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Coupling of definitizable operators in Krein spaces
V. Derkach1'2, C. Trunk3
department of Mathematics, Dragomanov National Pedagogical University, Pirogova 9, Kiev, 01601, Ukraine 2 Department of Mathematics, Vasyl Stus Donetsk National University, 600-Richchya Str 21, Vinnytsya, 21021,
Ukraine
3Institut fur Mathematik, Technische Universitat Ilmenau, Postfach 100565, D-98684 Ilmenau, Germany derkach.v@gmail.com, carsten.trunk@tu-ilmenau.de
DOI 10.17586/2220-8054-2017-8-2-166-179
Indefinite Sturm-Liouville operators defined on R are often considered as a coupling of two semibounded symmetric operators defined on R+ and R—, respectively. In many situations, those two semibounded symmetric operators have in a special sense good properties like a Hilbert space self-adjoint extension.
In this paper, we present an abstract approach to the coupling of two (definitizable) self-adjoint operators. We obtain a characterization for the definitizability and the regularity of the critical points. Finally we study a typical class of indefinite Sturm-Liouville problems on R. Keywords: self-adjoint extension, symmetric operator, Krein space, locally definitizable operator, coupling of operators, boundary triple, Weyl function, regular critical point.
Received: 18 January 2017 Revised: 1 February 2017
1. Introduction
Let K be a Hilbert space with the inner product (•, •) and let J be a linear operator in K, such that J = J* = J-1. The space K endowed with Hermitian sesquilinear form [•, = (J•, •) is called a Krein space and is denoted by (K, [•, .]K), for details see [1,2] or Section 2.1 below.
The Hermitian sesquilinear form [•, .]K induces in an obvious way a sign type spectrum for linear operators. In the last two decades, this notion was frequently used in theoretical physics in connection with PT-symmetric problems; here, we mention only [3-7] and in the study of PT-symmetric operators, we refer to [8-11].
A self-adjoint operator A in a Krein space (K, [., .]k) is said to be definitizable [12], if its resolvent set p(A) is nonempty and there exists a real polynomial p such that p(A) is nonnegative in (K, [., -]K). If a1 < a2 < • • • < aN is the set of all real zeros of p, then there exists a spectral function E(A) of A, which is defined on all intervals A, such that the endpoints of A do not belong to the set {aj}N=1, E(A) takes values in the set of orthogonal projections, commuting with A and E(A) is monotone on each interval (aj,aj+1). These intervals are classified in [12] as intervals of positive and negative type and the points aj which are spectral points of neither positive type nor negative type are called critical, see exact denitions in Section 2.2. A critical point a is called regular, if the operators E(A) are uniformly bounded for all small A containing a, otherwise it is called singular. The set of critical points of A is denoted by c(A), the set of regular (singular) critical points of A is denoted by cr(A) (cs(A), respectively). The notion of local definitizability of a self-adjoint operator A in a Krein space (K, [., .]k) was introduced in [13,14], see Section 3 below.
In the present paper, the following problem is studied: the problem of the definitizability of the coupling A of two symmetric operators A+ and A_ and the regularity of their critical points. Note the definition of the coupling from [15] adapted to the case of Krein spaces. Let a Krein space (K, [., -]K) be the orthogonal sum K = K+ [+]K_ of (K, [ •, • ]) of two Krein spaces (K+, [., -]K+) and (K_, [., -]K_), such that the subspaces:
D+ = {/ € K+ n (domA):A/ G K+} and D_ = {/ G K_n (domA):A/ G K_}
are dense in K+ and K_ and the restrictions:
A+ = A|d+ and A_ = A|D_
are symmetric operators with defect numbers (1,1) in the Krein spaces (K+, [ •, • ]K) and (K_, [ •, • ]K), respectively. The operator A is called a coupling of two symmetric operators A+ and A_. The coupling A of two symmetric operators A+ and A_ is not uniquely defined by the above definition. We will make this definition more precise in Theorem 4.4 by using the boundary triple approach developed in [16-19]. For differential operators with indefinite weights, the coupling method was used in [20], and also in [21-23] to study the similarity problem and in [24] to study definitizabilty.
The main result of the paper is Theorem 4.6, where conditions for regularity of the critical point to g c(A) are found under the assumptions that the symmetric operators A+ and A_ admit definitizable and semibounded extensions A+,0 and A_j0. The proof is based on the K. Veselic criterion of regularity [25,26] adapted to the case of definitizable operators in [27]. In the case when A+ and A_ are Hilbert space symmetric operators, similar results were obtained in [23] and [28].
Typically, such problems arise in the study of indefinite Sturm-Liouville operators:
i(f)(t) wtt (-1 (f) + q(t)f(t)) fora- 4 G r, (u)
where the coefficients r, q and w are real functions on R satisfying the conditions: (C1) r, q, w G ¿i1oc(R) and r, w > 0 a.e. on R, (C2) the expression i is in the limit point case at —to and at +to,
(C3) minimal differential operators B± generated by ±i in L2w (R±) are semibounded from below. The operator A generated by the differential expression (1.1) in the Krein space is the coupling of two semibounded symmetric operators A± := ±B±. In Proposition 5.1, it is shown that the operator A is definitizable over a vicinity of to and conditions (4.18) for to G cs(A) are formulated in terms of the m-coefficients for the operators B±. In the case w = 1, the conditions (4.18) are fulfilled automatically [28]. This fact was proved earlier by another method in [29].
