On a minimax principle in spectral gaps

The minimax principle for eigenvalues in gaps of the essential spectrum in the form presented by Griesemer, Lewis, and Siedentop in [Doc. Math. 4 (1999), 275--283] is adapted to cover certain abstract perturbative settings with bounded or unbounded perturbations, in particular ones that are off-diagonal with respect to the spectral gap under consideration. This in part builds upon and extends the considerations in the author's appendix to [J. Spectr. Theory 10 (2020), 843--885]. Several monotonicity and continuity properties of eigenvalues in gaps of the essential spectrum are deduced, and the Stokes operator is revisited as an example.


Introduction and main result
The standard Courant minimax values λ k (A) of a lower semibounded operator A on a Hilbert space H are given by x, Ax = inf for k ∈ N with k ≤ dim H, see, e.g., [21,Theorem 12.1] and also [27, Section 12.1 and Exercise 12.4.2]. Here, ·, · denotes the inner product of H, and a with a[x, x] = |A| 1/2 x, sign(A)|A| 1/2 x for x ∈ Dom(|A| 1/2 ) is the form associated with A.
The above minimax values have proved to be a powerful description of the eigenvalues below the essential spectrum of A; in fact, they agree with these eigenvalues in nondecreasing order counting multiplicities as long as they exists and else equal the bottom of the essential spectrum. A standard application in this context is that the eigenvalues below the essential spectrum exhibit a monotonicity with respect to the operator: for two lower semibounded self-adjoint operators A and B with A ≤ B in the sense of quadratic forms one has λ k (A) ≤ λ k (B) for all k, see, e.g., [27,Corollary 12.3].
Matters get, however, much more complicated when eigenvalues in a gap of the essential spectrum are considered. If A + is the (lower semibounded) part of A associated with its spectrum in an interval of the form (γ, ∞), γ ∈ R, then the minimax values for A + still describe the eigenvalues of A + below its essential spectrum, and thus the eigenvalues of A in (γ, ∞) below the essential spectrum of A above γ. However, the subspaces over which the corresponding infimum is taken are chosen within the spectral subspace for A associated with the interval (γ, ∞) and therefore usually depend on the operator itself rather than just its domain. This makes it difficult to the approach from [13] promise to be of independent interest. In any case, to the best of the authors knowledge, neither the main results presented here nor their applications have been stated explicitly anywhere before. Rare exceptions to the latter are commented on accordingly.
Main results. In order to formulate our main results, it is convenient to fix the following notational setup tailored towards spectral gaps to the right of 0; other gaps can of course always be reduced to this situation by spectral shift, cf. Remark 3.2 below. Hypothesis 1.1. Let A be a self-adjoint operator on a Hilbert space. Denote the spectral projections for A associated with the intervals (0, ∞) and (−∞, 0] by P + and P − , respectively, that is, Here, E A and E B stand for the projection-valued spectral measures for the operators A and B, respectively, and Ran P ± denotes the range of P ± . We have also used the notation I for the identity operator. Denoting the form associated with A by a, the minimax values of the positive part A| Ran P + of A can clearly be written as λ k (A| Ran P + ) = inf x, Ax = inf for k ∈ N with k ≤ dim Ran P + . The point of interest is now to find conditions on B under which the minimax values for the positive part B| Ran Q + of B admit the same representations with x, Ax and a[x, x] replaced by x, Bx and b[x, x], respectively, but with the infima taken over the same respective families of subspaces as for A above. It is natural to consider this in a perturbative framework where B is obtained by an operator or form perturbation of A and, thus, one has Dom(A) = Dom(B) and/or Dom(|A| 1/2 ) = Dom(|B| 1/2 ). In the situation of Hypothesis 1.1, a representation for the minimax values of B| Ran Q + of the above mentioned form is guaranteed by [23,Theorem 1] in the form setting with Dom(|A| 1/2 ) = Dom(|B| 1/2 ) if (1.1) sup or by [5,Theorem 1.1] in the operator setting with Dom(A) = Dom(B) if the analogous condition with D ± replaced by D ± is satisfied; as pointed out in [26], for the latter additionally the restriction P − B| Ran P − should be essentially self-adjoint on D − . In case of (1.1), the right-hand side of (1.1) then agrees with λ 1 (B| Ran Q + ), so that the strict inequality in (1.1) is also a necessary condition for such a representation to hold if B has a spectral gap to the right of zero. However, this strict inequality is not always very convenient to verify or it is sometimes not even entirely clear how to verify it, cf. Remarks 1.6 (2) and 2.2 (2) below. Instead of (1.1), [13] used the conditions (1.2) sup x − ] ≤ 0 and (|A| + I) 1/2 P + Q − (|A| + I) −1/2 < 1, cf. Remark 3.3 below, which the authors were able to handle in case of Dirac operators but where especially the second condition seems to be hard to deal with in a general abstract setting. However, although (1.1) can treat more Coulomb-like potentials than (1.2) in case of the Dirac operator, (1.2) does not seem to imply (1.1) directly.
