Spectra, Simple Operators, and Weyl Functions

In this chapter the spectrum of a self-adjoint operator or relation will be completely characterized in terms of the analytic behavior and the limit properties of the Weyl function.

the measure. The present interest is in an analytic description of these minimal supports in terms of the Borel transform. For the convenience of the reader, a brief review on Borel measures on R and some properties of their Borel transforms are recalled.
In the following let μ be a regular Borel measure on R and denote the Lebesgue measure on R by m. Recall that any Borel measure on R which is finite on compact sets is automatically regular. Associated with the regular Borel measure μ is the nondecreasing, left-continuous function x>0, 0, x= 0, −μ ([x, 0)), x < 0, (3.1.1) on R. Observe that ν μ is bounded if and only if μ is a finite measure, that the derivative ν μ of the nondecreasing function ν μ exists m-almost everywhere, and that μ([x, y)) = ν μ (y) − ν μ (x), x<y. (3.1.2) It is important to note that via (3.1.2) the function ν μ induces a Lebesgue-Stieltjes measure on R, which is a complete measure that coincides with the completion of μ. In the following it is often more convenient to work with this completion, which will also be denoted by μ, and the corresponding μ-measurable subsets of R.
The regular Borel measure μ has a Lebesgue decomposition with respect to the Lebesgue measure m: μ = μ ac + μ s , where the measure μ ac is absolutely continuous and the measure μ s is singular, each with respect to the Lebesgue measure. The singular measure μ s is further decomposed into the singular continuous part μ sc and the pure point part μ p , so that μ = μ ac + μ sc + μ p .
The corresponding nondecreasing, left-continuous functions ν μac , ν μsc , and ν μp defined via (3.1.1), are absolutely continuous, continuous with ν μsc = 0 m-almost everywhere, and a step function, respectively, and Furthermore, for all Borel sets B, and hence the derivative ν μ coincides with the Radon-Nikodým derivative of μ ac m-almost everywhere.
For x ∈ R the derivative μ (x) of the Borel measure μ with respect to the Lebesgue measure m is defined by : I x an interval containing x , (3.1.4) whenever the limit exists and takes values in [0, ∞]. It can be shown that the sets E 0 = x ∈ R : μ (x) exists finitely (3.1.5) and E = x ∈ R : μ (x) exists finitely or infinitely (3.1.6) are Borel sets, and for the set R \ E 0 on which the derivative μ does not exist finitely one has that m(R \ E 0 ) = 0, (3.1.7) while for the set R\E on which the derivative μ does not exist finitely or infinitely one has that m(R \ E) = 0 and μ(R \ E) = 0; (3.1.8) Recall also that the derivative ν μ of the function ν μ in (3.1.2) and the derivative μ in (3.1.4) of the measure μ coincide m-almost everywhere.
A μ-measurable set S ⊂ R is called a support of μ if μ(R \ S) = 0. In particular, this implies that μ(A) = μ(A ∩ S) for all μ-measurable sets A ⊂ R. A support S ⊂ R of μ is called minimal if for subsets S 0 ⊂ S that are μ-measurable and m-measurable, μ(S 0 ) = 0 implies m(S 0 ) = 0. A minimal support is not uniquely defined. The next auxiliary lemma provides some useful properties of minimal supports. (ii) If S is a minimal support for μ while μ(S \ S ) = 0 and m(S \ S) = 0, then S is a minimal support of μ. In particular, if S is a minimal support for μ and S ⊂ S is such that m(S \ S) = 0, then S is a minimal support of μ.
In particular, μ(S \ S ) = 0. Now S \ S ⊂ S is μ-measurable and m-measurable, and since S is a minimal support, it follows that m(S\S ) = 0. A similar argument shows that m(S \ S) = 0. Hence, m(SΔS ) = 0.
(ii) From R \ S = ((R \ S) ∪ (S \ S )) \ (S \ S) one concludes that Since S is a support of μ and it is assumed that μ(S \ S ) = 0, it follows that μ(R \ S ) = 0. Hence, S is a support of μ.
To prove that S is a minimal support for μ, let S 0 ⊂ S be μ-measurable and m-measurable, and assume that m(S 0 ) > 0. Since and m(S \ S) = 0 by assumption, it follows that m(S 0 ∩ S) = m(S 0 ) > 0. As S is a minimal support for μ, this implies μ(S 0 ∩ S) > 0. Therefore, (3.1.9) leads to Thus, S is a minimal support for μ.
Minimal supports for the parts of the spectrum in the Lebesgue decomposition can be expressed in terms of the behavior of the derivative μ ; cf. [335,Lemma 4] (see also [676,682]).
For practical reasons the attention is now restricted to finite Borel measures on R. The properties of such measures are reflected by the boundary behavior of their so-called Borel transform in a sense to be made precise; cf. Appendix A.
Definition 3.1.3. Let μ be a finite Borel measure on R. Then the Borel transform F of μ is the function F defined by (3.1.10) If for some x ∈ R the limit lim y ↓ 0 F (x + iy) exists and takes values in [0, ∞], it will be denoted by F (x + i0). The set of points in R where the limit of the imaginary part of F exists and takes values in [0, ∞] is denoted by F = x ∈ R : Im F (x + i0) exists finitely or infinitely . (3.1.11) It follows from the integral representation (3.1.10) that and hence, by dominated convergence, lim y ↓ 0 y Re F (x + iy) = 0 and lim y ↓ 0 y Im F (x + iy) = μ ({x}) (3.1.12) for all x ∈ R; cf. Lemma A.2.6. In particular, for all x ∈ R. Note also that the Borel transform F is a Nevanlinna function (see Definition A.2.3) and μ(R) = sup y>0 y Im F (iy). Conversely, every Nevanlinna function F with sup y>0 y Im F (iy) < ∞ and lim y→∞ F (iy) = 0 is the Borel transform of a finite Borel measure μ as in (3.1.10); cf. Proposition A.5.3. An important observation concerning the boundary values Im F (x + i0) is contained in the following theorem, which is formulated in terms of the symmetric derivative (Dμ)(x) = lim ↓ 0 μ((x − , x + )) 2 (3.1.14) of μ. Here the limit is assumed to take values in [0, ∞]. Note that if for some x ∈ R the derivative μ (x) in (3.1.4) exists with values in [0, ∞], then the same is true for the symmetric derivative (Dμ)(x). In particular, the following statements hold: (i) Im F (x + i0) and (Dμ)(x) exist simultaneously finitely m-almost everywhere and (3.1.15) holds; (ii) Im F (x + i0) and (Dμ)(x) exist simultaneously finitely or infinitely μ-almost everywhere and m-almost everywhere and (3.1.15) holds.
Proof. Assume first that the symmetric derivative (Dμ)(x) exists in [0, ∞) for some x ∈ R and choose c − , c + ∈ R with c − < (Dμ)(x) < c + . From the definition (3.1.14) it follows that there exists δ > 0 such that 2c − ≤ μ(I ) ≤ 2c + , I := (x − , x + ), (3.1.16) holds for all ∈ (0, δ]. In the following set K y (s) := y s 2 +y 2 for y > 0 and s ∈ R. Then one has for y > 0. First one estimates the second term on the right-hand side in (3.1.17). Since t ∈ R \ I δ , one has |t − x| ≥ δ, so that 0 ≤ K y (t − x) ≤ K y (δ). Then it is for y ↓ 0. In order to estimate the first integral on the right-hand side in (3.1.17) one uses the identity where Fubini's theorem on the triangle in the (t, )-plane given by = t − x, = x − t, with 0 ≤ ≤ δ, was used. Now integration by parts, the fact that (3.1.16), −K y ( ) ≥ 0 for , y > 0, and (3.1.19) give the estimate In the same way one verifies the estimate It follows that Next the case where the symmetric derivative (Dμ)(x) exists and equals ∞ for some x ∈ R is discussed. In this situation the above reasoning leads to It remains to show assertions (i) and (ii). Recall that if μ (x) exists at some point x ∈ R, then so does the symmetric derivative (Dμ)(x) and with equality in [0, ∞]. For (ii) the above reasoning implies that the set E in (3.1.6) is contained in the set F in (3.1.11) and hence μ(R \ F) = 0 and m(R \ F) = 0 by (3.1.8). Assertion (i) follows in the same way from (3.1.5) and (3.1.7).
It follows from Theorem 3.1.4 and (3.1.12) that Theorem 3.1.2 has a counterpart expressing minimal supports in terms of the Borel transform of μ.
Theorem 3.1.5. Let μ be a finite Borel measure and let F be its Borel transform. Then the sets are minimal supports for μ, μ ac , μ s , μ sc , and μ p , respectively.
Proof. Only statement (i) will be proved. The proofs of the other statements are similar. Let and note that M is a Borel set. Recall that, by Theorem 3.1.2 (i), M is a minimal support for μ. Now introduce the set which is also a Borel set, as Im F (x + iy), y > 0, and hence Im F (x + i0) are Borel measurable functions in x. Then Theorem 3.1.4 shows that M ⊂ M and furthermore one has Since m(R\E) = 0 according to (3.1.8), it follows that m(M \M) = 0 and as M ⊂ M , and M is a minimal support for μ, one concludes from Lemma 3.1.1 (ii) that M is a minimal support for μ.
Most of the results in this section have been stated in the context of finite Borel measures on R and their Borel transforms. They will be applied to study the spectrum of self-adjoint relations and operators in Section 3.6. However, it is also useful for later references to have similar results in the more general context of scalar Nevanlinna functions and the corresponding spectral functions; cf. Chapter 6 and Chapter 7. Let N be a scalar Nevanlinna function of the form where α ∈ R, β ≥ 0, and τ is a Borel measure on R which satisfies cf. Theorem A.2.5. Then the last condition implies that μ defined by is a finite Borel measure on R. Let F be the Borel transform of μ: The connection between N and F is given in the following lemma.
where a, b ∈ R. If x ∈ R, then the limits Im N (x + i0) and Im F (x + i0) exist simultaneously with values in [0, ∞], and in that case and lim Proof. It is an immediate consequence of the integral representation (3.1.20) that N can be rewritten as cf. Theorem A.2.4. This leads to (3.1.24). Note that for λ = x + iy one has Now observe that for each x ∈ R one has lim y↓0 y Re F (x + iy) = 0 by (3.1.12). Together with the previous identity this proves the assertion in (3.1.25). Furthermore, now one sees (3.1.27) directly; cf. (3.1.12). Finally, note that which together with (3.1.12) leads to the identity (3.1.26).
The next corollary deals with the existence of the limit lim ↓0 N (x + i ) for any scalar Nevanlinna function N .
Then it is easy to see that Im N (λ) ≥ 0 and Im (i N (λ)) ≥ 0 for λ ∈ C + and hence λ → N (λ) and λ → i N (λ) are scalar Nevanlinna functions when they are extended to C − by symmetry. It follows from (3.1.25) and Theorem 3.1.4 that the limits it follows that the limit in (3.1.28) exists finitely m-almost everywhere.
Let τ be the Borel measure on R in (3.1.20) which satisfies the condition (3.1.21). It has the Lebesgue decomposition where τ ac is absolutely continuous, τ s is singular, τ sc is singular continuous, and τ p is pure point. In the next corollary, which is a consequence of Theorem 3.1.5, (3.1.22), and (3.1.25), minimal supports for these measures are expressed in terms of the boundary behavior of N .

