilbert spaces, let H := H ф H ф H and let [H] be the set of all linear operators in H. We study the first-order symmetric differential system
Jy' - B(t)y = XA(t)y, t £ I, Л £ C, (1)
where В(t) = B*(t) and A(t) ^ 0 are [H]-valued functions defined on an interval I = [a,b), b ^ w, and integrable on each compact subinterval [a,[3] С I and
( 0 0 -IH\
J = I 0 HH 0 I : H ф H ф H ^ H ф H ф H. (2)
\Ih 0 0 )
Let H = L2a (I) be the Hilbert space of functions f : I ^ H such that
/(A(t)f (t),f (t)) dt< то i
and let У0(',А) be the [H]-valued solution of (1) with У0(^,А) = IH. An [H]-valued distribution function E(-) is called a spectral function of system (1) if the Fourier transform : H ^ b2(S; H) given by
(VnÍ)(s) = f(s) := J Y*(t, s)A(t)f (t) dt, f (■) £ H (3)
is an isometry. If E(-) is a spectral function, then the inverse Fourier transform is defined for each f £ H by
f (t) = j Yo(t,s) dE(s)f(s) (4)
(the integrals in (3) and (4) converge in the norm of L2(E; H) and H, respectively). If the operator A(t) is invertible a.e. on I, then spectral functions of system (1) exist. Otherwise the
В.И. МогилЕвский, О спектральных и псевдоспектральных функциях симметрических систем первого порядка. Поступила 20 октября 2014 г.
Fourier transform may have a nontrivial kernel ker Vs and hence the set of spectral functions may be empty [1, 2, 3]. The natural generalization of a spectral function to this case is an [H]-valued distribution function £(•) such that the Fourier transform Vs of the form (3) is a partial isometry. If £(•) is such a function, then the inverse Fourier transform (4) is valid for each f E H 0 ker V^. Therefore, the following problem seems to be interesting:
• To characterize [H]-valued distribution functions £(•) such that the corresponding Fourier transform Vs is a partial isometry with minimally possible kernel ker Vs and describe these functions in terms of boundary conditions.
In the paper we solve this problem applying the extension theory of symmetric linear relations to symmetric systems. As it is known, system (1) generates the minimal (symmetric) linear relation Tmin and the maximal relation Tmax(= Tmin) in H (for more details see Sect. 3.1). The domain domTmax of relation Tmax is the set of all absolutely continuous functions y E H satisfying
Jy' - B(t)y = A(t)f(t) (a.e. on X) (5)
with some /(•) E H. Moreover, the multivalued part mulTmin of Tmin is the set of all
/(•) E H such that the solution y of (5) with y(a) = 0 satisfies A(t)y(t) = 0 (a.e. on X)
and lim( Jy(t), z(t)) = 0, zE domTmax. t^b
Recall that system (1) is called regular if X is a compact interval and quasi-regular if for any A E C each solution y of (1) belongs to H. For a quasi-regular (in particular regular) system the integral in (3) converges in the norm of H and hence the Fourier transform /(•) of a function /(•) E H does not depend on a choice of a distribution function £(•). One can easily show that for a quasi-regular system
mulTmin = {/ E H : f(s) = 0, s E R} (6)
and hence mulTmin coincides with the subspace kerU defined in [4, 3].
The following theorem obtained in the paper plays a crucial role in our considerations.
Theorem 1.1. Let £(•) be an [H]-valued distribution function such that the Fourier transform Vs is a partial isometry from H to L2(£; H). Then
mulTmin C ker Vs. (7)
For quasi-regular systems formula (7) directly follows from (6). Moreover, under the additional condition ||Vs/1| = II f ||, f E domTmin, Theorem 1.1 can be derived from the results of [2] (see Remark 3.7 below).
The inclusion (7) makes natural the following definition.
Definition 1.2. An [H]-valued distribution function £(•) is called a pseudospectral function of system (1) if the Fourier transform Vs is a partial isometry with the minimally possible kernel ker Vs = mulTmin.
We call system (1) absolutely definite if the Lebesgue measure of the set {t E X : A(i) is invertible} is positive. The main result of the paper is a parametriza-tion of all pseudospectral and spectral functions of absolutely definite system (1) with deficiency indices n±(Tmin) of the minimal relation satisfying n-(Tmin) ^ n+ (Tmin). Such a parametriza-tion is given by the following theorem.
Theorem 1.3. Let system (1) be absolutely definite and assume for simplicity that n+(Tmin) = n-(Tmin). Then:
(1) There exist an auxiliary finite-dimensional Hilbert space operator functions
Ho( A)(e [H]), S (A)(e [H ®H H]) and a Nevanlinna operator function
M(A)(e [H © H © Hb]), A E C \ R, such that the identities
nr(A) = Qo( A) + S(A)(Co(A) - Ci(A)M(A))-1Ci(A)S*(A), A e C \ R
rs-S
Er(s)= lim lim — I Im QT(a + ie) da (9)
<S—+o £—+o n J-&
establish a bijective correspondence between all Nevanlinna pairs r = {C0( A),C1(A)}, Cj(A) E [H © H ©H], j E {0, —}, satisfying the admissibility conditions
lim 1 (Co(iy) - C\(xy)M(zy))-1Ci(zy) = 0 (10)
y—Y^O 'y
lim IM(iy)(Co(iy) - Ci(iy)M(iy))-1Co(iy) = 0 (11)
y—^ 'y
and all pseudospectral functions Er(-) of the system. Moreover, the above statement holds for arbitrary (not necessarily admissible) Nevanlinna pairs r if and only if lim 1M(iy) = 0 and
y—<x %y
lim y ■ Im(M(iy)h, h) = h = 0.
y—^
(2) In the case mulTmin = {0} (and only in this case) the set of spectral functions is not empty and statement (1) holds for spectral functions.
Note that operator function M(A) in (8) is defined in terms of the boundary values of respective operator solutions of (1) at the endpoints a and b, while Qo(A) and S(A) are defined in terms of M(A). Observe also that similar to (8), (9) parametrization of [H © H]-valued pseudospectral functions corresponding to self-adjoint extensions of Tmin can be found in recent works [5, 6].
Existence of pseudospectral functions follows also from the results of [2, 7]. In these papers all pseudospectral functions of regular system (1) are parametrized in the form close to (8), (9). Note that the proof of the results of [2] is not complete (for more details see Remark 3.23).
Recall that system (1) is called a Hamiltonian system if Hi = {0}. [H]-valued pseudospectral functions Eh(■) of a Hamiltonian system corresponding to a certain "truncated" Fourier transform are studied in [4, 1, 3]. In the case H = C existence of a scalar function Eh(■) is proved in [1]. A description of all pseudospectral functions Eh(■) of a regular Hamiltonian system is obtained in [4, 3]. Such a description is given in terms of a linear-fractional transform of a Nevanlinna operator pair, which plays a role of a parameter.
Our approach is based on concepts of a boundary triplet for a symmetric relation and the corresponding Weyl functions (see [8, 9, 10, 11, 12, 13, 14] and references therein). In the framework of this approach the operator M(A) in (8) is the Weyl function of an appropriate boundary triplet for Tmax. Moreover, conditions (10) and (11) are implied by results on n-admissibility from [11, 6].
In conclusion note that spectral functions of very general boundary problems were studied in the recent papers [15, 16].