In the main results below the aim is to discuss situations where the second condition in (1.2) can be replaced by P + Q − < 1, P + − Q + < 1, or by a certain explicit structural assumption on how B is related to A. Here, especially the first two conditions seem to be natural since they relate the subspaces Ran P + and Ran Q + . Four results in this direction are presented here, each addressing a different situation, which are not contained in the previously known results in the sense that their hypotheses do not seem to imply (1.1), its operator analogue, or (1.2) directly. We first treat the case of operator perturbations and start with the direct extension of [25,Theorem A.2] and if there is no such a for 0 < b < b * . If b * = 0, then V is called infinitesimal with respect to A.
Theorem 1.2. Assume Hypothesis 1.1. Suppose, in addition, that B is of the form B = A + V , Dom(B) = Dom(A), with some symmetric operator V that is infinitesimal with respect to A. Furthermore, suppose that we have P + Q − < 1 and that Then, x, Bx = inf It is worth to note that every operator of the form B = A + V as in Theorem 1.2 is automatically self-adjoint on Dom(B) = Dom(A) by the wellknown Kato-Rellich theorem. Two more remarks regarding Theorem 1.2 are in order: (1) also certain perturbations V that are not infinitesimal with respect to A can be considered here, but at the cost of a stronger assumption on P + Q − , see Remark 4.2 below; (2) the condition P + Q − < 1 is satisfied if the stronger inequality P + − Q + < 1 holds. In the latter case, the subspaces Ran P + and Ran Q + automatically have the same dimension, that is, dim Ran P + = dim Ran Q + , see Remark 3.5 (a) below.
The stronger condition P + − Q + < 1 just mentioned in fact also opens the way to employ a different approach than the one used to prove Theorem 1.2. This alternative approach has previously been used in the context of block diagonalization of operators and forms, see Section 5 below, and is particularly attractive if the unperturbed operator A is semibounded. Theorem 1.3. Assume Hypothesis 1.1. Suppose, in addition, that A is semibounded and that P + − Q + < 1.
for all k ≤ dim Ran P + = dim Ran Q + . If even Dom(A) = Dom(B) and x, Bx ≤ 0 for all x ∈ D − , then also It should be emphasized that the conditions Dom(A) = Dom(B) and x, Bx ≤ 0 for all x ∈ D − in Theorem 1.3 indeed imply that one has also Dom(|A| 1/2 ) = Dom(|B| 1/2 ) and b[x, x] ≤ 0 for all x ∈ D − , see Lemma 3.4 below. Note also that in contrast to Theorem 1.2, Theorem 1.3 makes no assumptions on how the operator B is related to A. The latter will, however, be relevant when the hypotheses of Theorem 1.3 are to be verified in concrete situations.
The condition x, Bx ≤ 0 for all x ∈ D − plays an important role in both Theorems 1.2 and 1.3. In the case where B = A + V with some A-bounded symmetric operator V , this condition is automatically satisfied if x, V x ≤ 0 for all x ∈ D − since x, Ax ≤ 0 holds for all x ∈ D − by definition. The latter is certainly the case for nonpositive V . Another instance of perturbations satisfying x, V x ≤ 0 for all x ∈ D − are so-called off-diagonal perturbations with respect to the decomposition Ran P + ⊕ Ran P − , in which case also the condition P + − Q + < 1 can be verified efficiently. In comparison with Theorem 1.2, we may even relax the assumption on the A-bound of V here. Theorem 1.4. Assume Hypothesis 1.1. Suppose, in addition, that B has the form B = A + V , Dom(B) = Dom(A), with some symmetric A-bounded operator V with A-bound smaller than 1 and which is off-diagonal on Dom(A) with respect to the decomposition Ran P + ⊕ Ran P − , that is, Then, one has dim Ran P + = dim Ran Q + and x, Bx = inf It is again worth to note that every operator of the form B = A + V as in Theorem 1.4 is automatically self-adjoint on Dom(B) = Dom(A) by the Kato-Rellich theorem. Moreover, although off-diagonal perturbations may seem a bit restrictive, they appear quite naturally when a general, not necessarily off-diagonal, perturbation is decomposed into its diagonal and off-diagonal parts. How Theorem 1.4 may then be applied is demonstrated in Proposition 2.12 below and the considerations thereafter.
The method of proof for Theorem 1.4 can to some extend be carried over to off-diagonal form perturbations, at least in the semibounded setting. The latter restriction is commented on in Section 5 below.
Theorem 1.5. Assume Hypothesis 1.1. Suppose, in addition, that B is semibounded and that its form b is given by where a is the form associated with A and v is a symmetric sesquilinear form satisfying with some constants a, b ≥ 0.