Growth points of finite Borel measures
Let μ be a finite Borel measure on R. In this section the set of its growth points σ(μ), defined by is studied. The growth points σ(μ) and the growth points σ(μ ac ), σ(μ s ), and σ(μ sc ) of the absolutely continuous, singular, and singular continuous part of μ will be located by means of the minimal supports expressed in terms of the Borel transform of μ.
There is an intimate connection between the set of growth points σ(μ) and supports for μ. Lemma 3.2.1. Let μ be a finite Borel measure on R. Then the following statements hold: (ii) The set σ(μ) is closed and it is a support of μ.
Proof. (i) Let S be a support of μ, so that μ(R \ S) = 0. Assume that x ∈ σ(μ), so that for any > 0 one has μ(( which implies that for any > 0 the set (x − , x + ) ∩ S is nonempty. Hence, there exists a sequence x n ∈ (x − 1/n, x + 1/n) ∩ S converging to x from inside S. This shows that σ(μ) ⊂ S.
For completeness it is noted that in general the set σ(μ) is not a minimal support of μ. Observe also that, by Lemma 3.2.1, the set of growth points σ(μ) has the following minimality property: each closed support S ⊂ R of μ satisfies σ(μ) ⊂ S. Therefore, one has the next corollary. The set of growth points of μ will now be described by means of the Borel transform of μ.
Theorem 3.2.3. Let μ be a finite Borel measure on R and let F be its Borel transform. Then Proof. With the notation it will be proved that σ(μ) = N. Recall first that, by Theorem 3.1.5 (i), the set is a (minimal) support for μ. Since M ⊂ N, it follows that N is also a support for μ. Hence, Lemma 3.2.1 (i) yields σ(μ) ⊂ N. For the inclusion N ⊂ σ(μ) it suffices to show N ⊂ σ(μ), since σ(μ) is closed; cf. Lemma 3.2.1 (ii). Assume that x ∈ σ(μ). Then there exists > 0 such that μ((x − , x + )) = 0 and it follows from that Im F (x + i0) = 0. This implies x ∈ N and hence N ⊂ σ(μ).
Analogous to Theorem 3.2.3 there are also results for the parts of the finite Borel measure μ on R in its Lebesgue decomposition. In order to describe these results one needs the following notions of closure.  Proof. First it will be shown that for any Borel set B ⊂ R both sets clos ac (B) and clos c (B) are closed.
To show that clos c (B) is closed, let x n ∈ clos c (B) converge to x ∈ R. Assume that x ∈ clos c (B). Then there is > 0 such that the set (x − , x + ) ∩ B is countable. For this there exist n 0 ∈ N and 0 > 0 with is countable, a contradiction, as x n0 ∈ clos c (B). Therefore, x ∈ clos c (B) and clos c (B) is closed.
To see the first inclusion in (3.2.2) assume that x ∈ clos ac (B). Then one has m((x− , x+ )∩B) > 0 for all > 0 and hence for all > 0 the set (x− , x+ )∩B is not countable. This implies clos ac (B) ⊂ clos c (B). Likewise, to see the second inclusion assume that x ∈ clos c (B) and that x ∈ B. Then there is > 0 such that (x − , x + ) ∩ B = ∅, a contradiction. Hence, clos c (B) ⊂ B.
(i) (⇒) Assume that clos ac (B) = ∅. This implies that for all x ∈ R there exists form an open cover for B. Therefore, there exists a finite subcover ( (ii) (⇒) Assume that clos c (B) = ∅. This implies that for all x ∈ R there exists Here is the promised treatment of the absolutely continuous, singular, and singular continuous parts of the Borel measure μ.
Theorem 3.2.6. Let μ be a finite Borel measure on R and let F be its Borel transform. Then the following statements hold: and note that M ac is a Borel set. It is claimed that σ(μ ac ) = clos ac (M ac ). (3.2.4) To verify the inclusion (⊂) in (3.2.4), assume that x ∈ clos ac (M ac ). Then there As μ ac is absolutely continuous with respect to the Lebesgue measure m, also Furthermore, by Theorem 3.1.5 (ii), the set M ac is a minimal support for μ ac and, in particular, μ ac (R \ M ac ) = 0. Hence, and from (3.2.5)-(3.2.6) one obtains μ ac ((x − , x + )) = 0. Hence, x ∈ σ(μ ac ). Thus, the inclusion (⊂) in (3.2.4) has been shown. For the converse inclusion (⊃), let x ∈ σ(μ ac ). Then there exists > 0 such that where in the last equality the Radon-Nikodým theorem was used; cf. (3.1.3) and note that ν μ = μ = Dμ m-almost everywhere. Due to Theorem 3.1.4 and the fact that Im F (t + i0) ≥ 0 for all t ∈ F, one concludes that This implies m((x− , x+ )∩M ac ) = 0 since Im F (t+i0) is positive on M ac . Hence, x ∈ clos ac (M ac ). Thus, the inclusion (⊃) in (3.2.4) has been shown. Therefore, the equality (3.2.4) has been established, which gives the assertion (i).
(ii) According to Theorem 3.1.5 (iii) the set {x ∈ F : Im F (x + i0) = ∞} is a minimal support for the singular part μ s of μ. Since σ(μ s ) is contained in the closure of this set by Lemma 3.2.1 (i), the assertion follows.
(iii) By Theorem 3.1.5 (iv) and (3.1.13), the Borel set is a minimal support for μ sc and hence, in particular, μ sc (R \ M sc ) = 0. Let clos c (M sc ) be the continuous closure of M sc , which is a Borel set, as it is closed; cf. Lemma 3.2.5. It will be shown that clos c (M sc ) is a support for μ sc , that is, since this implies that σ(μ sc ) ⊂ clos c (M sc ); cf. Lemma 3.2.1 (i) and Lemma 3.2.5. In fact, for x ∈ R \ clos c (M sc ) by definition there exists > 0 such that This yields μ sc (K) = 0 for each compact set K ⊂ R \ clos c (M sc ) and hence, by the (inner) regularity of the finite measure μ sc , (3.2.7) follows.