1.1. Notations and preliminaries
By C+, we denote the set of all z g C with positive imaginary part and we set C := C U {to} and R := R U {to}.
A complex function m is called a Nevanlinna function if m is holomorphic at least on C \ R and satisfies the following two conditions:
m(z) = m(z) and Im m(z) > 0, for all z G C+. (1.2)
For information on Nevanlinna functions, we refer readers to [30] and [31, Chapter II].
All operators in this paper are closed densely defined linear operators. For such an operator T, we use the common notation p(T), dom(T), ran(T) and ker(T) for the resolvent set, the domain, the range and the nullspace, respectively, of T. We define the extended spectrum <r(A) of A by a(A) := <r(A) if A is bounded and a(A) := a(A) U {to} if A is unbounded and we set p(A) := C \ a(A).
2. Definitizable operators in Krein spaces 2.1. Krein spaces
We recall standard notation and some basic results on Krein spaces. For a complete exposition on the subject (and the proofs of the results below) see the books by Azizov and Iokhvidov [1] and Bognar [2]. A vector space K with a Hermitian sesquilinear form [., .]K is called a Krein space if there exists a so-called fundamental decomposition
K = K+ + K—,
such that (K+, [., .]K) and (K_, —[., .]K) are Hilbert spaces which are orthogonal to each other with respect to [., .]K. Those two Hilbert spaces induce in a natural way a Hilbert space inner product (.,.) and, hence, a Hilbert space topology on the Krein space K. Observe that the indefinite metric [., .]K and the Hilbert space inner product (.,.) of K are related by means of a fundamental symmetry, i.e. a unitary self-adjoint operator J which satisfies
(x,y) = [Jx,y],c for x,y G K. (2.1)
If H and K are Krein spaces and T : H ^ K a bounded operator, the adjoint operator T + of T with respect to the Krein spaces H and K is defined by:
T + := JhT J
where JH and JK are the fundamental symmetries associated with H and K, respectively; the operator T + satisfies [Tx, y]K = [x, T+y]K for all x g H, y G K. If A is a densely defined operator in K then the adjoint A+ of A with respect to [ •, • ]K is defined analogously. In fact, if J is a fundamental symmetry on (K, [ •, • ]K) and (.,.) is the corresponding Hilbert space inner product (2.1), then A+ = JA* J. The operator A+ satisfies the following:
[Ax,y]K = [x, A+y]K for all x G dom(A), y G dom(A+).
By analogy with the definitions in Hilbert spaces, A is symmetric in (K, [ •, • ]K if A+ is an extension of A and A is self-adjoint in (K, [ •, • ]K) if A = A+.
A densely defined operator A is called nonnegative in (K, [ •, • ]K) if [Af, f> 0 for all f e dom(A). A nonnegative self-adjoint operator in a Krein space can have an empty resolvent set; a specific example is given in [12, Section 1.2] and [2, Example VII.1.5]. But if a nonnegative self-adjoint operator in a Krein space also has a nonempty resolvent set, then it has real spectrum only.
An operator A is called semibounded from below in the Krein spaces (K, [ •, • ]/c), if there exists a e R such
that:
[Af, f ]k > a[f, f ]k , f e dom(A).
2.2. Definitizable operators
In this section, we recall some facts on definitizable operators in Krein spaces. For an overview, we refer to [32], see also [33]. For this purpose, it is convenient to introduce in Definition 2.1 below the notion of sign-type spectra, cf. [34-37].
Let A be a closed operator in a Krein space (K, [ •, • ]K). A point A0 e C is said to belong to the approximative point spectrum aap(A) of A if there exists a sequence (xn) in dom(A) with ||xn|| = 1, n = 1,2,..., and || (A - A0)xn|| ^ 0 if n ^ to. For a self-adjoint operator A in (K, [ •, • ]K), all real spectral points of A belong to a0p(A) (see e.g. [2, Corollary VI.6.2]).
Definition 2.1. For a self-adjoint operator A in (K, [ •, • ]K) a point A0 e a(A) is called a spectral point of positive (negative) type of A if A0 e aap(A) and for every sequence (xn) in dom(A) with ||xn|| = 1, n = 1,2,..., and ||(A - A0)xn|| ^ 0 for n ^ to, we have:
liminf[xn,i„|k > 0 (resp. limsup [xn, i„|k < 0).
The point to is said to be a point of positive (negative) type of the extended spectrum of A if A is unbounded and for every sequence (xn) in dom(A) with lim ||xn|| = 0 and ||Axn|| = 1, n =1,2,..., we have:
liminf [Axn, Axn]/c > 0 (resp. limsup [Axn, Axn]/c < 0).
We denote the set of all points of 5(A) of positive (negative) type by a++(A) (resp. a__(A)). Points from
5(A) of neither positive nor negative type are called critical. In the following proposition, we collect some properties. For a proof, we refer to [34].
Proposition 2.2. (i) The sets a++ (A) and a__(A) are contained in R.
(ii) The non-real spectrum of A cannot accumulate to a++(A) U a__(A).
(iii) The sets a++(A) and a__(A) are relatively open in 5(A).
(iv) Let A0 be a point of a++(A) (a__(A), respectively). Then there exists an open vicinity U in C of A0
and a number M > 0 such that:
M —
||(A - A)-1|| < |ima for all A e U\R.
We shall say that an open subset A of R is of positive type (negative type) with respect to A if:
A n 5(A) c a++(A) (resp. A n 5(A) c a__(A)).