Then, one has dim Ran P + = dim Ran Q + and The semiboundedness of B in Theorem 1.5 forces A to be semibounded as well, see the proof of Theorem 1.5 below. In this regard, Theorem 1.5 can be interpreted as a particular case of the first part of Theorem 1.3 with Dom(|A| 1/2 ) = Dom(|B| 1/2 ), in which the remaining hypotheses are automatically satisfied due to the structure of the perturbation. Remark 1.6. (1) If B in Theorem 1.5 is lower semibounded, then the operator (|B| + I) 1/2 Q − is everywhere defined and bounded, and so is the operator x] > 0 for x ∈ D + , Theorem 1.5 therefore reproduces in this situation a particular case of the earlier result [9, Theorem 3].
(2) If in Theorems 1.4 or 1.5 the unperturbed operator A has a spectral gap to the right of 0, then considerations as in part (1) show that (1.1) or the corresponding analogue in the operator framework is satisfied; cf. also Corollaries 2.5 and 2.7 (a) below. In this regard, Theorems 1.4 and 1.5 then can also be deduced from [5] and [23], respectively. However, if A does not have such a gap, it is a priori not clear how to derive the two theorems from [5,23].
The rest of this note is organized as follows. In Section 2 we discuss applications of the main theorems and revisit the Stokes operator as an example in the framework of Theorem 1.5. It is also explained there how the framework for off-diagonal perturbations in Theorem 1.4 can be applied to general, not necessarily off-diagonal, perturbations by decomposing the perturbation into its diagonal and off-diagonal parts. Section 3 is devoted to an abstract minimax principle based on [13]. Two approaches are then used to verify the hypotheses of this abstract minimax principle, the graph norm approach and the block diagonalization approach, respectively, which are discussed separately in Sections 4 and 5 below. Theorem 1.2 is proved in Section 4, which is based on the author's appendix to [25] and extends the corresponding considerations to certain unbounded perturbations. Theorems 1.3-1.5 are proved in Section 5, which builds upon recent developments on block diagonalization of operators and forms from [22] and [11], respectively. Finally, Appendix A reproduces the proof from [13] for the abstract minimax principle discussed in Section 3, and Appendix B provides some consequences of the well-known Heinz inequality that are used at various spots in this work and are probably folklore.

Applications and examples
In this section, we use the main results from Section 1 to prove monotonicity and continuity properties of minimax values in gaps of the essential spectrum in various situations and also revisit the well-known Stokes operator in the framework of Theorem 1.5 as an example. We finally discuss how to apply the off-diagonal framework from Theorem 1.4 to general, not necessarily off-diagonal, perturbations.
We first consider the situation of indefinite or semidefinite bounded perturbations, which has partially been discussed in a slightly different form in [25]. For a bounded self-adjoint operator V we define bounded nonnegative operators V (p) and V (n) with V = V (p) − V (n) via functional calculus by We clearly have V (p) ≤ V and V (n) ≤ V . The following result can be proved in several ways. The proof below is based on Theorem 1.2 and is in its core close to the proofs of Theorems 3.14 and 3.15 in [25]. An alternative proof for part (b) based on Theorem 1.4 is discussed after Remark 2.13 below. The result itself extends Theorem 5 in [9], which was formulated there for bounded nonpositive V that are relatively compact with respect to the unperturbed operator A.
x, Ax for all k ∈ N with k ≤ dim Ran P .
In particular, the subspaces Ran P and Ran Q have the same dimension; cf. also Remark 3.5 (1) below.
In light of part (a), the claim now follows from Theorem 1.2 with P + = P and Q + = Q upon a spectral shift by γ.
(c). Similarly as in (b), pick γ ∈ (c + V (p) + V (n) , d). We then have E A−γ ((0, ∞)) = P and for some chosen ρ ∈ (c + V (p) , d − V (n) ) also E A+V −ρ ((0, ∞)) = Q . Moreover, for x ∈ E − we have In light of part (a), the claim now follows analogously from Theorem 1.2 with switched roles of A and A + V and with P + = Q and Q + = P .
Remark 2.2. (1) A corresponding representation of the minimax values in terms of the forms associated with A + V and A, respectively, as in Theorems 1.2-1.4 holds here as well. However, for the sake of simplicity and since this is not needed in Corollaries 2.3 and 2.4 below, this has not been formulated in Proposition 2.1.
(2) Part (b) of Proposition 2.1 can also be deduced from [5]. Indeed, for y ∈ D + we analogously have which together with (2.2) implies that the operator analogue to condition (1.1) is satisfied. However, the situation is less clear for part (c) of Proposition 2.1. Here, for y ∈ E + we get 3), this requires the stronger assumption V (p) + V (n) < (d − c)/2. The latter can be somehow remedied with a continuity argument, but in the end the approach presented in the proof of Proposition 2.1 above is just more convenient.