Spectra of self-adjoint relations
The spectrum of a self-adjoint relation or operator in a Hilbert space will be studied in terms of its spectral measure. In particular, a division of the spectrum into absolutely continuous and singular spectra will be introduced based on the Lebesgue decomposition of a finite Borel measure; cf. Section 3.1.
Let A be a self-adjoint relation in the Hilbert space H. Then σ(A) ⊂ R by Theorem 1.5.5 and σ r (A) = ∅, and hence σ( cf. (1.5.6). First the parts σ p (A) and σ c (A) of the spectrum σ(A) will be characterized in terms of the spectral measure E(·). These results will play an important role in the further development; cf. Section 3.5 and Section 3.6. The facts in Proposition 3.3.1 are immediate consequences of the orthogonal decomposition where H op = dom A and H mul = mul A, of the self-adjoint relation A (see Theorem 1.5.1) and the properties of the spectral measure of A op .

Proposition 3.3.1. Let A be a self-adjoint relation in H with spectral measure E(·).
Then the following statements hold: A further subdivision of the spectrum will be introduced analogous to the Lebesgue decomposition of a finite Borel measure on R; cf. Section 3.1. This requires another description of the spectrum via the introduction of a collection of finite Borel measures induced by the spectral function. Let A be a self-adjoint relation in H with spectral measure E(·). For each h ∈ H, define μ h by It will be shown that the spectrum of A is made up of the growth points of μ h for a dense set of elements h ∈ H. Furthermore, the statement in the following proposition is in a local sense, namely, it concerns the spectrum of A relative to an open interval Δ ⊂ R; cf. Definition 3.4.9.

Proposition 3.3.2. Let A be a self-adjoint relation in H, let Δ ⊂ R be an open interval, and assume that D Δ is a subset of the closed subspace E(Δ)H such that
Then the following identities hold: Proof. First it will be shown that Hence, the inclusions (3.3.4) follow. Next it will be shown that which, together with (3.3.4), yields (3.3.3). For this purpose, assume that for all h ∈ D Δ , and hence for all h ∈ span D Δ . Since by assumption span D Δ is dense in E(Δ)H, it follows that (3.3.5) holds for all h ∈ E(Δ)H and hence again by Proposition 3.3.1 (i), The collection of Borel measures μ h , h ∈ H, as defined in (3.3.2), is now used to introduce a number of subspaces of H. Definition 3.3.3. Let A be a self-adjoint relation in H. The pure point subspace, the absolutely continuous subspace, and the singular continuous subspace corresponding to A op are defined by In conjunction with the orthogonal decomposition (3.3.1), these subspaces span the original Hilbert space and lead to invariant parts of the self-adjoint relation, see, e.g., [649,Theorem VII.4]

and the restrictions
By means of these subspaces one defines, in analogy with the case of finite Borel measures, the singular subspace and the continuous subspace corresponding to A op by respectively. The restrictions of A op to these subspaces are denoted by A s op and A c op , respectively, and it follows that Note that for the pure point part A p op one only has σ p (A) = σ(A p op ). The spectral measures of the self-adjoint operators A ac op , A sc op , and A s op in the Hilbert spaces H ac (A op ), H sc (A op ), and H s (A op ), are given by the corresponding restrictions of the spectral measure E(·) of A. These spectral measures will be denoted by E ac (·), E sc (·), and E s (·), respectively.
The following corollary relates the absolutely continuous, singular continuous, and singular spectrum of A in an open interval Δ with the growth points of the absolutely continuous, singular continuous, and singular parts of the measures μ h .

Corollary 3.3.6. Let A be a self-adjoint relation in H, let Δ ⊂ R be an open interval, and assume that D Δ is a subset of the closed subspace E(Δ)H such that
Denote by μ h,ac , μ h,sc , and μ h,s the absolutely continuous, singular continuous, and singular part in the Lebesgue decomposition of the Borel measure μ h in (3.3.2). Then the following identity holds: i= ac, sc, s.
Proof. Observe first that the absolutely continuous, singular continuous, and singular part of the Borel measure μ h , h ∈ H, are given by , is the Borel measure defined with the help of the spectral measures E i (·) of A i op , i = ac, sc, s, then Definition 3.3.5, Proposition 3.3.2 and (3.3.6) yield for i = ac, sc, s. Here it was also used that the linear span of the set Example 3.3.7. Let μ be a Borel measure on R and consider the maximal multiplication operator by the independent variable in L 2 μ (R), given by The operator A is self-adjoint in L 2 μ (R) and for every Borel set B ⊂ R the spectral measure of A is given by For the spectral subspaces of A in Definition 3.3.3 one has H i (A) = L 2 μi (R), i = ac, sc, s, and this implies σ ac (A) = σ(μ ac ), σ sc (A) = σ(μ sc ), and σ s (A) = σ(μ s ).