2. Preliminaries
2.1. Notations. The following notations will be used throughout the paper: H, % denote Hilbert spaces; [H1, %2] is the set of all bounded linear operators defined on the Hilbert space with values in the Hilbert space H2; [H] := [H,H]; Pc is the orthoprojection in H onto the subspace £ C H; C+ (C-) is the upper (lower) half-plane of the complex plane.
Recall that a closed linear relation from Ho to H1 is a closed linear subspace in Ho ©H1. The set of all closed linear relations from Ho to H1 (in H) will be denoted by C(Ho,H1) (C(H)). A closed linear operator T from Ho to H1 is identified with its graph grT E C(Ho, H1).
For a linear relation T E C(Ho,H1) we denote by domT, ranT, kerT and mulT the domain, range, kernel and the multivalued part of T respectively. Recall that mul T ia a subspace in
linear relations of T are the relations T i G Ko) and T* G (^(Ki, Ko) defined by
T-1 = {{hi, ho} G Ki ©Ko : {ho,hi} G T} T* = {{ki,ko} G Ki © Ko : (ko, ho) — (ki,hi) = 0, {ho, hi} G T}.
Recall also that an operator function $(•) : C \ R ^ [K] is called a Nevanlinna function if it is holomorphic and satisfies Im A • Im$(A) ^ 0 and $*(A) = $(A), A G C \ R.
2.2. Symmetric relations and generalized resolvents. Recall that a linear relation A G C(H) is called symmetric (self-adjoint) if A C A* (resp. A = A*). For each symmetric relation A G C(H) the following decompositions hold
where mul A = {0} © mul A and A0 is a closed symmetric not necessarily densely defined operator in Ho (the operator part of A). Moreover, A = A* if and only if A0 =
Let A = A* G C(H), let B be the Borel a-algebra of R and let Eo(-) : B ^ [Ho] be the orthogonal spectral measure of A0. Then the spectral measure Ea(-) : B ^ [H] of A is defined as Ea(B) = Eo(B)Ph0, B GB.
Definition 2.1. Let A = A* G C(H) and let H be a subspace in H. Relation A is called H-minimal if span{H, (A - X)-1H : A G C \ R} = H.
Definition 2.2. The relations T G C(Hj), j G {1, 2}, are said to be unitarily equivalent (by means of a unitary operator U G [Hi, H2]) if T2 = UTx with U = U ©U G [Hi, H2].
Let A G C(H) be a symmetric relation. Recall the following definitions and results.
Definition 2.3. A relation A = A* in a Hilbert space H ^ H satisfying A C A is called an exit space self-adjoint extension of A. Moreover, such an extension A is called minimal if it is H-minimal.
In what follows we denote by Self (A) the set of all minimal exit space self-adjoint extensions of A. Moreover, we denote by Self (A) the set of all extensions A = A* G C(H) of A (such an extension is called canonical). As is known, for each A one has Self (A) = 0. Moreover, Self (A) = 0 if and only if A has equal deficiency indices, in which case Self (A) C Self (A).
Definition 2.4. Exit space extensions Aj = A* G C(Hj), j G {1, 2}, of A are called equivalent (with respect to H) if there exists a unitary operator V G [Hi © H, H2 © H] such that Ai and A2 are unitarily equivalent by means of U = Ih © V.
Definition 2.5. The operator functions R(-) : C \ R ^ [H] and F(■) : R ^ [H] are called a generalized resolvent and a spectral function of A respectively if there exists an exit space extension A of A (in a certain Hilbert space H D H) such that
Here Ph is the orthoprojection in H onto H and E(•) is the spectral measure of A.
In the case A G Self (A) identity (12) defines the canonical resolvent R(A) = (A — A)-i of A.
H = Ho © mul A, A = gr Ao © mul A,
R(A) = Ph(A — A)-i \ H, A G C \ R F(t) = PhE((—to, t)) \ H, tG R.
(12)
(13)
Proposition 2.6. Each generalized resolvent R(A) of A is generated by some (minimal) extension A e Self (A). Moreover, the extensions Ai, A2 e Self (A) inducing the same generalized resolvent R(-) are equivalent.
In the sequel we suppose that a generalized resolvent R(-) and a spectral function F(•) are generated by an extension A e Self (A). Moreover, we identify equivalent extensions. Then by Proposition 2.6 identity (12) gives a bijective correspondence between generalized resolvents R(A) and extensions A e Self (A), so that each A e Self (A) is uniquely defined by the corresponding generalized resolvent (12) (spectral function (13)).
It follows from (12) and (13) that the generalized resolvent R(-) and the spectral function F (•) generated by an extension A e Self (A) are related by
R(A) = f , A e R.
JR t — A
Moreover, setting Ho = H © mul A one gets from (13) that
F(ro)(:= s - lim F(t)) = P»P^o \ H. (14)
t—JU
2.3. The spaces £2(E; %) and L2(E; %). Let % be a finite dimensional Hilbert space. A non-decreasing operator function £(•) : R ^ [%] is called a distribution function if it is left continuous and satisfies E(0) = 0.
Theorem 2.7. [17, ch. 3.15], [18] Let E(-) : R ^ [%] be a distribution function. Then:
(1) There exist a scalar measure a on Borel sets of R and a function ^ : R ^ [%] (uniquely defined by a up to a-a.e.) such that ^(s) ^ 0 a-a.e. on R, a([a,/)) < ro and E(/) — E(a) = J s) da(s) for any finite interval [a, /) C R.
i«,/3)
(2) The set £2(E; %) of all Borel-measurable functions /(•) : R ^ % satisfying
ii/ii/W) = i(dE( 8 )/( s ),/( s )) :=/(*( s )f(s), f( 8 ))H da( s ) < ro
RR
is a semi-Hilbert space with the semi-scalar product
(f, g)c*^H) = i (dE( s )/( S ), g( s )) := i (*( S )/( S ), g( s ))Hda(s), f,g e £2(E; %).
RR
Moreover, different measures a from statement (1) give rise to the same space £2(E; %).
Definition 2.8. [17, 18] The Hilbert space T2(E; %) is a Hilbert space of all equivalence classes in £2(E; %) with respect to the seminorm || • ||£2(S;%).
In the following we denote by the quotient map from £2(E; %) onto L2(E; %). Moreover, we denote by C'^oc(E; %) the set of all functions g e £2(E; %) with the compact support and we put L2oe(E; %) :=^E^2oe(E; %).
With a distribution function E(^) one associates the multiplication operator A = As in L2(E; %) defined by
domAs = {Je L2(E; %) : s/(s) e £2(E; %) for some (and hence for all) /(•) e /}
As7 =Msf(s)), 7e domAs, /(•) e 7. (15)
As is known, AS = As and the spectral measure Es of As is given by
Es(B)7 =ns(XB(•)/(•)), B eB, 7e L2(E; %), /(•) e 7, (16)
where xB(•) is the indicator of the Borel set B.
Let K, K,' and H be finite dimensional Hilbert spaces and let £(s)(e [H]) be a distribution function. For Borel functions Y(s)(e [H, £]) and g(s)(e H) we let
i Y(s)dE(s)g(s) := / Y(s)^(s)g(s) da(s) (e £) (17)
Jr Jr
where a and are defined in Theorem 2.7, (1).