An open set A of R is called of definite type if A is of positive or of negative type with respect to A. If we relate Definition 2.1 to nonnegative operators in Krein spaces (cf. Section 2.1), we obtain from the properties of the spectral function of a nonnegative operator in a Krein space, see, e.g., [1,32,38], and [34, Proposition 25] the following.
Proposition 2.3. Let A be a nonnegative operator with p(A) = 0 in a Krein space (K, [ •, • ]K). Then c(A) c {0, to} and
a(A) n (0, to) c a++(A) c R \ (-to, 0), a(A) n (-to, 0) c a__(A) C R \ (0, to). In particular, we have:
c(A) = 5(A) \ (a++(A) U a__(A)). (2.2)
A generalization of the class of nonnegative operators in Krein spaces is given by the class of definitizable operators. Recall, that a self-adjoint operator A in a Krein space (K, [ •, • ]K) is called definitizable if p(A) = 0 and if there exists a rational function p = 0 having poles only in p(A) such that [p(A)x,x]K > 0 for all x e K. Such a function p is called definitizing function for A. Then the spectrum of A is real or its non-real part consists
of a finite number of points. Inspired by Proposition 2.3 we introduce the set of critical points of a definitizable operator A via:
c(A):= 5(A) \ (a++(A) U a__(A)). (2.3)
It is known (cf. [32]) that c(A) is contained in {t g R : p(t) = 0} U {to}.
For the definitizable operator A, the spectral function E(A) can be introduced for every interval A such that the endpoints of A belong to intervals of definite type, see [32], [14]. We mention only that E(A) is defined and is a self-adjoint projection in (K, [•, -]K) for every such interval. Moreover,
(E(A)K, [ •, • ]K) is a Hilbert space whenever A c {t G R : p(t) > 0}. (2.4)
If a critical point a is the endpoint of two intervals (A1,a) and (a, A2) of the definite type, then the sequences E([A^t]) and E([t, A2]) are monotone in (A1,a) and (a, A2), resp. The point a is called a regular critical point of A, if the limits ( ) ( )
lim E([Ai,t]) and lim E([t,A2]) (2.5)
tfa t\.a
exist in the strong operator topology. A critical point of A which is not regular is called singular critical point of A. The set of all singular critical points of A is denoted by cs(A).
In Subsection 4.2, we essentially use the following resolvent criterion of K. Veselic [25,26] for to G cs(A). We state a special case of this criterion as it has appeared in [27, Corollary 1.6].
Theorem 2.4. Let A be a definitizable self-adjoint operator in a Krein space (K, [ •, • ]). Then: (a) to G cs(A) if and only if there is no > 0, such that the set of numbers:
n
J Re [(A — iy)-1/, f]Kdy (n G (no, to))
no
is bounded for every / G K. (b) Let £0 G R. Then £0 G cs(A) and ker(A — £0) = ker(A — £0)2 if and only if there is no > 0, such that the set of numbers:
' l o
J Re [(A — £0 — iy)-1 /, /]Kdy (n G (0, %))
n
is bounded for every / G K.
A characterization of definitizable operators via their sign-type spectrum together with some growth conditions for the resolvent is provided by the following theorem. Its proof follows from [35, Definition 4.4 and Theorem 4.7]).
Theorem 2.5. Let A be a self-adjoint operator in the Krein space (K, [ •, • ]K). Then A is definitizable if and only if the following holds.
(i) The non-real spectrum <r(A) \ R consists of isolated points which are poles of the resolvent of A, and no point of R is an accumulation point of the non-real spectrum <r(A) \ R of A.
(ii) There is an open vicinity U of R in C and numbers m > 1, M > 0 with
||(A — A)-1|| < M(|A| + 1)2m-2|ImA|-m for all A gU\R.
(iii) Every point A G R has an open connected vicinity I\ in R such that both components of I\ \ {A} are of definite type with respect to A.
3. Locally definitizable operators and their direct sum
3.1. Locally definitizable operators in Krein spaces
In view of Theorem 2.5, it is natural to introduce a local version of definitizability which will play an important role in the following. The next notion is due to P. Jonas, see [13,14], we mention also the overview in [39].
Definition 3.1. Let Q be a domain in C which is symmetric with respect to R such that Q n R = 0 and the intersections with the open upper and lower half-plane are simply connected. Let A be a self-adjoint operator in the Krein space (K, [ •, • ]K). The operator A is called definitizable over Q if the following holds:
(i) The non-real spectrum in Q, i.e. ^(A) n (Q \ R), consists of isolated points which are poles of the resolvent of A, and no point of Q n R is an accumulation point of the non-real spectrum a(A) \ R of A.
(ii) For every closed subset A of Q n R there exist an open vicinity U of A in C and numbers m > 1, M > 0 such that
||(A - A)-1|| < M (|A| + 1)2m-2|Im A|-m for all A eU\R.
(iii) Every point A g Q n R has an open connected vicinity I\ in R such that both components of I\ \ {A} are of definite type with respect to A.
Let A be definitizable over Q. Similar as in (2.3) we call a point t g Q n R a critical point of the operator A if there is no open subset A of definite type with t g A. The set of critical points of A is denoted by c(A). As in Section 2.1, critical points admit a classification into singular and regular critical points: If for some A g c(A) \ {to} the limits analogous to (2.5) exist, then A is called a regular critical point of A. If to is a critical point of A and the limits (2.5) exist in the strong operator topology for some A1, A2 g R \ {0}, then to is called regular critical point of A. A critical point of A which is not regular is called singular critical point of A. The set of all singular critical points of A is denoted by cs(A).