Simple symmetric operators
It will be shown that any closed symmetric relation in a Hilbert space can be decomposed into the orthogonal componentwise sum of a closed simple, i.e., completely non-self-adjoint, symmetric operator, and a self-adjoint relation. Criteria for the absence of the self-adjoint relation in this decomposition will be given, and a local version of simplicity will be studied.
First some attention is paid to the notions of invariance and reduction. These notions appeared already in the self-adjoint case in the previous section when subdividing the spectrum, and are also important in the description of self-adjoint extensions of symmetric relations. Let S be a closed symmetric relation in the Hilbert space H. Decompose H as H = H ⊕ H , let P and P be the orthogonal projections onto H and H , and define The closed symmetric relation S gives rise to the restrictions Assume, in addition, that S is self-adjoint. Then (iii) S = P S or, equivalently, S = P S implies that S and S are self-adjoint in H and H , respectively.
The assumption that S is self-adjoint in H implies {P f, P f } ∈ S . Therefore, P S ⊂ S . This implies S = P S and (i) yields S = P S.
(iii) According to (i), either of the conditions S = P S or S = P S implies that S = S ⊕ S . Since S is self-adjoint, this shows that S is self-adjoint in H and that S is self-adjoint in H .
Before introducing the notion of simplicity in Definition 3.4.3 below, the following lemma on symmetric and self-adjoint extensions of symmetric relations that contain a self-adjoint part is discussed.
Lemma 3.4.2. Let S be a closed symmetric relation in H whose defect numbers are not necessarily equal and assume that there are orthogonal decompositions such that S is closed and symmetric in H and S is self-adjoint in H . Then every closed symmetric (self-adjoint ) extension A of S in H admits the decomposition Proof. Observe that the inclusion S ⊂ A and the decomposition (3.4.3) imply that Therefore, the assumption that S is self-adjoint in H shows that the closed symmetric relation A∩(H ) 2 is actually self-adjoint in H and that This completes the proof.
The notion of simplicity or complete non-self-adjointness is defined next.
Definition 3.4.3. Let S be a closed symmetric relation in H whose defect numbers are not necessarily equal. Then S is simple if there is no orthogonal decomposition Every closed symmetric relation S in H has the orthogonal componentwise decomposition S = S op ⊕ S mul , where S mul is a purely multivalued self-adjoint relation in the closed subspace H mul = mul S; cf. Theorem 1.4.11. Hence, a closed simple symmetric relation is necessarily an operator. A similar argument shows that a closed simple symmetric relation does not have any eigenvalues; cf. Lemma 3.4.7.
Any closed symmetric relation S in H has a decomposition as in (3.4.4), where S is simple in H and S is self-adjoint in H . To see this, define the closed and the closed subspace K = R ⊥ , so that It follows from Lemma 1.6.11 that the set C \ R in (3.4.6), and hence in the intersection in (3.4.5), can be replaced by any subset of C \ R which has an accumulation point in C + and an accumulation point in C − . Then the relation S admits the orthogonal decomposition where S is a closed simple symmetric operator in K and S is a self-adjoint relation in R. Proof.
Step 1. First it will be shown that R satisfies the following invariance property: for any which proves the inclusion in (3.4.9).
Step 2. Next it will be shown that the relation S ∩ R 2 is self-adjoint. Fix some λ 0 ∈ C \ R and define the relation S first by Therefore, h ∈ R and hence {f, f } ∈ S , so that S ∩ R 2 ⊂ S . This leads to the equality S = S ∩ R 2 in (3.4.7); in particular, S in (3.4.10) does not depend on the choice of λ 0 ∈ C \ R. From S ⊂ S it follows that S is symmetric and from (3.4.10) one obtains that ran (S − λ 0 ) = R. Since S is independent of the choice of λ 0 , it follows that ran (S − λ) = R holds for every λ ∈ C \ R. Hence, S = S ∩ R 2 is a self-adjoint relation in R by Theorem 1.5.5. Now Lemma 3.4.1 (i)-(ii) imply (3.4.8).
Step 3. In order to show that S = S ∩ K 2 is simple in the Hilbert space K, assume that there is an orthogonal decomposition K = K 1 ⊕ K 2 and a corresponding orthogonal decomposition S = S 1 ⊕ S 2 such that S 2 is self-adjoint in K 2 . Then ran (S 2 − λ) = K 2 for all λ ∈ C \ R and thus According to (3.4.5), this implies K 2 ⊂ R while K 2 ⊂ K = R ⊥ . Thus, K 2 = {0}, so that S is simple. The set C \ R on the right-hand side can be replaced by any set U which has an accumulation point in C + and in C − .
Proof. It follows from Theorem 3.4.4 and the definition of K in (3.4.6) that the equality (3.4.11) holds if and only if S is simple. The last assertion in the corollary follows from Lemma 1.6.11.
Corollary 3.4.6. Let S be a closed symmetric relation in H. Then the following statements are equivalent: The set C \ R on the right-hand side in (i) can be replaced by any set U which has an accumulation point in C + and in C − .
The assumption implies in the context of Theorem 3.4.4 that R = mul S, so that which is a self-adjoint relation in mul S. Hence, by the decomposition S = S ⊕ S in Theorem 3.4.4 it follows that S = S op in H op = H mul S.  Proof. By Lemma 3.4.2 and Theorem 1.4.11, it suffices to consider the case where S is a closed symmetric operator and A is a self-adjoint extension of S. Now assume that S is not simple, so that by Theorem 3.4.4 there are nontrivial decompositions H = H ⊕ H and S = S ⊕ S with S closed, simple and symmetric in H , and S self-adjoint in H . Then A decomposes accordingly as A = A ⊕ S with A self-adjoint in H by Lemma 3.4.2. Now σ(A) = σ p (A) implies that S and thus S has a nontrivial point spectrum, which gives a contradiction.
The notion of simplicity of a closed symmetric relation S in H will now be specified in a local sense. This will be done relative to a Borel set Δ ⊂ R and by means of a self-adjoint extension A of S and its spectral measure E(·). Then H admits the orthogonal decomposition which leads to the orthogonal componentwise decomposition of A into self-adjoint components: Definition 3.4.9. Let S be a closed symmetric relation in H and let A be a selfadjoint extension of S with spectral measure E(·). Let Δ ⊂ R be a Borel set. Then S is said to be simple with respect to Δ ⊂ R and the self-adjoint extension A if In the next proposition this local notion and some of its consequences are discussed.
Proposition 3.4.10. Let S be a closed symmetric relation in H and let A be a selfadjoint extension of S with spectral measure E(·). Assume that S is simple with respect to the Borel set Δ ⊂ R and the self-adjoint extension A. Then the following statements hold: (3.4.13) (ii) There is no point spectrum of S in Δ: (3.4.14) Proof. (i) First note that the inclusion (⊃) in (3.4.13) holds. To see the converse inclusion, let f ∈ E(Δ )H. As Δ ⊂ Δ, one has and hence f ∈ E(Δ)H. By assumption, the identity (3.4.12) holds and so, in the linear span of there exists a sequence (f n ), that converges to f . Then (E(Δ )f n ) is a sequence in the linear span of This shows the inclusion (⊂) in (3.4.13).
Further, since f ∈ E(Δ)H, one concludes that which shows that f = 0. Thus, S does not possess any eigenvalues in Δ.
(iii) The inclusion (⊃) in (3.4.14) is clear. In order to prove the identity, fix μ ∈ U and recall from Lemma 1.4.10 that the operator for all g μ ∈ N μ (S * ) and all λ ∈ U. Since for each g μ ∈ N μ (S * ) the function is analytic on ρ(A), it follows from (3.4.15) and the assumption that U has an accumulation point in each connected component of ρ(A) that this function is identically equal to zero. Hence, (E(Δ)k, E(Δ)f ) = 0 for all k ∈ N λ (S * ) and λ ∈ C \ R. Now (3.4.12) yields E(Δ)f = 0 and (iii) follows.
The connection with the global notion of simplicity is given in the following corollary.