2.4. The classes R+(H0, Hi) and R(H). Let Ho be a Hilbert space, let Hi be a subspace in H0 and let r = {r+, r_j be a collection of holomorphic functions r±(-) : C± ^ C(H0, Hi). In the paper we systematically deal with collections r = {r+, r_j of the special class R+(H0, Hi). Definition and detailed characterization of this class can be found in our paper [6] (see also [19, 20, 5], where the notation R(H0,Hi) were used instead of R+(H0,Hi)). If dimHi < to, then according to [6] the collection r = {r+, r_j e R+(H0, Hi) admits the representation
r+(A) = {( C0( A),Ci(A)); H0}, A e C+; r_(A) = {(D0(A),Di(A)); Hij, A e C_ (18)
by means of two pairs of holomorphic operator functions
( C0(A),Ci(A)): H0 ©Hi ^H0, A e C+, and (D0(A),Di(A)) : H0 ©Hi ^ Hi, A e C_
(more precisely, by equivalence classes of such pairs). Identities (18) mean that
r+( A) = {{^0, hij e H0 © Hi : C0(A)^0 + Ci(A)hi = 0j, A e C+ t_(A) = {{h0, hij e H0 © Hi : D0(A)h0 + Di(A)hi = 0j, A e C_.
In [6] the class R+ (H0, Hi) was characterized both in terms of C(H0, Hi)-valued functions r±(^) and in terms of operator functions Cj(•) and Dj(•), j e {0,1j, from (18).
If Hi = H0 =: H, then the class R(H) := R+ (H, H) coincides with the well-known class of Nevanlinna C(H)-valued functions r(-) (see, for instance, [11]). In this case the collection (18) turns into the Nevanlinna pair
r( A) = {( C0(A), Ci(A)); Hj, A e C \ R, (19)
with C0(A),Ci(A) e [H]. Recall also that the subclass R0(H) C R(H) is defined as the set of all rO e R(H) such that r(A) = 6(= 9*), A e C \ R. This implies that r(^) e -R0(H) if and only if
r( A) = {( C0,Ci); Hj, A e C \ R, (20)
with some operators C0,Ci e [H] satisfying Im(CiC(*) = 0 and 0 e p(C0 ± zCi) (for more details see e.g. [5, Remark 2.5]).
2.5. Boundary triplets and Weyl functions. Here we recall definitions of a boundary triplet and the corresponding Weyl function of a symmetric relation following [8, 9, 13, 12, 10, 14, 21,6].
Let A be a closed symmetric linear relation in the Hilbert space H, let = ker (A* — A)
( A e C) be a defect subspace of A, let Na(A) = {{f,Af j : fe NA(A)j and let n±(A) := dimNa(A) ^ to, A e C±, be deficiency indices of A.
Next, assume that H0 is a Hilbert space, Hi is a subspace in H0 and H2 := H0 ©Hi, so that H0 = Hi © H2. Denote by Pj the orthoprojection in H0 onto Hj, j e {1, 2j.
Definition 2.9. A collection n+ = {H0 © Hi, r0, ri j, where r : A* ^ Hj, j e {0,1 j, are linear mappings, is called a boundary triplet for A*, if the mapping r : f ^ {r0 f, ri fj, f e A*, from A* into H0 © Hi is surjective and the following Green's identity holds
( f', 9) — (f, g') = (ri f ] r0^)Wo — (r0 f ] ri^)Wo + Kp2r0 f ] p2r0 holds for all f = {f, f' j, g = {g,g'j e A*.
According to [21] a boundary triplet n+ = {%0 © %1, r0, r1} for A* exists if and only if n-(A) ^ n+(A), in which case dim%1 = n-(A) and dim%0 = n+(A).
Proposition 2.10. [21] Let n+ = {%o © %i, ro, ri} be a boundary triplet for A*. Then the identities
ri \ Na(A) = M+(A)r \ Na(A), A e C+ (ri + iP2ro) \ Na(A) = M-(A)Piro \ Na(A), A e C-
well define the (holomorphic) operator functions M+(-) : C+ ^ [%o, %i] and M-0 : C- ^ [%i,%o] satisfying M+ (A) = M-(A), A e C-.
Definition 2.11. [21] The operator functions M±(-) defined in Proposition 2.10 are called the Weyl functions corresponding to the boundary triplet n+.
Theorem 2.12. [21] Let A be a closed symmetric linear relation in H, let n+ = {%o ©%i, ro, ri} be a boundary triplet for A* and let M+(-) be the corresponding Weyl function. If t = {r+, 7-} e R+(%o, %i) is a collection of holomorphic pairs (18), then for every g e H and A e C \ R the abstract boundary value problem,
{f,Af + g} e A* (21)
Co(A)ro{f,Af + g} — Ci(A)ri{f,Af + g} = 0, A e C+ (22)
Do(A)ro{ f, Af + g} — Di(A)ri{ f, Af + g} = 0, A e C- (23)
has a unique solution = ( , A) and the identity R( A) := ( , A) defines a generalized resolvent R(A) = RT(A) of A. Moreover, 0 e p(r+(A) + M+(A)) and the following Krein-Naimark formula for resolvents is valid:
Rt(A) = (Ao — A)-i — 7+(A)(r+(A) + M+(A))-i7-(A), A e C+ (24)
Conversely, for each generalized resolvent R(A) of A there exists a unique r e R+(%o, %i) such that R(A) = RT (A) and, consequently, identity (24) is valid.
Remark 2.13. It follows from Theorem 2.12 that the boundary value problem (21)-(23) as well as formula for resolvents (24) give a parametrization of all generalized resolvents
R(A) = Rt(A) = Ph(At — a)-1 \ H, a e C \ R, (25)
and, consequently, all extensions A = AT e Self (A) of A by means of an abstract boundary parameter r e R+(%o, %1).
Definition 2.14. An extension A e Self(A) (A e Self (A)) is referred to the class Selfo(A) (resp. Selfo(A)) if mulAl = mulA.
Theorem 2.15. Let under the assumptions of Theorem 2.12 r = {r+, t-} e R+(%o, %1) be a collection of holomorphic pairs (18) and let AT e Self(A) be the corresponding extension of A (see Remark 2.13). Then: (1) Identities
$t( a) := Pi (Co (A) — Ci(A)M+(A))-1Ci(A), A e C+ (26)
$t( A) = M+(A)(Co(A) — Ci(A)M+(A))-1Co(A) \ %i, A e C+ (27)
define holomorphic [Hi]-valued functions $t(0 and gr(•) on C+ satisfying Im$T( A) ^ 0 and Imgr( A) ^ 0, A e C+ . Hence there exist strong limits
BT := S— lim iPi(C0(iy) — Ci(ty)M+(ty))_iCi(iy) (28)
y—^+^0 'f
Br := s — lim iM+(iy)(C0(iy) — Ci(iy)M+(iy))_iC0(iy) \ Hi (29)
y—+TO 'y
(2) The inclusion AT e Self0(A) holds if and only if Br = BT = 0
Proof. Statement (1) for $ T( A) was proved in [6, Theorem 4.8]. Next assume that
C0(A) = (C0i(A),C02(A)) : Hi ©H2 ^ H0, D0(A) = (D0i(A),D02(A)) : Hi ©H2 ^ Hi
M+(A) = (M(A),N+(A)) : Hi © H2 ^ Hi, M_(A) = (M(A),N_(A))T : Hi ^ Hi © H2
are the block-matrix representations of C0(A), D0(A) and M±(A). Moreover,let
¿70(A) = (Ci(A), C02(A)) : Hi © H2 ^ H0; Ci(A) = —C0i(A), A e C+
M+(A) = (—M_i(A), —M_i(A)N+ (A)) : Hi © H2 ^ Hi, A e C+
Then according to [6], the identities
$r ( A):=Pi(^0(A) — Ci (A)M+(A))_iCi(A), A e C+; gr (A) := g * (A), A e C_ (30)
define a Nevanlinna function gT(•) : C\R ^ [Hi] (i.e., a holomorphic function gT(•) such that Im A • ImgT( A) ^ 0 and g*( A) = gT( A), A e C \ R). The immediate checking shows that
( P2 — M+(A))_i = —M_i(A)Pi — M_i(A)N+(A)P2 + P2
and, consequently, Pi(P2 — M+(A))_i = M+(A) (here M+(A) is considered as the operator in H0). This and (30) imply that for each A e C+
$r( A) = —Pi (Ci(A)Pi + C02(A)P2 + C0i(A)Pi(P2 — M+(A))_i)_i C0i(A) =
—Pi( P2 — M+(A))((Ci(A)Pi + C02 (A) P2)(P2 — M+(A)) + C0i(A)Pi)_iC0i (A) =
M+(A)(C02(A)P2 — Ci(A)M+(A) + C0i(A)Pi)_iC0i(A) =
M+(A)(C0(A) — Ci(A)M+(A))_iC0(A) f Hi.