Theorem 2.4 has a counterpart for locally definitizable operators: Let A be definitizable over a vicinity Q of to. Then, A admits an orthogonal decomposition into two operators: a definitizable one with spectrum in A and a self-adjoint one with spectrum outside A, where A(c Q) is a vicinity of to, for details we refer to [35, Theorem 4.8]. Then, the following theorem follows easily from this decomposition and Theorem 2.4:
Theorem 3.2. Let a self-adjoint operator A in a Krein space (K, [ •, • ]) be locally definitizable over a neighborhood Q of to. Then to G cs (A) if and only if there is no > 0, such that the set of numbers:
1
J Re [(A - iy)-1/, /],cdy (n € (no,
is bounded for every f gK.
Similarly, if £0 G R and A is locally definitizable over a vicinity Q of £0, then £0 G cs(A) and ker(A — £0) ker(A — £0)2 if and only if there is n0 > 0, such that the set of numbers:
J Re [(A - £o - iy)-1 /, /]Kdy (n € (0, no))
n
is bounded for every / € K.
Roughly speaking, the property of an operator to be definitizable or to be locally definitizable is stable under finite rank perturbations. This is made more precise in the following theorem which is taken from J. Behrndt [40, Theorem 2.2]:
Theorem 3.3. Let A0 and A1 be self-adjoint operators in a Krein space (K, [ •, • ]K) with p(A0) n p(A1) = 0 and assume that for some A0 G p(A0) n p(A1) the difference:
(A0 — A0)-1 — (A1 — A0)-1
is a finite rank operator. Then, A0 is definitizable over Q if and only if A1 is definitizable over Q.
Moreover, if A0 is definitizable over Q and 5 c Q n R is an open interval with endpoint ^ G Q n R and the spectral points of A0 in 5 are only of positive type (negative type), then there exists an open interval 5', 5' c 5, with endpoint ^ such that the spectral points of A1 in 5' are only of positive type (negative type, respectively).
Theorem 3.3 also holds for definitizable operators as the class of definitizable operators over C coincides with the class of definitizable operators ( [35, Theorem 4.7]). For definitizable operators, this fact is already contained in [41].
3.2. Local definitizability of the direct sum of two operators
In this section, we characterize the definitizability of an operator which is the direct sum of two definitizable operators. For this, we provide the following definition:
Definition 3.4. We shall say that the sets S1 and S2, S1, S2 c R, are separated by a finite number of points if
— TO = «0 < «1 < • • • < ®-N < «N+1 = +TO,
there exists a finite ordered set {a3- }N=1, N g N:
such that one of the sets Sj, j = 1, 2, is a subset of [ak , ak+1] and the other one is a subset of
k is even
[ak,ak+1 ]. Here, we agree that 0 is even, [a0,a1] stands for (—to, a1] U {to} and [aN,aN+1] for
k is odd
[aN, to) U {to}.
The following theorem can be considered as a refinement of [42, Theorem 3.6]:
Theorem 3.5. Consider two operators A and B where A is self-adjoint in the Krein space (K+, [ •, • ]/c+) and B in (K_, [ •, • ]/c_). Let the direct sum of the two Krein spaces:
K = K+[+]K_,
be endowed with the natural inner product:
[f,g]K := [P+ f,P+g]K+ + [P_f,P_g]K- (f,g e K), (3.1)
where P± are the orthogonal projections onto K±. Then, the sum of the operators A[+]B is self-adjoint in the direct sum of the Krein spaces K with the natural inner product from (3.1). We set the following:
S+ := a++(A) U a++(B) and S_ := a__(A) U a__(B).
Then, A[+]B is definitizable if and only if the operators A and B are definitizable and S+ and S_ are separated by a finite number of points.
Proof. The non-real-spectrum of A[+]B coincides with the union of the non-real spectra of A and of B. Therefore, if A[+]B is definitizable, then item (i) of Theorem 2.5 holds for A and for B. Conversely, if A and B are both definitizable, then (i) of Theorem 2.5 holds for A[+]B. Therefore, it is no restriction to assume that A[+]B, A, and B have real spectrum only.
If A[+]B is definitizable, then by the definition of the inner product in K = K+[+]K_ a definitizing function p for A[+]B is also a definitizing function for A and for B. From (2.4), we deduce:
{t e R : p(t) > 0} c a++(A) U p(A), {t e R : p(t) < 0} c a__(A) U p(A), {t e R : p(t) > 0} c a++(B) U p(B), {t e R : p(t) < 0} c a__(B) U p(B),
and, hence, the zeros of p are the points separating S+ and S_, cf. Definition 3.4.
It remains to prove the converse. We assume that S+ and S_ are separated by the points {a0,..., aN+1}, cf. Definition 3.4, then we have:
S+ n S_ c {a0,. .., aN+1}.
Note that S+ and c(A) may have a non-empty intersection (and the same applies to S+ n c(B), S_ n c(A), and S_ n c(B)). Indeed, let A e a++(B) (and, hence, A e S+) such that A is an isolated spectral point of A which belongs to c(A). Then, A e S+ n c(A) and, moreover as A e S_, we have in addition A e {a0,..., aN+1}. We define:
A := {ao,..., aw+1} U c(A) U c(B), and for A e S+ \ A, the following statements are true:
(i) A e a++(A) U a++(B) (as A e S+),
(ii) A e a__(a) U a__(b) (as A e S_),
(iii) A e c(A) U c(B) (as A e A).