Corollary 3.4.11. Let S be a closed symmetric relation in H and let A be a selfadjoint extension of S with spectral measure E(·). Then S is simple if and only if S is simple with respect to any Borel set Δ ⊂ R and the self-adjoint extension A.
Proof. Assume that S is simple. Then (3.4.11) holds and hence (3.4.12) holds with Δ = R. Then Proposition 3.4.10 (i) implies that S is simple with respect to any Borel set Δ ⊂ R and the self-adjoint extension A. Conversely, if (3.4.12) holds for any Borel set Δ ⊂ R, then (3.4.12) also holds for Δ = R, and hence reduces to (3.4.11), that is, S is simple.
In the following lemma the eigenspace of A corresponding to an eigenvalue x is described in the case where x is not an eigenvalue of S. In particular, this observation leads to a characterization of local simplicity if the Borel set Δ ⊂ R in Definition 3.4.9 is a singleton; cf. Corollary 3.4.13.
Lemma 3.4.12. Let S be a closed symmetric relation in H, let A be a self-adjoint extension of S with spectral measure E(·), and let x ∈ R. Then for some, and hence for all λ ∈ C \ R, if and only if x ∈ σ p (S).
Proof. Assume first that (3.4.16) holds for some fixed λ ∈ C \ R. Assume that Conversely, assume that x ∈ σ p (S) and let λ ∈ C \ R. The inclusion (⊃) in (3.4.16) is clear and since both subspaces in (3.4.16) are closed, it suffices to verify that As f ∈ E({x})H, this implies (f, k λ ) = 0 and hence f ∈ ran (S − λ). Choose {g, g } ∈ S such that g − λg = f . Then and it follows that {f, xf } ∈ S. Since x ∈ σ p (S) by assumption this yields f = 0.
The above lemma together with Proposition 3.4.10 (ii) implies that S is simple with respect to a point x ∈ R if and only if x is not an eigenvalue of S.

Corollary 3.4.13. Let S be a closed symmetric relation in H, let A be a self-adjoint extension of S with spectral measure E(·), and let x ∈ R. Then
holds if and only if x ∈ σ p (S).