Thus the restriction of $r(•) on C+ admits representation (27), which yields statement (1) for g r (A).
It was shown in [6] that the second identity in (30) can be written as
gr( A) := M(A)(D0i(A) — Di(A)M(A) — W02(A)N_(A))_iD0i(A), A e C_ Therefore, by (29) one has
BT = s — lim T(¿y) = s — lim T(¿y) =
r y—гy ni// y—_^ ty nw
s— lim iyM(¿y)(D0i(iy) — Di(zy)M(¿y) — ^(^N^r^^).
Now statement (2) follows from [6, Theorem 4.9]. □
Remark 2.16. (1) If H0 = Hi := H, then the boundary triplet in the sense of Definition 2.9 turns into the boundary triplet n = {H, r0, rij for A* in the sense of [12, 9]. In this case n+ (A) = n_(A)(= dimH) and M±(-) turn into the Weyl function M(•) : C \ R ^ [H] introduced in [10, 14]. Moreover, in this case M(•) is a Nevanlinna operator function.
In the sequel a boundary triplet n = {H, r0, rij in the sense of [12, 9] will be called an ordinary boundary triplet for A*.
(2) Let n+(A) = n-(A), let n = {'H, r0, ri} be an ordinary boundary triplet for A* and let M(•) be the corresponding Weyl function. Then an abstract boundary parameter r in Theorem 2.12 is a Nevanlinna operator pair r G R(H) of the form (19) and identities (28) and (29) become
BT = s - lim i( Co(iy) - Ci(iy)M(iy))-1Ci(iy) (31)
BT = S - lim ¿M(iy)(CQ(iy) - C1(iy)M(iy))-iCQ(iy). (32)
Note that for this case Theorem 2.15 was proved in [11, 22].
3. PSEUDOSPECTRAL AND SPECTRAL FUNCTIONS OF SYMMETRIC SYSTEMS
3.1. Symmetric systems. Let H and H be finite dimensional Hilbert spaces, let
Ho := H ©H, H := Ho © H = H © H © H (33)
and let J G [H] be operator (2). A first order symmetric system of differential equations on an interval X = [a, b), -<ro < a < b ^ ro, (with the regular endpoint a) is of the form
Jy'(t) - B(t)y(t) = AA(t)y, t G X, A G C, (34)
where B(•) and A(-) are the [H]-valued functions on X integrable on each compact interval [a,p] c X and such that B(t) = B*(t) and A(t) ^ 0 (a.e. on X).
An absolutely continuous function y : X ^ H is a solution of (34) if identity (34) holds a.e. on X. An operator function Y(•, A) : X ^ [K, H] is an operator solution of equation (34) if y(t) = Y(t, A)h is a solution of this equation for every h G K (here K is a Hilbert space with dim K < ro).
The following lemma will be useful in the sequel.
Lemma 3.1. Let K be a finite dimensional Hilbert space, let Y(•, •) : X x R ^ [K, H] be an operator function such thatY(•, s) is a solution of (34) andY(a, •) is a continuous function on R and let £(•) : R ^ be a distribution function. Then for each function g G C^oc(E; the identity
f(t)= [y (t, s)dE(s)g(s), t gX (35)
jr
defines an absolutely continuous function /(•) such that
f(t) = -J f (B(t) + sA(t))Y(t, s) dE(s)g(s) (a.e.on X). (36)
Jr
Proof. In accordance with (17), identity (35) means
f(t)= [y (t, s)*(s)g(s)da(s), t gX , (37)
jr
where ^ and a are defined in Theorem 2.7, (1). Since Y(t, s) satisfies
Y (t, s) = Y (a, s) -J (B(u) + sA(u))Y (u, s)du, t gX , (38)
J [a,t)
it follows that Y(•, •) is a continuous function on X x R. Moreover, one can easily prove that f ||^(s)g(s)|| da(s) < ro. Therefore, the integral in (37) exists and
R
J ||(B(u) + sA(u))Y(u, ss)g(s)Hduda(s) < ro. (39)
[a, i)xR
It follows from (39) and the Fubini theorem that
i(i (B (u) + s A(u))Y (u, s s )#( s ) du)da( s )= (40)
Jr\ J [a, t) J
/ ( (B (u) + s A(u))Y (u, s s)sr( s ) d<r( s) ) du.
J [a, t)\ JR J
Now combining (37) with (38) and taking (40) into account, one gets
/(i) = C - J / (/ (B(u) + sA(u))Y(u, s)^(s)5-(s) d<r(s) ] du,
./[a,i) VJR J
where C = / Y (a, s)^(s)<jf(s) d<r(s). Hence (36) holds. □
R
Denote by £A(X) the semi-Hilbert space of Borel measurable functions /(•) : X ^ H such that
J(A(i)/(i), /(¿))h < ^ and let H := pA(X) be the Hilbert space of all equivalence classes in i
£A(X) [17, Chapter 13.5]. Denote also by the quotient map from £A(X) onto ¿A(X). For each system (34) the identities
T^ax = |{y, /} S (£A(X))2 : y is absolutely continuous and
Jy'(i) - B(i)y(i) = A(i)/(i) a.e. on X}
and Tmax = (^a © ^a)Tmax define the linear relations 7max in ^A(X) and Tmax in H. Moreover, the identity
[y, z]6:=lim( Jy(i), z(i)), y, z S dom7max. (41)
tffe
well defines the skew-Hermitian bilinear form [■, •]& on dom 7max. By using this form one defines the relations Ta in ^A(X) and Tmin in H via
Ta = {{y, /} S 7max : y(a) = 0 and [y, z]6 = 0 for every z S dom^ax}
and Tmin = (^A ©^A)Ta. It turns out that Tmin is a closed symmetric linear relation in H with finite deficiency indices n±(Tmin) and Tmin = Tmax (see [23] for regular and [1, 24, 25, 26] for general systems). The relations Tmin and Tmax are called the minimal and maximal relations respectively.
The following assertion is immediate from definitions of Tmin and Tmax.
Assertion 3.2. (1) The multivalued part mulTmin of the minimal relation Tmin is the set of all / S ¿A(X) such that for some (and hence for all) / S / the solution y of the equation
jy'-B(i)y = A(i)/(i), ieX
with y(a) = 0 satisfies A(i)y(i) = 0 (a.e on X) and [y, z]b = 0, z S dom7max-
(2) The identity mulTmin = mulTmax holds if and only if for each function y S dom 7max the identity A(i)y(i) = 0 (a.e. on X) yields y(a) = 0 and [y, z]b = 0, z S dom7max.