Thus, by (2.2) applied to both A and B, we obtain:
A e a++(A) U 5(A) and A e a++(B) U 5(B).
This implies:
A e a++(A[+]B),
and we obtain:
S+ \ A c a++(A[+]B), (3.2)
and with similar arguments:
S_ \ A c a—(A[+]B). (3.3)
From (2.2), we conclude:
5(A[+]B) = 5(A) U 5(B)
= a++(A) U c(A) U a—(A) U a++(B) U c(B) U a__(B) (3.4)
= S+ U c(A) U c(B) U S_ c S+ U S_ U A.
Obviously, for the operator A[+]B the statements (i) and (ii) from Theorem 2.5 are satisfied as A and B are definitizable operators. It remains to show (iii). Clearly, for A e C \ 5(A[+]B) (iii) in Theorem 2.5 is satisfied. Let A e 5(A[+]B). If A e (S+ U S_) \ A we deduce from (3.2) and (3.3) that either A e a++(A[+]B) or A e
a__(A[+]B). As the sets a++(A[+]B) and a__(A[+]B) are relatively open in 5(A[+]B) (cf. Proposition 2.2),
(iii) follows. By (3.4), it remains to consider A e A. For A e {a0,..., aN+1} (iii) follows from (3.2) and (3.3). Therefore, consider A e c(A) U c(B). It is sufficient to consider A e c(A) \ {a0,..., aN+1}. It follows from the definition of the points {a0,..., aN+1} and the fact that A e {a0,..., aN+1} that there exists open connected vicinities /a, Ja in R of A with:
(/A \{A}) n 5(A) c a++(A) and (J \ {A}) n 5(B) c a++(B)
or
(/a \{A}) n 5(A) c a—(A) and (J \ {A}) n 5(B) c a__(B). This shows (/a n Ja \ {A}) n 5(A[+]B) is a subset of a++(A[+]B) or of a__(A[+]B) and (iii) follows.
□
Corollary 3.6. Let A+ and A_ be self-adjoint and semibounded from below in the Krein spaces (K+, [ •, • ]K+) and (K_, [ •, • ]/c_), respectively:
[A±f±, f±]K± > a±[f±, f±]K±, f± e dom(A±), (3.5)
for some a± e R. Let p(A+) = 0, p(A_) = 0. Then, their direct sum A+ [+]A_ is definitizable over:
Q := C \ [min{a+, a_}, max{a+, a_}], (3.6)
in the direct sum of the Krein spaces K = K+[+]K_. In particular, A+[+]A_ is definitizable if and only if the sets S+ and S_ from Theorem 3.5 are separated by a finite number of points. This is fulfilled in the following special cases:
(I) a_ = a+.
(II) a_ < a+ and either a(A+) n (a_, a+) is finite or a(A_) n (a_, a+) is finite.
(III) a+ < a_ and either a(A+) n (a+, a_) is finite or a(A_) n (a+, a_) is finite.
Proof. The assumptions on A± imply that A+ - a+ and A_ - a_ are nonnegative operators and, hence, A± are definitizable operators. Then, with Proposition 2.3, we see that:
(a±, to) n a(A±) c a++(A±) and (—to, a±) n a(A±) c a__(A±) (3.7)
and properties (i)—(iii) from Definition 3.1 for the operator A+[+]A_ and Q as in (3.6) are easily shown, cf. Proposition 2.2. Therefore, A+[+]A_ is definitizable over Q.
The statements on the definitizability of the operator A+[+]A_ now follow directly from (3.7) and Theorem 3.5. □
4. Coupling of definitizable operators in Krein spaces
4.1. Boundary triples and Weyl functions of symmetric operators
Starting from this section, we will denote by A a closed densely defined symmetric operator in a Krein space (K, [ •, • ]/c). Let p~(A) denote the set of points of regular type of A, see [43], and let Nz denote the defect subspace of the operator A:
Nz := He ran(A - z) = ker(A+ - z), z e 5(A). In what follows, we assume that the operator A admits a self-adjoint extension A in (K, [ •, • ]/c) with a nonempty resolvent set p(A). Then, for all z e p(A), we have:
dom(A+) = dom(A) + Nz direct sum in H. (4.1)
This implies, in particular, that the dimension dim(Nz) is constant for all z e p(A). Definition 4.1. Let r0 and r1 be linear mappings from dom(A+) to Cd such that:
(i) the mapping Г : f ^ {Го/, Г/} from dom(A+) to C2d is surjective;
(ii) the abstract Green's identity:
[A+f,g]K - [f,A+g]K = (Го^)Ф(Г1/) - (Г^ПГо/) (4.2)
holds for all f, g G dom(A+).
Then, the triplet П = { Cd, Г0, Г1} is said to be a boundary triple for A+, see [19,44,45, Sect.3.1.4] for a much more general setting.
It follows from (4.2) that the extensions A0, A1 of A defined as restrictions of A+ to the domains:
dom(A0) := ker(r0) and dom(A1) := ker(r1) (4.3)
are self-adjoint extensions of A.