Eigenvalues and eigenspaces
Let S be a closed symmetric relation in a Hilbert space H and let {G, Γ 0 , Γ 1 } be a boundary triplet for S * with A 0 = ker Γ 0 , and corresponding γ-field γ and Weyl function M . The purpose of the present section is to characterize eigenvalues and the associated eigenspaces of the self-adjoint relation A 0 by means of the corresponding Weyl function M .
Recall that the Weyl function M can be expressed in terms of the γ-field and the resolvent of the self-adjoint relation A 0 ; cf. Proposition 2.3.6 (v). In particular, for λ = x + iy, y > 0, and λ 0 ∈ ρ(A 0 ) one has This formula will be used to study the behavior of the Weyl function M at a point x ∈ R. In the next proposition it turns out that the strong limit of iyM (x + iy), y ↓ 0, is closely connected with the eigenspace of A 0 at x. Here the spectral measure E of A 0 is not used explicitly in the assertion; the orthogonal projection onto the eigenspace N Proposition 3.5.1. Let S be a closed symmetric relation in H, let {G, Γ 0 , Γ 1 } be a boundary triplet for S * , let A 0 = ker Γ 0 , let M and γ be the corresponding Weyl function and γ-field, and let x ∈ R. Then for each λ 0 ∈ ρ(A 0 ) and all ϕ ∈ G one has lim Proof. For x ∈ R and λ 0 ∈ ρ(A 0 ), it follows from (3.5.1) that As the first and second terms on the right-hand side tend to 0 as y ↓ 0, one obtains for all ϕ ∈ G. Since x ∈ R is fixed and y ↓ 0, one has that where the approximating functions are uniformly bounded by 1. The spectral calculus for the self-adjoint relation A 0 in Lemma 1.5.3 yields Now the assertion follows from (3.5.3).
Definition 3.5.2. Let S be a closed symmetric relation in H, let {G, Γ 0 , Γ 1 } be a boundary triplet for S * , let A 0 = ker Γ 0 , and let M be the corresponding Weyl function. For x ∈ R the operator R x : G → G is defined as the strong limit Observe that R x in Definition 3.5.2 is a well-defined operator in B(G); indeed, this is clear from the identity (3.5.2). It also follows from (3.5.2) that R x = 0 when x ∈ ρ(A 0 ) ∩ R.
Remark 3.5.3. If x ∈ R is an isolated singularity of the function M , then x is a pole of first order of M ; cf. Corollary 2.3.9. Moreover, in a sufficiently small punctured disc B x \ {x} centered at x such that M is holomorphic in B x \ {x}, one has a norm convergent Laurent series expansion of the form It follows that R x coincides with the residue of M at x, i.e., where C denotes the boundary of B x .
In the following let x ∈ R and recall that the corresponding eigenspaces of S and A 0 are given by The main interest will be in the closed linear subspace N x (A 0 ) N x (S), which is the orthogonal complement of N x (S) in N x (A 0 ). Similarly, the orthogonal com- Lemma 3.5.4. Let λ 0 ∈ ρ(A 0 ), let x ∈ R, and let P x be the orthogonal projection from H onto N x (A 0 ) N x (S). Then the operator R x has the representation Proof. First, recall from Corollary 2.3.3 that for x ∈ R and {h, xh} ∈ A 0 one has
In the following theorem the eigenvalue x ∈ R and the corresponding eigenspace of A 0 are characterized by means of the Weyl function M and the operator R x . Later it will be shown how to distinguish between isolated and embedded eigenvalues of A 0 ; cf. Theorem 3.6.1.
Theorem 3.5.5. Let S be a closed symmetric relation in H, let {G, Γ 0 , Γ 1 } be a boundary triplet for S * , let A 0 = ker Γ 0 , let M and γ be the corresponding Weyl function and γ-field, and let x ∈ R. Then the mapping is an isomorphism. In particular, Proof. Let x ∈ R and define K x = N x (A 0 ) N x (S). The mapping Γ 1 : S * → G is continuous and, in particular, its restriction to K x ⊂ S * is continuous. The proof consists of three steps. In Step 1 it will be shown that the restriction of Γ 1 to K x is injective and in Step 2 it will be shown that it has closed range. Then it follows from Step 3 that τ in (3.5.6) is an isomorphism.
Step 1. The restriction of the mapping Step 2. The range of the restriction of Γ 1 to K x is closed. In fact, let (ϕ n ) be a sequence in ran (Γ 1 K x ) such that ϕ n → ϕ ∈ G. Then there exists a sequence ( f n ) in K x such that Γ 1 f n = ϕ n and as f n ∈ A 0 one has Γ 0 f n = 0. Therefore, Γ f n = {0, ϕ n } → {0, ϕ}. Recall from Proposition 2.1.2 that the restriction of Γ to S * S is an isomorphism onto G × G. It follows that f n converge to some element f , which belongs to the closed subspace K x . This yields Γ 1 f = ϕ and hence ran (Γ 1 K x ) is closed.
Step 3. The linear space so that Γ 1 f = 0, and hence f = 0 by Step 1.
Step 4. The mapping in (3.5.6) is an isomorphism. To see this, observe that (3.5.7) The first inclusion in (3.5.7) follows directly from (3.5.4). From the same identity one also sees that Hence, the second inclusion in (3.5.7) follows from Step 3 and the boundedness of Γ 1 . It is clear from (3.5.7) and Step 2 that ran (Γ 1 K x ) = ran R x , and hence, due to Step 1, the mapping in (3.5.6) is an isomorphism.
The statement of Theorem 3.5.5 can be simplified if x is not an eigenvalue of the symmetric relation S, that is, S satisfies a local simplicity condition at x ∈ R; cf. Corollary 3.4.13.
Corollary 3.5.6. Assume that x is not an eigenvalue of the closed symmetric relation S in Theorem 3.5.5. Then Now the behavior of M at ∞ will be considered and the multivalued part of A 0 will be described. First recall that the self-adjoint relation A 0 is decomposed into the orthogonal sum (3.5.8) where A 0,op is a self-adjoint operator in the Hilbert space and A 0,mul is the purely multivalued self-adjoint relation in H mul = mul A 0 . Then the resolvent of A 0 has the form (3.5.11) In order to use this formula for large y decompose the term γ(λ 0 ) * γ(λ 0 ) as where P op denotes the orthogonal projection from H onto H op , ι op is the canonical embedding of H op into H, and I − P op is viewed as an orthogonal projection in H. From the representation of the resolvent of A 0 in terms of the resolvent of A 0,op in (3.5.10) it follows that (3.5.11) may be rewritten as for all y > 0. This formula will be used to study the behavior of M at ∞. It turns out that the strong limit 1 iy M (iy), y → +∞, is closely connected with the multivalued part of A 0 . Proposition 3.5.7. Let S be a closed symmetric relation in H, let {G, Γ 0 , Γ 1 } be a boundary triplet for S * , let A 0 = ker Γ 0 , and let M and γ be the corresponding Weyl function and γ-field. Then for each λ 0 ∈ ρ(A 0 ) and ϕ ∈ G one has Proof. It follows from (3.5.12) with λ 0 ∈ ρ(A 0 ) that It suffices to show that the first and the third term on the right-hand side converge to 0 strongly. This is obvious for the first term on the right-hand side. For the third term note that for y → +∞ one has and hence the spectral calculus for A 0,op shows that for y → +∞ tends to zero for all ϕ ∈ G; cf. Lemma 1.5.3. This leads to (3.5.13).
Definition 3.5.8. Let S be a closed symmetric relation in H, let {G, Γ 0 , Γ 1 } be a boundary triplet for S * , let A 0 = ker Γ 0 , and let M be the corresponding Weyl function. The operator R ∞ : G → G is defined as the strong limit It follows from Proposition 3.5.7 that R ∞ ∈ B(G). For the following properties of R ∞ recall the notations The next lemma can be viewed as a variant of Lemma 3.5.4 for x = ∞. Here the main interest is in the closed subspace Lemma 3.5.9. Let λ 0 ∈ ρ(A 0 ) and let P ∞ be the orthogonal projection from H onto Then the operator R ∞ has the representation (3.5.14) Proof. First recall from Corollary 2.3.3 that for {0, h } ∈ A 0 one has Now let ϕ ∈ G and consider h = (I − P op )γ(λ 0 )ϕ ∈ mul A 0 . According to Proposition 3.5.7, Since {0, P N∞(S) γ(λ 0 )ϕ} ∈ S and S = ker Γ 0 ∩ ker Γ 1 by Proposition 2.1.2 (ii), it follows that Γ 1 0, P N∞(S) γ(λ 0 )ϕ = 0 and hence (3.5.15) leads to (3.5.14).
In the next theorem the multivalued part of A 0 is characterized by means of the Weyl function M and the operator R ∞ .
Theorem 3.5.10. Let S be a closed symmetric relation in H, let {G, Γ 0 , Γ 1 } be a boundary triplet for S * , let A 0 = ker Γ 0 , and let M and γ be the corresponding Weyl function and γ-field. Then the mapping is an isomorphism. In particular, Proof. The proof follows a strategy similar to the one used in the proof of Theorem 3.5.5. To simplify notation, set The mapping Γ 1 : S * → G is continuous and, in particular, its restriction to K ∞ ⊂ S * is continuous.
Step 1. The restriction of the mapping Γ 1 to K ∞ is injective.
Step 2. The range of the restriction of Γ 1 to K ∞ is closed. In fact, let (ϕ n ) be a sequence in ran (Γ 1 K ∞ ) such that ϕ n → ϕ ∈ G. Then there exist ( f n ) in K ∞ such that Γ 1 f n = ϕ n and as f n ∈ A 0 one has Γ 0 f n = 0. Thus, Γ f n = {0, ϕ n } → {0, ϕ}, and since the restriction of Γ to S * S is an isomorphism onto G × G, it follows that f n converge to some element f , which belongs to the closed subspace K ∞ . Therefore, Γ 1 f = ϕ and hence ran (Γ 1 K ∞ ) is closed.
Step 3. The linear space To see this, let f ∈ K ∞ be orthogonal to all elements {0, P ∞ γ(λ 0 )ϕ}, ϕ ∈ G. Then it follows from f = {0, f } and Corollary 2.3.3 that for all ϕ ∈ G one has so that Γ 1 f = 0, and hence f = 0 by Step 1.
Step 4. The mapping in (3.5.16) is an isomorphism. To see this, observe that (3.5.17) The first inclusion in (3.5.17) follows from (3.5.14). From the same identity one also sees that Hence, the second inclusion in (3.5.17) follows from Step 3 and the boundedness of Γ 1 . It is clear from (3.5.17) and Step 2 that ran (Γ 1 K ∞ ) = ran R ∞ , and hence, due to Step 1, the mapping in (3.5.16) is an isomorphism.

Spectra and local minimality
As in Section 3.5, let S be a closed symmetric relation in H, let {G, Γ 0 , Γ 1 } be a boundary triplet for S * with A 0 = ker Γ 0 , and corresponding γ-field γ and Weyl function M . The spectrum of the self-adjoint extension A 0 and its division into absolutely continuous and singular spectra (cf. Section 3.3) will now be discussed in detail in terms of the boundary behavior of M . For this purpose it is assumed that S either is simple or satisfies a local simplicity condition with respect to an open interval Δ ⊂ R and the self-adjoint extension A 0 ; see Definition 3.4.9 for the notion of local simplicity.
The following theorem describes the point spectrum and the continuous spectrum of A 0 in terms of the boundary behavior of the Weyl function M ; cf. Proposition 3.3.1.
Theorem 3.6.1. Let S be a closed symmetric relation in H, let {G, Γ 0 , Γ 1 } be a boundary triplet for S * with A 0 = ker Γ 0 , let M and γ be the corresponding Weyl function and γ-field, and let R x = lim y↓0 iyM (x + iy), x ∈ R, be the operator in Definition 3.