3.2. g-pseudospectral and spectral functions. Denote by Hb the set of all / S H with the following property: there exists [j S X such that for some (and hence for all) function / S / the identity A(i)/(i) = 0 holds a.e. on ([j, 6). Moreover, denote by Y0(-,A) the [H]-valued
operator solution of (34) satisfying Y(a, A) = /H. With each / S Hb we associate the function /(•) : R ^ H given by
/( s) = jf YW, s)A(i)/(i)di, /(•) S /. (42)
By using the well-known properties of the solution Y(■, A), one can easily prove that /(•) is a continuous (and even holomorphic) function on R.
Recall that an operator V G [Hi,H2] is called a partial isometry if ||V/|| = ||/|| for all f G Hi 0 ker V.
Definition 3.3. A distribution function £(•) : R ^ [H] will be called a g-pseudospectral function of the system (34) if f G £2(£; H) for all f G Hb and the operator Vbf := f, f G Hb, admits a continuation to a partial isometry V = Vs G [H; ¿2(S; H)].
The operator V = Vs will be called the Fourier transform corresponding to £(•).
Clearly, if E(-) is a g-pseudospectral function, then for each /(•) G £A(I) there exists a unique <?(= Vsf) G ¿2(E; H) such that for each function g(-) G <7 one has
lim
Ptb
- / Y*(t, •)A(t)f(t)dt
J[a,ß)
= 0.
£2(£;H)
Proposition 3.4. Let £(•) be a q-pseudospectral function and letV = V^ be the corresponding Fourier transform. Then for each Tj G Lfoc(£; H) the function
k(t)= [Yo(t, s)dE(s)g(s), g(-) G £ Jr
belongs to and V*g = /<?(")• Therefore,
V*g = ^aQ^o(-, s)dE(s)g(s)^ , g G L2(S; H), <?(•) G £ where the integral converges in the seminorm of £A(I).
Proof. According to Lemma 3.1 /gr(-) is a continuous H-valued function on I and by (17)
h(t) = / Yo(i, s)^(s)y(s) da(s), g(-) G y, (43)
./R
where a and ^ are defined in Theorem 2.7, (1).
Let /*(•) G £A(X) be a function such that nAf*(■) = V*g. Moreover, let h G H, let 8 C X be a compact interval and let f(t) = (t)h(G £A(X)). We show that
/( f(t), A(t)fg(t))H dt= i( f(t), A(t)f*(t))mdt. (44)
Jx Jx
In view of (43) one has
jf(f(t), A(t)fg(t))mdt = ^ ^(A(t)f(t),Yo(t, s)^(s)g(s))mda(s)^ dt. (45)
Since Y0(■, ■) is a continuous function on Ix R, it follows that
J l(A(t) f(t),Yo(t, s)^(s)g(s))ml dtda(s) < rc.
R
Therefore, by the Fubini theorem one has
jf (j(W)/, s)^(S)i7(S))Hd<7(S^ di
w)/w, yo(i, s)*(s)</(s))h dA d<r(s)
/
s)Fo*(i, s)A(i)/(i), </( s))Hdi) da(s) =
/ (*( s ) /V0*(i, s)A(i)/(i)di, da(S) = (V^A/,flOL2(S;H) = ./r V ./i / h
(vta/,V*5)H = j£(/(*), A(t)/*(t))di.
Combining these relations with (45) one gets identity (44).
It follows from (44) that A(i)/^(i) = A(i)/*(i) (a.e. on X). Hence /^(-) G ¿A(X) and ^A /?(•) = /*(■) = V*£. □
Let Vs be the Fourier transform corresponding to the g-pseudospectral function £(■) and let H0 = H e ker Vs, L0 = VsH(= VsH0) and LQ = L2(E; H) © L0. Then
H = kerVs © H0, L2(E; H) = L0 © LQ. (46)
Assume also that
h
Ho := H0 ©LQ, H :=kerVs © H0 ©LQ = H © LQ = kerVs © H0 (47)
and let V G [H0, L2(E; H)] be a unitary operator of the form
V/' = (Vs r H0, ^): H0 © LQ ^L2(E; H), (48)
where is an embedding operator from LQ to L2 (E; H). Since H C H, one may consider Tmin as a linear relation in H.
Lemma 3.5. Let £(■) be a q-pseudospectral function of the system (34) and let V' be a unitary operator (48). Moreover, let (Tmin)~ G C(H) be a linear relation adjoint to Tmin in H and let A = As be the multiplication operator in L2(E; H). Then the identities
/ =( V"')*£, To7 =(V/')*A<7, g G domA (49)
define a self-adjoint operator T0 in H0 such that T0 C (Tmin)~.
Proof. It is easily seen that (Tmin)~ = Tmax © (LQ)2. Moreover, in view of (48) one has
( )*£ = V*£ + 0, 0 GL2(E;H).
Therefore, (49) can be written as
7 = VS7 + To7 = V*A£ + A<7, £ G domA.
Thus to prove the inclusion T0 C (Tmin)~ it is sufficient to show that {V*¿T, V*Ag} G Tmax for all Tj G domA.
Let Tj G domA, <?(■) G <7 and let E(■) = E^(■) be the spectral measure of A. Then by (16) and (15) for each compact interval 8 C R one has E($)<? = ^(x<s(■)<?(■)) and
AE(8)g = Xs(s)g(s))• Therefore, according to Proposition 3.4 V*E(8)g = nAy(■) and V*AE(8)g = W (■), where
y(t)= i Yo(t, s)dE(s)xs(s)g(s), f(t)=[ sY0(t, s)dE(s)xs(s)g(s). Jr Jr
It follows from Lemma 3.1 that y(-) is absolutely continuous and
y'(t) = -J i(B(t) + sA(t))Yo(t, s) dE(s)xs(s)g(s) (a.e.on X). Jr
Therefore,
Jy'(t) - B(t)y(t) = A(t) [ sYo(t, s) dE(s)xs(s)g(s) = A(t)f(t) (a.e.on X)
max
and, consequently, [y, f] G ^ax- Hence [ VS*E(8)g,V£AE(5)g}(= [tïaV0,W(■)}) G T and passage to the limit when 8 ^ R yields the required inclusion [V*g, V*Ag] G Tmax. □
Theorem 3.6. For each q-pseudospectral function £(•) of the system (34) the corresponding Fourier transform Vs satisfies
mulTmin C kerVs (50)
(for mulTmin see Assertion 3.2, (1)).
Proof. Let T0 = T* be the operator in H0 defined in Lemma 3.5 and let (To)-, be the linear relation adjoint to T0 in H' . Then (T0)~, = T0 © (ker Vs)2 and the inclusion T0 C (Tmin)~ yields
Tmin C T0 © (ker Vs)2. (51)
Let n G mulTmin. Then [0,n] G Tmin and by (51) [0,n] G T0 © (kerVs)2. Therefore, there exist f G domT0 and g, g' G ker^ such that
f + 9 = 0 + g' = n.
Since f G H0, g G kerVs and H0 ^ ker^ (see (47)), it follows that f = g = 0. Therefore, T0/ = 0 and hence n = g' G ker V:. This yields the inclusion (50). □
Remark 3.7. According to [2, Lemma 5], the identity
$af = J Y£(t, s)A(t)f(t)dt, f G domTmin n Hb, S G R (52)
defines a directing mapping $ of Tmin in the sense of [2]. By using this fact and Theorem 1 from [2] one can prove the inclusion (50) for g-pseudospectral functions £(•) satisfying the additional condition ||Vs/|| = ||fH, f G domTmin.