If A has a self-adjoint extension A, with p(A) = 0, then the operator A+ admits a boundary triple {Cd, Г0, Г1}, such that A0 = A and d = dimNz (z G p(A0)). In this case, for every z G p(A0), the decomposition (4.1) holds with A = A0 and the mapping r0|Nz is invertible for every z g p(A0). Therefore, the operator-function:
7(z):=(r0|Nz )-1, (4.4)
is well defined and takes values in the set of bounded operators from Cd to Nz. The operator-function 7(z) is called the 7—field of A, associated with the boundary triple П. Notice, that y(z) satisfies the equality:
7 (z) = (A0 — z0)(A0 — z)-17(z0) (z, z0 G p(A0)).
Definition 4.2. The matrix valued function M : p(A0) ^ Cdxd is defined by the equality:
M(z)r0fz =r1fz, fz G Nz, z G p(A0). (4.5)
The matrix valued function M is called the Weyl function of A corresponding to the boundary triple П = {Cd, Г0, Г1}.
Clearly,
M (z) = r17(z), z G p(A0), (4.6)
and hence M(z) is well defined and takes values in Cdxd. It follows from the identity that the Weyl function M (A) satisfies the identities:
M(z) — M(w)* = (z — W)y(w)+y(z), z,w G p(A0). (4.7)
With w = z the identity (4.7) yields that the Weyl function M satisfies the symmetry condition:
M (z)* = M (z) for all z G p(A0). (4.8)
The identity (4.7) was used in [46] as a definition of the Q-function. In the case when (K, [ •, • ]/c) is a Hilbert space, it follows from (4.7) and (4.8) that M is a Nevanlinna matrix valued function cf. (1.2). In what follows, the function:
f(z) := [/,y(z)]k (f GK, z G p(A0))
is called the generalized Fourier transform of f associated with the boundary triple {C, Г0, Г1}. A motivation for this name is hidden in the fact, that the mapping f ^ f is a unitary mapping from K to a reproducing kernel
Krein space with the kernel M(z)—M(w) (see [28] for the Hilbert space case).
z — wz
Proposition 4.3. [44-46] Let A1 be the self-adjoint extension of A with the domain defined in (4.3) and let d =1. For every z G p(A0), the following equivalence holds:
z g p(A1) ^ M(z) = 0,
and the resolvent of A1 can be found by the formula:
(A1 — z)-1/ = (A0 — z)-1/ — ^Mz) 7(z),
for all f G H and all z G p(A0) П p(A1).
4.2. Construction of the coupling of two self-adjoint operators in a Krein space
In this section, we consider two Krein spaces (K+, [ •, • ]K ) and (K_, [ •, • ]). Let their direct sum:
K = K+[+]K_,
be endowed with the natural inner product (3.1). Consider two closed symmetric densely defined operators A+ and A_ with defect numbers (1,1) acting in the Krein spaces (K+, [•, • ]/c ) and (K_, [ •, • ]/c_). Let {C, r±,r±} be a boundary triple for A±. Let M± be the corresponding Weyl function and ya± the 7-field. By A±,0, we denote the self-adjoint extension of A± which is defined on:
dom(A±,o) = ker(r±) by A±,o = A±|ker(r±
and assume that p(AIj0) n p(A-j0) = 0. Then, the functions M± are defined and holomorphic on p(A±j0). The following theorem is the indefinite version of a result from [47] (see also [28]).
Theorem 4.4. Under the general assumptions of this subsection we have:
(a) The linear operator A defined as the restriction of A+ [+]A- to the domain:
dom(A)=f: if™=0; f± e<to(A4 (4-9)
is closed, densely defined and symmetric with defect numbers (1,1) in the Krein space K.
(b) The adjoint A+ of A is the restriction of A+ [+] A- to the domain:
dom(A+) = j f j : r+(/+) - r-(f-) = 0, f± G dom(A±) j . (4.10)
(c) A boundary triple {C, r0, ri} for A+ is given by:
rof = r+/+, rif = r+/+ +r-/_, f = f+j Gdom(A+). (4.11)
(d) The Weyl function M (z) and the Y-field of A relative to the boundary triple {C, r0, r1} are given by:
M (z) = M+(z)+ M (z), y (z) = (7A+ (zM zG C \ R. (4.12)
(z)y
(e) The self-adjoint extension A1 of A such that dom(A1) = ker(r1) coincides with the restriction of A+[+]A- to the domain:
dom(A1»={(f-): if !- i-!f-ï=0: f±G dom(A±»} • (4.13)
and is called a coupling of A+ and A_ relative to the boundary triples {C, r1, r+} and {C, r-, r-}.
(f) The self-adjoint extension A0 of A coincides with the direct sum A+ 0[+]A- 0 and p(A 0) = p(A+ 0) n p(A-,o) = 0.
(g) The resolvent set p(A1) is nonempty if and only if
M+ + M- ^ 0.
^ z G P(A1) n P(A0) and f = (f- )G K=KI[+]K-the reso'vent of A1 is gh'e'by:
(.1 _z)-1/=,.0 -z)-1/- M+izi; M:i;; yW, (4.14)
where:
/a+ (z):=[fi,YA+ (z)]k+ , fA: (z):=[f- ,YA- (z)]/C: . (4.15)
Proof. (a)-(c) Since {C, r±, r±} is a boundary triple for A±, it follows from (4.2) that for all /± G dom(A±):
[A+/+,g+],c+ - [/+,A+g+],c_ + [Ai/_,g_]K_ - [/-, A-g-]k_ + + + (4.16)
= (r+g+)(r+/+) - (r+g+)(r+/+) + (r_g_)(r_/_) - (r_g_)(r_/_).