Let Δ ⊂ R be an open interval and assume that the condition
is the spectral measure of A 0 . Then the following statements hold for each x ∈ Δ:  ((a, b)) of A 0 corresponding to the interval (a, b) is given by Stone's formula (1.5.7) where the integral on the right-hand side is understood in the strong sense. For ν ∈ C \ R and ϕ ∈ G this implies (3.6.2) and the identities from Proposition 2.3.6 (vi) (with λ = t ± iδ and μ = ν) together with the holomorphy of M in O yield that the integral on the right-hand side of (3.6.2) is zero.
(ii)-(iii) According to Proposition 3.4.10 (ii), the condition (3.6.1) implies that S does not have eigenvalues in Δ. Hence, items (ii) and (iii) follow immediately from item (i) and Corollary 3.5.6.
(iv) Assume that x ∈ Δ is an isolated eigenvalue of A 0 . Then by Proposition 2.3.6 (iii) or (v) there exists an open neighborhood O of x such that M is holomorphic on O \ {x}. Since x ∈ σ p (S) by Proposition 3.4.10 (ii), it follows from Corollary 3.5.6 that there exists ϕ ∈ G such that (3.6.3) This implies that M has a pole at x, which is of first order; cf. Corollary 2.3.9. By Remark 3.5.3 the residue of M at x is given by R x . Conversely, if M has a pole (of first order) at x, then (3.6.3) holds for some ϕ ∈ G. Thus, x is an eigenvalue of A 0 by Corollary 3.5.6 and from item (i) it follows that there exists an open Under the condition that S is simple the spectrum of A 0 can be described completely in terms of the Weyl function M .
Corollary 3.6.2. Let S be a closed symmetric relation in H, let {G, Γ 0 , Γ 1 } be a boundary triplet for S * , let A 0 = ker Γ 0 , and let M and γ be the corresponding Weyl function and γ-field. Assume that S is simple. Then the assertions (i)-(iv) in Theorem 3.6.1 hold for all x ∈ R.
To describe the absolutely continuous, singular, and singular continuous parts of the spectrum of A 0 in terms of the boundary behavior of the Weyl function M , some preliminary lemmas are needed.
For x = λ 0 it follows from (3.5.1) that The first term on the right-hand side clearly goes to 0 as y ↓ 0. For the third term on the right-hand side, observe that for y ↓ 0 one has and since the approximating functions are uniformly bounded, the spectral calculus for A 0 (see Lemma 1.5.3) yields Hence, also the third term on the right-hand side goes to 0 as y ↓ 0. Furthermore, |x − λ 0 | 2 − y 2 → |x − λ 0 | 2 > 0 as y ↓ 0. Therefore, Im (M (x + iy)ϕ, ϕ) converges as y ↓ 0 if and only if Im (A 0 − (x + iy)) −1 γ(λ 0 )ϕ, γ(λ 0 )ϕ converges as y ↓ 0. In addition, it is clear that the identity in the lemma for the limits is satisfied.
Recall that the self-adjoint extension A 0 generates a collection of finite Borel measures on R: for each h ∈ H the finite Borel measure μ h in (3.3.2) is defined by cf. Definition 3.1.3. In particular, if λ = x + iy, where x ∈ R and y > 0, then one has and (3.6.5) By means of Lemma 3.6.3 the boundary values of the Borel transform F h for a class of elements h ∈ H are expressed in terms of the boundary values of the Weyl function M . Lemma 3.6.4. Let Δ ⊂ R be an open interval and let λ 0 ∈ C \ R. Then for elements of the form h = E(Δ)γ(λ 0 )ϕ, ϕ ∈ G, the following statements hold: exist simultaneously, and Proof. It follows from (3.6.4) that for all h ∈ H the (possibly improper) limits Im F h (x + i0) and Im ((A 0 − (x + i0)) −1 h, h) exist simultaneously and coincide. For the choice h = γ(λ 0 )ϕ, ϕ ∈ G, it follows from Lemma 3.6.3 that the (possibly improper) limits Im F h (x + i0) and Im (M (x + i0)ϕ, ϕ) exist simultaneously, and then for x ∈ Δ the spectral calculus implies while for x ∈ Δ it follows that This shows the assertions in (i) and (ii). Now the absolutely continuous spectrum, the singular spectrum, and the singular continuous spectrum (cf. Section 3.3) of A 0 can be described in terms of the boundary behavior of the Weyl function M , still under the assumption of local simplicity. The results are essentially consequences of Theorem 3.2.3, Theorem 3.2.6, and Corollary 3.3.6. Theorem 3.6.5. Let S be a closed symmetric relation in H, let {G, Γ 0 , Γ 1 } be a boundary triplet for S * with let A 0 = ker Γ 0 , and let M and γ be the corresponding Weyl function and γ-field. Let Δ ⊂ R be an open interval and assume that the condition is satisfied, where E(·) is the spectral measure of A 0 . Then the absolutely continuous spectrum of A 0 in Δ is given by (3.6.7) If S is simple, then (3.6.7) holds for every open interval Δ, including Δ = R.
Proof. By assumption, the span of the set is dense in E(Δ)H and hence Corollary 3.3.6 implies the identity According to Theorem 3.2.6 (i) (where the set F was replaced by R), by Lemma 3.6.4. This yields (3.6.7).
The next lemma is of similar nature as Lemma 3.6.4. Here the limits exist for all x ∈ R by (3.1.12)-(3.1.13) and Proposition 3.5.1.
Next some inclusions for the singular and singular continuous spectra of A 0 will be shown.
is satisfied, where E(·) is the spectral measure of A 0 . Then the following statements hold: If S is simple, then (i) and (ii) hold for every open interval Δ, including Δ = R.
Proof. By assumption, the span of the set (i) Recall that by Corollary 3.3.6 one has and according to Theorem 3.2.6 (ii) (with F replaced by R) For h = E(Δ)γ(ν)ϕ ∈ D Δ this gives, via Lemma 3.6.4, Hence, the set σ s (A 0 ) ∩ Δ in (3.6.11) is contained in which yields the assertion in (i).
An immediate corollary of the previous theorem and Lemma 3.2.5 (ii) is a sufficient condition for the absence of the singular continuous spectrum in terms of the limit behavior of the function M .
Corollary 3.6.9. Let A 0 and M be as in Theorem 3.6.8 and let Δ ⊂ R be an open interval such that the condition (3.6.10) is satisfied. Assume that for each ϕ ∈ G there exist at most countably many x ∈ Δ such that If S is simple, then the assertion holds for every open interval Δ, including Δ = R.
As a further corollary of the theorems of this section sufficient conditions are provided for the spectrum of A 0 to be purely absolutely continuous or purely singularly continuous, respectively, in some set.
Corollary 3.6.10. Let A 0 and M be as in Theorem 3.6.5 or Theorem 3.6.8 and let Δ ⊂ R be an open interval such that the condition (3.6.6) or (3.6.10) is satisfied. Assume that for all ϕ ∈ G and all x ∈ Δ lim y ↓ 0 yM (x + iy)ϕ = 0. (3.6.13) Then the following statements hold: (i) If for each ϕ ∈ G there exist at most countably many x ∈ Δ such that If S is simple and Δ is an open interval such that (3.6.13) holds for all ϕ ∈ G and all x ∈ Δ, then (i) and (ii) are satisfied.

Limit properties of Weyl functions
Let S be a closed symmetric relation in a Hilbert space H, let {G, Γ 0 , Γ 1 } be a boundary triplet for S * , let A 0 = ker Γ 0 , and let M be the corresponding Weyl function. The aim of this section is to relate limit properties of the imaginary part Im M of the Weyl function with defect elements in dom A 0 and dom |A 0 | 1 2 , and ran (A 0 − x) and ran |A 0 − x| 1 2 , x ∈ R, respectively, where with respect to the usual decomposition H = H op ⊕ H mul . This also leads to necessary and sufficient conditions for S to be a densely defined operator in terms of the Weyl function.