Definition 3.8. A g-pseudospectral function £(•) of the system (34) will be called a pseudospectral function if the corresponding Fourier transform Vs satisfies ker Vs = mulTmin.
Definition 3.9. A distribution function £(•) : R ^ [H] will be called a spectral function of the system (34) if for every f G Hb the inclusion f G £2(£; H) holds and the Parseval identity ||7|£2(s;H) = ||7||h is valid (for / see (42)).
It follows from Theorem 3.6 that a pseudospectral function is a g-pseudospectral function £(•) with the minimally possible ker VS. Moreover, the same theorem yields the following assertion.
Assertion 3.10. A distribution function £(•) : R ^ [H] is a spectral function of the system, (34) if and only if it is a pseudospectral function with ker VS (= mulTmin) = [0] (that is, with the isometry Vs).
In the following we put Ho := H 0 mulTmin, so that
H = mulTmin © Ho. (53)
Moreover, for a pseudospectral function £(•) we denote by V0 = V0,s the isometry from Ho to L2(E; H) given by
V>,s := Vs r Ho. (54)
Clearly, Vv admits the representation
Vs = (0, Vo,s) : mulTmin © Ho ^ L2(E; H) (55)
3.3. Pseudospectral functions and extensions of the minimal relation. Recall that system (34) is called definite if for some (and hence for all) À G C there exists only the trivial solution y = 0 of this system satisfying A(i)y(i) = 0 a.e. in X. We also introduce the following definition.
Definition 3.11. System (34) will be called absolutely definite if the Lebesgue measure of the set (i G X : A(i) is invertible} is positive.
Remark 3.12. (1) Clearly, each absolutely definite system is definite. Moreover, one can easily construct definite, but not absolutely definite system (34) (even with B(i) = 0 and continuous A( )).
(2) It is known (see e.g. [24]) that the maximal relation Tmax induced by the definite symmetric system (34) possesses the following property: for any (y, /} G Tmax there exists a unique absolutely continuous function y G £^(X) such that y G y and (y, /} G 7max for any / G /. Below we associate such a function y with each pair (y, /} G Tmax.
Similarly to [5, Proposition 6.9] one proves the following proposition.
Proposition 3.13. Let £(•) be a q-pseudospectral function of the definite system and let Lo be a subspace in L2(E; H) given by Lo = VsH. Then the multiplication operator A^ is Lo-minimal (in the sense of Definition 2.1).
For a Hilbert space H ^ H we put Ho := H 0 mulTmin, so that
H = mulTmin © Ho. (56)
It is clear that Ho C Ho (for Ho see (53)).
Let T G Selfo(Tmin) be a linear relation in a Hilbert space H ^ H and let H be decomposed as in (56) (for the class Selfo see Definition 2.14). In the sequel we denote by To the operator part of T. Since mulT = mulTmin, it follows that To is a self-adjoint operator in Ho. Let Eo(-) be the orthogonal spectral measure of To and let Fo(-) : R ^ [Ho] be a distribution function given by
Fo(i) = Pho£o((-^, i)) r Ho, t G R, (57)
where PH0 is the orthoprojector in Ho onto Ho. It is clear that a spectral function F(■) of Tmin generated by T is of the form
F(i) = diag (Fo(i), 0) : Ho © mulTmin ^ Ho © mulTmin. (58)
Proposition 3.14. Let system (34) be definite. Then for each pseudospectral function £(•) of this system there exists a unique (up to the equivalence) exit space extension T G Selfo(Tmin) such that the corresponding spectral function F(■) of Tmin satisfies
(( F (// ) -F (a)) J, 7)h =/ (dE(s)/(s), /(s)), 7 G H, -œ <a<^< to. (59)
J [«,/?)
Moreover, if T is a linear relation in a Hilbert space H ^ H, then there exists a unitary operator F G [Ho, L2(£; H)] such that V r Ho = and the operators To and A^ are unitarily equivalent by means of F.
Proof. For a given pseudospectral function £(•) we put Lo = VsHo and L^ = L2(£; H) 0 Lo, so that L2(£; H) = Lo © L^. Assume also that
Ho := Ho © L^, H := mulTmin © Ho © L^ = mulTmin © Ho (60)
and let F G [Ho, L2 (£; H)] be a unitary operator given by
V = (Vo,s, IL±) : Ho © L^ ^ L2(S; H). (61)
Since kerVs = mulTmin, it follows that Ho = Ho, H o = H o and V = F (see (46), (47) and (48)). Therefore, by Lemma 3.5 identities (49) with V' = F define a self-adjoint operator To in Ho. Moreover, in view of (49) the operators To and A = A^ are unitarily equivalent by means of F. Hence the spectral measure Eo(-) of To satisfies
£o([a,/)) = ^*#s([a,/))V, -œ < a < / < œ. (62)
Observe also that FHo = V^H = Lo and by Proposition 3.13 operator A^ is Lo-minimal. Therefore, the operator To is Ho-minimal.
It follows from the second identity in (60) that T := ((0} © mulTmin) © To is a self-adjoint linear relation in H with the operator part To and mulT = mulTmin. Moreover, (0} © mulTmin C Tmin C (Tmin)h and by Lemma 3.5 To C (Tmin)h. Hence T C (Tmin)h and, consequently, Tmin C T. Observe also that relation T is H-minimal, since operator To is Ho-minimal. Hence T G Selfo(Tmin).
Next we assume that F(■) is a spectral function of Tmin generated by T and let Fo(-) be given by (57). By using (62) and (61) one can easily show that
Fo(/3) - Fo(a) = i5ho^o([a,3)) r Ho = Vo*s£s([a,/))Vo,s, -œ < a < / < œ. Therefore, by (58) and (55) one has
F (// ) - F (a) = Vs*£s([a, /)) Vs, -œ < a < / < œ,
which is equivalent to (59). Finally, uniqueness of T directly follows from (59) and H-minimality of T. □
The following corollary is immediate from Proposition 3.14.
Corollary 3.15. Let £(•) be a pseudospectral function of the definite system (34). Then Vo,£ is a unitary operator from Ho onto L2(£;H) if and only if n+(Tmin) = n_(Tmin) and the corresponding extension T from Proposition 3.14 is canonical, that is T G Selfo(Tmin). If these conditions are satisfied, then operators To (the operator part ofT) and A^ are unitarily equivalent by means ofVo,^.
Remark 3.16. Applying [2, Theorem 1] to the directing mapping (52) one can give another proof of Proposition 3.14.
The following theorem is well known (see e.g. [27, 28, 29]).
Theorem 3.17. For each generalized resolvent p(À) of Tmin there exists a unique operator function fi(-) : C \ R ^ [H] such that for each / G LA(X) and À G C \ R
fl(À) 7 = ^A Q^Vo(-, À)(fi(À) + 1 sgn(i - s) J)Fo*(i ,Â)A(t)/(t) dtj, /G 7 (63)
Moreover, Q(-) is a Nevanlinna operator function.
Definition 3.18. [27, 29] The operator function Q(-) is called the characteristic matrix of the symmetric system (34) corresponding to the generalized resolvent R(A).
Remark 3.19. For a much more general situation formula (63) is obtained in [30, 31].
Since Q(-) is a Nevanlinna function, it follows that the identity (the Stieltjes formula)
1 Fs-s
Eq(s )= lim lim — / Im Q(a + ie) da. (64)
-J-s
defines a distribution [H]-valued function Eq(-). This function is called a spectral function of fi(-).