We denote by T the restriction of A+[+]A_ to the set of the right hand side of (4.10). If
/ = fj , g = j e dom(T ) then r+/+ = ro_/_ and r+g+ = r_g_, and hence, one obtains from (4.16):
[T/,g]K - [/,Tg]K = r+g+(r+ /+ +r_/_) - (r+g+ +r_g_)r+/+. (4.17)
Now, it follows from (4.17) that A is a closed, densely defined and symmetric operator in the Krein space K, T = A+ and a boundary triple for A+ can be chosen in the form (4.11).
(d) The formulas for M and 7 are implied by (4.11), (4.4) and (4.5).
(e) & (f) As {C, r0, r 1} is a boundary triple for A+, the extension A 1 with dom(A 1 ) = ker(r 1) being a restriction of A+[+]A;. The formula (4.13) for the domain follows from A 1 c A+ (see (4.10)) and dom(A 1 ) = ker(r1). The statement (f) is immediate from (4.10) and (4.11).
(g) The statement (g) is implied by (4.12) and Proposition 4.3. □
Remark 4.5. The construction in Theorem 4.4 shows that the coupling of two self-adjoint operators A+,0 and A_,0 is not uniquely defined. Namely, let the boundary triples n_ = {C, r_, r_} and II_ = {C, r_, r_} be related by
r_ = cr_ r_ = c^^ for some non-zero c e C, c =1. Then, the extension A1 defined as the restriction of A+[+]A+ to the domain:
dom(A1> = ilM : r+J/;»-:r_//:=0= 0, /± e dom(A±)
is also a coupling of A- and A+ with = Ai.
However, when the boundary triples {C, r±, r±} are fixed, then the coupling A1 of the operators A± is uniquely defined by the formula (4.13) and is called the coupling of the operators A±,0 relative to the boundary triples {c, r±, r±}.
Let us suppose that the operators A±,0 are semibounded from below, that is there exists G R such that (3.5) holds. Then, the results of Section 3.2 allow us to show that the coupling A1 of the operators A+,0 and A-j0 is at least locally definitizable in a vicinity of to. In the next theorem, sufficient conditions for regularity of the critical point to are given.
Theorem 4.6. Under the general assumptions of this subsection, we assume that the operators A±,0, the 7—fields Y± and the Weyl functions M± satisfy the following assumptions:
(A1) The operators A±,0 are semibounded from below, p(A±,0) = 0, and
to G cs(A±,o).
(A2) (w(z) :=)|M+(z) + M_(z)| ^ 0 on p(A+,o) n p(A_,o). (A3) There is y1 > 0, such that for all /a± G K±:
t» ^ t» ^
f I/a± (iy)|2 d < » I/A± (—iy)|2 d < (418)
-^^dy < to, -—-dy < to, (4.18)
J w(«y) J w(«y)
yi yi
where the generalized Fourier transforms /A+ and /A- are defined by (4.15).
Then, the coupling A1 of the operators A+,0 and A-j0 is definitizable over Q, where Q is as in (3.6). Moreover, we have:
to G cs(A1).
Proof. By Corollary 3.6, the operator A0 = A+j0[+]A_j0 is definitizable over Q. In view of Theorem 4.4, the assumption (A2) yields p(A1) = 0. Since the operator A1 is a two-dimensional perturbation of A0, by Theorem 3.3, the operator A1 is also definitizable over Q.
Clearly, to e cs(A0) and it follows from Theorem 3.2 that there is y2 > y1 > 0, such that:
J |Re [(Ao - iy)-1f,f],c|dy < to for all f e K.
Let us set:
We show:
A(f,iy) :=
(/a+ (iy) + fA- (iy))(fA+ ( —iy) + fA- ( —iy)) M+(iy) + M- (iy) ■
(4.19)
C»
J |A(f, iy)|dy < to for all f e K.
^2
It follows from (A3) that for every fA± e K±
fA± (iy)fA± ( —iy)
dy
w(iy)
<
fA± (iy)
dy w(iy)
1/2
fA± ( —iy)
! dy w(iy)
1/2
<.
Similarly, one obtains for all fA± e K±:
fA+ (iy)fA- ( —iy)
dy
o(iy)
<.
(4.20)
(4.21)
^2
Combining (4.20) and (4.21), one obtains from (4.19) for all f e K
w
/ |A(f, iy)|dy =
(fA+ (iy) + fA- (iy))(J'A+ (-iy) + fA- (-iy))
M+(iy) + M-(iy)
dy < to.
□
Now the statement to e cs(A1) is implied by Theorem 2.4 and (4.14).
Theorem 4.7. Under the assumptions of this subsection we assume that the operators A±,0, the 7-fields y± and the Weyl functions M± satisfy the following conditions: (A1') The operators A±,0 are semibounded from below, p(A±,0) = 0, one of the conditions (i), (ii) or (iii) of Corollary 3.6 holds, and a := min{a_, a+} satisfies:
a e cs(A±io).
(A2) (w(z) :=)|M+(z) + M_(z)| ^ 0 on p(A+,o) n p(A_,o). (A3') There is y1 > 0, such that for all fA± e K±:
yi ^ o yi ^
f |fA± (a + iy)|2,^ f |fA± (a - iy) |2
w(a + iy)
-dy < to,
w(a + iy)
-dy < to.
00 Then, the coupling A1 of the operators A+,0 and A_,0 is a definitizable operator and
a e cs(A).
Proof. In view of Corollary 3.6, the operator A0 := A+ 0[+]A_ 0 is definitizable. By Theorem 4.4, the assumption (A2') implies p(A1) = 0. Then by [41] the operator A1 is also definitizable.