The first result connects limit properties of the Weyl function at ∞ with elements in dom
The representation (3.5.12) of the Weyl function M in terms of the extension A 0 = ker Γ 0 will now be used; cf. (3.5.8)-(3.5.9). For simplicity one takes λ 0 = i in (3.5.12), which leads to the representation for all y > 0. The spectral calculus for the self-adjoint operator A 0,op applied to (3.7.2) shows that for ϕ ∈ G and y > 0 (3.7.6) It is clear that the left-hand side of (3.7.6) has a finite limit for y → +∞ if and only if (I − P op )γ(i)ϕ = 0 and which follows from the monotone convergence theorem. In other words, the lefthand side of (3.7.6) has a finite limit for y → +∞ if and only if γ(i)ϕ ∈ dom A 0 .
In this case, S * is an operator and all intermediate extensions of S are operators.
Proof. Note that Proposition 3.5.7 and the fact that γ(λ 0 ) * (I − P op )γ(λ 0 ) in (3.5.13) is a nonnegative operator in G show that condition (i)  In the next result, which is parallel to Proposition 3.7.1, the limit properties of the Weyl function at x ∈ R will be connected with elements in ker (S * − λ) ∩ ran (A 0 − x) and ker (S * − λ) ∩ ran |A 0 − x| 1 2 .
For this reason the representation (3.5.1) expressing the Weyl function M in terms of the self-adjoint relation A 0 = ker Γ 0 will be used. For simplicity one takes λ 0 ∈ C \ R such that Re λ 0 = x in (3.5.1), which leads to It follows by means of the spectral calculus applied to (3.7.8) that for x ∈ R and ϕ ∈ G one has (3.7.9) Proposition 3.7.4. Let S be a closed symmetric relation in H, let {G, Γ 0 , Γ 1 } be a boundary triplet for S * , let A 0 = ker Γ 0 be decomposed as in (3.7.1), and let M and γ be the corresponding Weyl function and γ-field. Then the following statements hold for x ∈ R and ϕ ∈ G: Proof. (i) It will first be shown that for λ, it follows that Thus, γ(λ 0 )ϕ ∈ ran (A 0 − x) implies that γ(λ)ϕ ∈ ran (A 0 − x). Since λ 0 and λ in the above argument can be interchanged, it is clear that γ(λ)ϕ ∈ ran (A 0 − x) if and only if γ(λ 0 )ϕ ∈ ran (A 0 − x).
To verify the remaining assertion in (i) with λ = λ 0 , note first that the limit as y ↓ 0 in (3.7.10) is finite if and only if the limit of the integral in the second term in (3.7.9) is finite. An application of the monotone convergence theorem shows that the limit as y ↓ 0 of the integral in the second term in (3.7.9) is finite if and only if where the definition of the spectral measure E(·) of A 0 via the spectral measure E op (·) of A 0,op was used. Therefore, the limit as y ↓ 0 in (3.7.10) is finite if and only if P op γ(λ 0 )ϕ ∈ dom (A 0,op − x) −1 = ran (A 0,op − x), that is, if and only if γ(λ 0 )ϕ ∈ ran (A 0 − x).
(ii) As in (i), it will first be shown that P op γ(λ)ϕ ∈ ran |A 0,op − x| 1 2 if and only if P op γ(λ 0 )ϕ ∈ ran |A 0,op − x| 1 2 for λ, λ 0 ∈ ρ(A 0 ). Assume that It follows from the functional calculus for unbounded self-adjoint operators that Since λ 0 and λ in the above argument can be interchanged, it is clear that P op γ(λ 0 )ϕ ∈ ran |A 0,op − x| 1 2 holds if and only if P op γ(λ)ϕ ∈ ran |A 0,op − x| 1 2 holds. To verify the remaining assertion in (ii), it is convenient to fix λ = λ 0 ∈ C \ R such that |Im λ 0 | > 1. One then concludes from (3.7.9) that the integral in (3.7.11) converges if and only if the integral

Spectra and local minimality for self-adjoint extensions
In this section the results on eigenvalues, eigenspaces, continuous, absolutely continuous and singular continuous spectra from Section 3.5 and Section 3.6 will be explicitly formulated for arbitrary self-adjoint extensions of a symmetric relation.
Let S be a closed symmetric relation in H and let {G, Γ 0 , Γ 1 } be a boundary triplet for S * with γ-field γ and Weyl function M . Consider a self-adjoint extension According to Section 2.2, the self-adjoint extensions A Θ in (3.8.1) can also be written in the form In order to describe the spectrum of A Θ consider the boundary triplet {G, cf. Corollary 2.5.11. Then one has A Θ = ker Γ 0 , (3.8.3) and the corresponding Weyl function and γ-field will be denoted by M Θ and γ Θ . respectively; cf. (2.5.17) and (2.5.18). From (3.8.3) it is clear that the spectrum of A Θ can be described by means of the Weyl function M Θ . Therefore, the earlier results expressing the spectrum of A 0 in terms of the Weyl function M (and the γ-field γ) can now be simply translated to the present context. The main results will be listed below; it is left to the reader to formulate analogs of the results in Section 3.7 in the present setting. First the analogs of Theorem 3.5.5 and Theorem 3.5.10 will be described. For this purpose define the operators R Θ x , x ∈ R, and R Θ ∞ similar to Definition 3.5.2 and Definition 3.5.8: As in Section 3.5, one has that R Θ x , R Θ ∞ ∈ B(G). In terms of the boundary triplet {G, Γ 0 , Γ 1 } in (3.8.2) and the corresponding Weyl function M Θ in (3.8.4), Theorem 3.5.5 and Corollary 3.5.6 read as follows.
Corollary 3.8.1. Let S, A Θ , and M Θ be as above and let x ∈ R. Then the mapping is an isomorphism. In particular, and if x ∈ σ p (S), then x ∈ σ p (A Θ ) if and only if R Θ x = 0. Similarly, Theorem 3.5.10 and Corollary 3.5.11 take the following form. For the next results the local simplicity condition appearing in many of the results in Section 3.6 has to be reformulated with respect to A Θ . According to Definition 3.4.9, the closed symmetric relation S is simple with respect to Δ ⊂ R and the self-adjoint extension A Θ if E Θ (Δ)H = span E Θ (Δ)γ Θ (ν)ϕ : ν ∈ C \ R, ϕ ∈ G , (3.8.5) where E Θ (·) is the spectral measure of A Θ .
Then Theorem 3.6.1 yields the following statement. Finally, the corresponding results for the absolutely continuous, singular, and singular continuous spectra will be formulated; it is left to the reader to state the analogs of Corollaries 3.6.6, 3.6.9, and 3.6.10.
In the present situation Theorem 3.6.5 reads as follows. For the singular and singular continuous spectra one obtains the following version of Theorem 3.6.8. Finally, the special case where the self-adjoint relation Θ in (3.8.1) is a bounded self-adjoint operator will be briefly discussed. In this situation there is a more natural choice of the transformed boundary triplet {G, Γ 0 , Γ 1 } above. In fact, if S is a closed symmetric relation, {G, Γ 0 , Γ 1 } is a boundary triplet for S * with γ-field γ and Weyl function M , and Θ ∈ B(G) is self-adjoint, then, by Corollary 2.5.7, the mappings Γ 0 = Γ 1 − ΘΓ 0 and Γ 1 = −Γ 0 lead to a boundary triplet {G, Γ 0 , Γ 1 } for S * such that ker Γ 0 = ker Γ 1 − ΘΓ 0 = A Θ . respectively. Then the above results in Corollaries 3.8.1-3.8.5 remain valid with the function M Θ in (3.8.7) and the mapping f → A * Γ 0 f +B * Γ 1 f in Corollary 3.8.1 and Corollary 3.8.2 replaced by f → −Γ 0 f .

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