Theorem 3.20. Assume that system (34) is absolutely definite. LetT G Self0(Tmin), let F(■) and R(-) be the spectral function and the generalized resolvent of Tmin respectively generated by T, let Q(-) be the characteristic matrix corresponding to R(-) and let Eq(-) be the spectral function of H(-). Then E(-) = Eq(-) is a unique pseudospectral function of the system (34) such that (59) holds.
Proof. (1) Assume that T is a linear relations in the Hilbert space H D H. By using (63) and the Stieltjes-Livsic inversion formula one proves identity (59) for E(-) = Eq(■) in the same way as Theorem 4 in [29].
Next assume that H and H are decomposed as in (53) and (56), respectively. Since mulT = mulTmin, it follows from (59) and (14) that for any f G H one has f G £2(E;H) and ||flic2(s;H) = II Pfi0/iih ^ Il/iih Hence the operator Vbf := -sf, f G Hb, admits a continuation to an operator V G [H,L2(E; H)] satisfying
IlVfl|L2(S;H) = iiphoI\ihh, / G H. (65)
It follows from (65), (56) and the inclusion Ho C H0 that Vf = 0, f G mulTmin, and ||V/||L2(S;H) = ||/iih = il/iih, f G H0. Thus V is a partial isometry with kerV = mulTmin and, consequently, E(-) = Eq(■) is a pseudospectral function of the system (34) such that (59) holds.
(2) Next we show that each pseudospectral function E(-) satisfying (59) coincides with Eq(-). So, let E(-) be such a function, let Vs be the corresponding Fourier transform and let Es be spectral measure (16). Then by (59) for each finite interval 8 = [a,0) C R one has
F (fi) - F (a) = VS*Es(8)Vs (66)
and Proposition 3.4 yields
( F (f) -F (a)) J =ka^Jy0(; s)dE(s) f(s)^ , 8=[a,f ) C R, f G H. (67) Substituting (42) into (67) and then using the Fubini theorem one can easily show that
( F (f ) -F(a))f = K^^Ks,s(-,u)A(u)f(u)du^, 8=[a,f) C R, Jg H, /g f, (68) where
Ks,s(t,u) = jY0(t, s)dE(s)Y0*(u, s), t,u G X. (69)
Let Ks,sn(t,u) be given by (69) with E(s) = Eq(s) and let Ks(t,u) = Ks,s(t,u) - Ks,sn(t,u), t,u G X. It follows from Theorem 2.7 that there exist a scalar measure a on B and functions
tf, tfn : R ^ [H] such that
E(/3) - E(a) = / tf( s) da( s) and En(/3) - En(a) = / tfn(s) d<r(s) (70)
for any finite 8 = [a,/). Let tf(s) = tf(s) — tfn(s). Then in view of (69) one has
KS (i ,«) = jfy0(t, s)tf (s)y0*(u, s) ^(s), i,« el, 5= [a,/) C R. (71)
Since En(-) also satisfies (59), identity (68) holds with KS,£n in place of KS,£. Hence
n^J Ks(;u)A(u)f(u) d«^ =0, ¿=[a,/) C R, / e ¿A(X), e H, (72)
Denote by F (F') the set of all finite intervals 8 = [a,/) C R (resp. = [a',/') C X) with rational endpoints. Moreover, let {ej}™ be a basis in H. It follows from (72) that for any 8 e F, i'eF' and ej there exists a Borel set B = B(ej) C X such that ^i(X\ B) = 0 and
/ A(i)KS(i,«)A(«)ejdw = 0, ieB. (73)
JS'
(here is the Lebesgue measure on X). For each 8 e F put
Ks(i,«) = A(i)Ks(i,«)A(«) = jiA(i)yo(i, s)tf (s)y0*(«, s)A(«)d<r(s) (74)
and let Bs = {{i,«} e X x X : Ks(i,«) = 0}, Bo = f| Bs. It follows from (73) that
sef
^2(X x X \ BS) = 0, 8 e F, and hence ^2(X x X \ B0) = 0 (here is the Lebesgue measure on X x X). Let XA = {i e X : A(i) is invertible}. Since system (34) is absolutely definite, it follows that ^1(XA) > 0. Hence ^2(XA x XA) > 0 and, consequently, (XA x XA) n B0 = 0. Therefore, there exist i0 and «0 in I such that the operators A(i0) and A(«0) are invertible and the identity
Ks(i0,«0) ^A(i0)lti(i0, s)tf (s)F0*(«0, s)A(«0) d<r(s) = 0
holds for all i e F. Hence A(i0)y0(i0, s)tf (s)y0*(«0, s)A(«0) = 0 (a-a.e. on R) and invertibility of y0(i0, s) and y0*(M0, s) yields tf (s) = 0 (a-a.e. on R). Thus, tf(s) = tfn(s) and by (70) E( s ) = En( s ). □
Now combining Proposition 3.14, Theorem 3.20 and Corollary 3.15 we arrive at the following theorem.
Theorem 3.21. Let system (34) be absolutely definite. Then:
(1) Identities (63) and (64) give a bijective correspondence E(-) = Ef(-) between all extensions T e Self0(Tmin) and all pseudospectral functions E(-). More precisely, letT e Self0(Tmin), let R(-) = R^ (■) be the generalized resolvent of Tmin induced by T, let Q(-) = n,f(-) be the characteristic matrix corresponding to Rf (■) and let E^(-) be the spectral function of Qy(-). Then E,f(-) is a pseudospectral function of system (34). And vice versa, for each pseudospectral function E(-) of system (34) there exists a unique (up to equivalence) T e Self0(Tmin) such that
E(-) = EtO_
(2) If T e Self 0(Tmin) and E(-) = E^(-), then operators T0 (the operator part ofT) and As are unitarily equivalent and hence they have the same spectral properties. In particular this implies that the spectral multiplicity of T0 does not exceed dimH.
(3) V0,s is a unitary operator from Ho onto F2(E; H) if and only if n+(Tmin) = n_(Tmin) and E(-) = E,f with T e Self0(Tmin). In this case the operators T0 and A^ are unitarily equivalent by means of V0,s.
Next, combining the results of this subsection with Assertion 3.10 one gets the following theorem.
Theorem 3.22. The set of spectral functions of system (34) is not empty if and only if mulTmin = {0}. If this condition is satisfied, then the sets of spectral and pseudospectral functions of system (34) coincide and hence Proposition 3.14, Theorems 3.20, 3.21 and Corollary 3.15 are valid for spectral functions (instead of pseudospectral ones). Moreover, in this case statements of Proposition 3.14 and Theorem 3.21 hold with T and Vs in place of T0 and V^;S respectively.
Remark 3.23. For a not necessarily absolutely definite system Theorem 3.21 could be easily obtained from Theorem 1 in [2] applied to directing mapping (52). For this purpose it would be needed one of the statements of the mentioned Theorem 1, which is not proved in [2] (namely, uniqueness of a spectral function V of (S; $) for a given extension S = S* of S, where the notations are taken from [2]). In fact, we do not know whether Theorem 3.21 is valid for not absolutely definite systems.