By the assumption (A1') a e cs(A±,0), then a e cs(A0). Since by Theorem 2.4 there is y2 e (0, y1), such
that:
y2
1
it remains to show that:
J |Re [(Ao - a - iy)-1f, f],c|dy < to for all f e K,
o
J |A(f, a + iy) |dy < to for all fe K,
2
where A is defined as in (4.19). The proof of this inequality is similar to that in Theorem 4.6 and is based on the assumption (A3'). □
5. Application to Sturm-Liouville operators with indefinite weights
Consider the differential expression:
i(/)(t):=S|y (-dt (rfdt) + q(t)/(t)) for aa. * e R, (5.1)
where the coefficients r, q and w are real functions on R satisfying the conditions:
(C1) r, q, w e i11oc(R) and r, w > 0 a.e. on R,
(C2) the expression i is in the limit point case at -to and at
Let H± = LW(R±) be the standard weighted L2-space with the positive definite inner product:
(/,g)± = J /(t)g(i)w(t)dt (/,g e LW(R±)).
R±
Consider minimal differential operators generated by ±i in L:W± (R±), here w± denotes the restriction of w to R±. Since we assume that i is in the limit point case at ±to, the operator is a densely defined symmetric operator with defect numbers (1,1) in the Hilbert space L2W± (R±) and:
dom(B±) = {/ e LW± (R±) : /, (r_1/' e ACoc[0, ±to), i(/) e LW± (R±)}, dom(B±) = {/ e dom(B±) : /(0) = /'(0) = 0},
B±/ := ±i(/), / e dom(B±). (5.2)
In addition to (C1), (C2), we assume that:
(C3) B+ and B_ are semibounded from below in L?w+ (R+) and L2W_ (R_), respectively. Let z e C \ R and denote by z) and <£>(•, z) the unique solutions of the equation:
-(r_1 / ' )' + q/ = zw/
satisfying the boundary conditions:
^(0, z) = 1, (r_1 ^')(0, z) = 0 and tf(0, z) = 0, (r_1 ^')(0, z) = 1, respectively. Since we assume that ±i are in the limit point case at ±to, for each z e C \ R there is a unique solution:
^±(i,z)= ^(t,z) ± m±(z)tf(t,z), t e R±, (5.3)
of the restriction of ±i(/) = z/ to R± which belongs to LW± (R±). Relation (5.3) defines the function m± : C \ R ^ C uniquely. The function m± is called the Dirichlet m-coefficient of the restriction of the expression ±i to R±.
A boundary triple for is {C, r±, r±}, where:
r±/ := /(0±), r±(/) = ±(r_1 /')(0±), / e dom(B±). (5.4)
It follows from (4.6) and (5.4) that the Dirichlet m-coefficient m± defined by (5.3) coincides with the Weyl function of the operator in (5.2) relative to the boundary triple in (5.4).
It is natural to consider the expression i in the Krein space (K, [•, -]K), where K = LW(R) is the standard weighted L2-space endowed with the indefinite inner product:
[/,g]K = (J/,g)LW(R) = J sgnt/(t)g(t)dt, /,g e L
R
and the operator:
(J/)(t) = (sgnt)/(t), / e L
is a fundamental symmetry on (K, [•, -]K). We set:
K± = {/ e LW(R) : / = 0 a.e. on RT}. Then K = K+[+]K_ is the fundamental decomposition corresponding to J.
Let the operators A± := ±B± be considered as semibounded symmetric operators in the Krein spaces (R±), ±0, 0l2±(k±)) . Then, the triples (5.4) are boundary triples for A±. The corresponding Weyl functions of the operators A+ and A_ take the form:
M+(z) = m+(z), M_(z) = m_(-z).
Consider a symmetric operator A in the Krein space (K, [•, determined by the conditions (4.9). Then the domain of the adjoint operator A+ is characterized by the boundary condition (4.10), which in view of (5.4), takes the form:
f (0+) = f (0-).
Consider the coupling A1 of A+ and A_ relative to the boundary triples (5.4). A1 is characterized by the boundary conditions (4.13), which now can be rewritten as:
f (0+) = f (0-), (r_1f')(0+) = (r_1f ')(0-).
Therefore, the operator A1 is associated with the expression in (5.1) in the Hilbert space LwW (R); that is A1f = t(f) for all:
f e dom(A1) = {f e Lw (R) : f,r_1f' e ACioc(R), t(f) e LW (R)}. Notice, that the assumption (A1) of Theorem 4.6 is satisfied in view of (C3) and the assumption (A2) is satisfied since if m+(z) + m_(-z) = 0 then m+(z) = -m_(-z) is holomorphic on the half-line (-/#_, to), what is impossible for the m-coefficient of the Sturm-Liouville operator. These considerations and Theorem 4.6 justify the following:
Proposition 5.1. Let the differential operation t satisfy (C1), (C2) and let the minimal differential operators B± generated by ±t in LW(R±) satisfy (C3) and let m± be the Dirichlet m-functions of B±. Then, the coupling A1 of A+ and A_ is locally definitizable in the Krein space (K, [•, -]k). If, in addition, m+ and m_ satisfy the condition (4.18), then to e cs(A1).
Acknowledgements
The research of the first author was supported by the Deutsche Forschungsgemeinschaft (DFG) under grant no. TR 903/16-1 and Ministry of Education and Science of Ukraine (projects # 0115U000136, 0115U000556).
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