4. PARAMETRIZATION OF PSEUDOSPECTRAL AND SPECTRAL FUNCTIONS
Proposition 4.1. [5] Let system (34) be definite and let n-(Tmin) ^ n+(Tmin). Then: (1) There exist a finite dimensional Hilbert space Tib, a subspace Hb C H and a surjective linear mapping
r = (T0b, r, r16)T : dom Tmax ^ H © H © H (75)
such that for all y,z G dom 7max the following identity is valid
[y, z]b = (T0by, ri6z) - (Tiby, T06z) + i(PH±T06y, PH±z) + i(fby, fbz) (76)
(here H^ = H ©H).
(2) If r is a surjective linear mapping (75) satisfying (76), then a collection n+ = {H0 ©Hi, r0, ri} with
H0 = H ©H ©HLh = H0 © H, Hi = H ©H © H = H0 © H (77)
T0{y, 7} = {-yi(a), i(y(a) - Tby), ^y} G H ©H ©Hb (78)
Ti {y, 7} = {110(a), 2 (y(a)+f6 y), -r^ y} G H ©H ©H (79)
is a boundary triplet for Tmax (in (78) and (79) y G dom 7max is a function corresponding to {y, f} G Tmax in accordance with Remark 3.12, (2)). If in addition n+(Tmin) = n-(Tmin), then
Hb = Hb, H0 = Hi =: H = H0 © Hb (80)
and n+ turns into an ordinary boundary triplet n = {H, r0, ri} forTmax with H defined by the second identity in (80).
The boundary triplet n+ constructed in Proposition 4.1 is called a decomposing boundary triplet for Tmax.
Below we suppose that the following assumptions are satisfied: (A1) System (34) is absolutely definite and n-(Tmin) ^ n+(Tmin)
(A2) Hb and H(C Hb) are finite dimensional Hilbert spaces and r is a surjective linear mapping (75) satisfying (76).
(A3) H0 and Hi are finite dimensional Hilbert spaces (77)
(A4) n+ = {H0©Hi, r0, ri} is the decomposing boundary triplet (78), (79) for Tmax and M+(-) is the Weyl function of n+ in the sense of Definition 2.11.
Definition 4.2. A boundary parameter r is a collection r = [r+, r_} G R+ (H0, H^ of the form (18).
In the case of equal deficiency indices n+ (Tmin) = n_(Tmin) identities (80) hold and a boundary parameter is an operator pair r G R(H) defined by (19). If in addition r G R0(H), then a boundary parameter will be called self-adjoint. In this case r admits the representation as a self-adjoint operator pair (20).
It follows from Theorem 2.15 that for each boundary parameter r = |r+, r_} defined by (18) there exist the limits Br and BT of the form (28) and (29).
Definition 4.3. A boundary parameter r will be called admissible if BT = BT = 0.
The following proposition is immediate from the results of [6].
Proposition 4.4. (i) If lim (iy) \ Hi = 0, then the boundary parameter r is admissi-
y^x 1 y
ble if and only if BT = 0.
(ii) Every boundary parameter is admissible if and only if mulTmin = mulTmax (see Assertion 3.2, (2)) or equivalently, if and only if lim :1yM+ (iy) \ Hi = 0 and
limy (lm( M+ (iy)ho,ho)no + 2 =ho G Hq, ho = 0, (81)
where P2 is the orthoprojector in H0 onto H2 = H0 0 Hi.
In the following theorem we describe all pseudospectral functions of the system (34) in terms of the boundary parameter r.
Theorem 4.5. Let the assumptions (A1)-(A4) be satisfied. Moreover, let
M+(A)= (^A) M2+(A)) : ^ Ho®H}, A G C+ (82)
be the block-matrix representation of the Weyl function M+(-) and let
A) = ( TpA\ 2 0^°) : ^ A G C \ R (83)
2 H H
S1(A) = ft M+A)) : HKm ^ , A G C+
&(A) = (m0Ml+2P8 I0H0) : ^ , A G C+,
H Hi
where Ph0,h G [Ho, H] is the orthoprojector in H0 onto H, Ih,h0 G [H, H0] is the embedding operator of H into H0 and Ph G [H0] is the orthoprojector in H0 onto H (see (33)). Then the identity
QT( A) = Qq( A) + Si(A)(C0(A) - Ci(A)M+ (A))-1^)^), A g C+ (84)
together with the Stieltjes inversion formula (9) establishes a bijective correspondence between all admissible boundary parameters r = {r+, r_} defined by (18) and all pseudospectral functions £(•) = £r(■) of the system (34). Moreover, statement of the theorem is valid for arbitrary (not necessarily admissible) boundary parameters r if and only if lim — M+(zy) \ Hi = 0 and
identity (81) is satisfied.
Proof. Application of Theorem 2.12 to the decomposing boundary triplet n+ for Tmax shows that the boundary problem (21)-(23) with operators r0 and r1 of the form (78) and (79) gives a parametrization R(A) = Rr(A) of all generalized resolvents of Tmin by means of a boundary parameter r. Denote by Tr(e Self(Tmin)) the extension of Tmin generating Rr(■) and by Qr(-)(= (■)) the characteristic matrix corresponding to Rr(■). Clearly, the identities T = Tr and Q(-) = Hr(-) give a parametrization of all extensions T e Self(Tmin) and all characteristic matrices Q(-) by means of a boundary parameter r. Moreover, representation (84) of Hr(-) was obtained in [32, Theorem 4.6]. Observe also that according to Theorem 2.15 Tr e Self0(Tmin) if and only if r is admissible. Combining these facts with Theorem 3.21 we arrive at the first statement of the theorem. The second statement is implied by the first one and Proposition 4.4. □
Remark 4.6. The entries of the matrix in (82) can be defined in terms of boundary values of respective operator solutions of (34) at the endpoints a and b (for more details see [5, Proposition 4.5]).
Assume now that Tmin has equal deficiency indices n+(Tmin) = n_(Tmin). Then identities (28) and (29) take a simpler form (31) and (32), where M(■) is the Weyl function of an (ordinary) decomposing boundary triplet n for Tmax.
Theorem 4.7. Let in addition to the assumptions of Theorem 4.5 the identity ^+(Tmin) = (Tmin) holds. Moreover, let
M(A)=(M°((A) M2(A)) : ^ , a e C \R
w w
be the block-matrix representation of the Weyl function M(■), let Q0(A) be given by (83) and let 5(A) = AM20(A)) : ^ ^003 A e C \ R.
Then the identity
À) = ^o( À)+ 5(À)(Co(À) -C1(À)M(À))_1C1(À)5*(Â), À G C \ R (85)
together with the Stieltjes formula (9) establishes a bijective correspondence between all admissible boundary parameters r of the form (19) and all pseudospectral functions £(•) = £r(-) of system (34). Moreover, Vo,s(G [Ho,L2(S; H)]) is a unitary operator if and only if £(•) = £r(-) with a self-adjoint (admissible) boundary parameter .
The above statements are valid for arbitrary (not necessarily admissible) boundary parameters if and only if
lim —M(¿y) r U = 0 and lim y ■ Im(M(¿y)fr, fe) = +œ, h G U, h = 0.
Proof. According to [32, Theorem 4.9] in the case n+(Tmin) = n_(Tmin) identity (84) admits representation (85). Combining of this fact with Theorem 4.5 and Theorem 3.21, (3) yields the required statements. □
The following corollary is immediate from Theorem 3.22.
Corollary 4.8. If mulTmin = (0}, then Theorems 4.5 and 4.7 are valid for spectral functions £(•) (instead of pseudospectral ones).
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Vadim Iosifovich Mogilevskii,
Department of Differential Equations, Bashkir State University, 32 Zaki Validi, Ufa, 450076, Russia
E-mail: vadim.mogilevskii@gmail.com