The above proposition includes the particular cases where V satisfies V < (d − c)/2 and where V is semidefinite with V < d − c, which in the context of part (c) have essentially been discussed in the proofs of Theorems 3.14 and 3.15 in [25]. However, Proposition 2.1 allows also certain indefinite perturbations V with (d − c)/2 ≤ V < d − c that were not covered before and may thus be used to refine the results in [25].
As immediate corollaries to Proposition 2.1, we obtain the following monotonicity and continuity statements for the minimax values in gaps of the essential spectrum, which, in essence, reproduce particular cases of results in [34].
3. Let A be as in Proposition 2.1, and let V 0 and V 1 be bounded self-adjoint operators on the same Hilbert space satisfying the condition max{ V is Lipschitz continuous with Lipschitz constant V .
Proof. Taking into account that for all x ∈ Dom(A), the claim follows immediately from Proposition 2.1.
It should again be mentioned that the above statements include the particular cases where the norm of the perturbations is less than (d − c)/2 or where the perturbations are semidefinite with a norm less than d − c. These cases have essentially been discussed in [25]. There, especially lower bounds on the movement of eigenvalues in gaps of the essential spectrum under certain conditions and the behaviour of edges of the essential spectrum have been studied. However, since this is not the main focus of the present work, this is not pursued further here.
As a consequence of Theorem 1.4, we obtain the following lower bound for the minimax values in the setting of off-diagonal operator perturbations.
Corollary 2.5. In the situation of Theorem 1.4, we have Taking the infimum over all such subspaces M + proves the claim by Theorem 1.4 and the standard minimax values for A| Ran P + .
As in Corollary 2.4, we also obtain a continuity statement in the situation of Theorem 1.4 with bounded off-diagonal perturbations. Here, however, we do not have to impose any condition on the norm of the perturbation.
Corollary 2.6. Let A and V be as in Theorem 1.4, and suppose that V is bounded.
is Lipschitz continuous with Lipschitz constant V .
In the particular case where B is semibounded, Theorem 1.5 allows us to extend Corollaries 2.5 and 2.6 to some degree to off-diagonal form perturbations. Recall here, that semiboundedness of B implies that also A is semibounded, see the proof of Theorem 1.5 below.
Corollary 2.7. Assume the hypotheses of Theorem 1.5.
(a) For each k ∈ N with k ≤ dim Ran P + = dim Ran Q + one has Then, for each k ≤ dim Ran E A ((0, ∞)) = dim Ran E Bt ((0, ∞)), the mapping is locally Lipschitz continuous.
Proof. (a). Taking into account that v[x, x] = 0 for all x ∈ D + by hypothesis, the inequality λ k (A| Ran P + ) ≤ λ k (B| Ran Q + ) is proved by means of Theorem 1.5 in a way analogous to Corollary 2.5. (b). Recall that each B t is indeed a semibounded self-adjoint operator with Dom[b t ] = Dom(|B t | 1/2 ) by the well-known KLMN theorem, and note that each tv satisfies the hypotheses of Theorem 1.5. Pick t, Consider first the case where A (and hence a) is lower semibounded with lower bound m ∈ R. We then have |a[ and, hence, and, therefore, This proves that t → λ k (t) is continuous on (−1/b, 1/b) and, in particular, bounded on every compact subinterval of (−1/b, 1/b). In turn, it then easily follows from (2.4) that this mapping is even locally Lipschitz continuous, which concludes the case where A is lower semibounded. If A is upper semibounded with upper bound m ∈ R, we proceed similarly.
Analogously as above, we then eventually obtain again (2.4), which proves the claim in the case where A is upper semibounded. This completes the proof.
for all x ∈ Dom[a], leading to for all x ∈ Dom[a]. If, in addition, b ≤ 1, this then leads to An example. The Stokes operator. We now briefly revisit the Stokes operator in the framework of Theorem 1.5. Here, we mainly rely on [12], but the reader is referred also to [11,Section 7], [28,Chapter 5], [8], and the references cited therein.
Let Ω ⊂ R n , n ≥ 2, be a bounded domain with C 2 -boundary, and let ν > 0 and v * ≥ 0. On the Hilbert space H = H + ⊕ H − with H + = L 2 (Ω) n and H − = L 2 (Ω), we consider the closed, densely defined, and nonnegative form a with Dom[a] := H 1 0 (Ω) n ⊕ L 2 (Ω) and Clearly, a is the form associated to the nonnegative self-adjoint operator A := −ν∆ ⊕ 0 on the Hilbert space , where ∆ = ∆ · I C n is the vector-valued Dirichlet Laplacian on Ω. Moreover, are the orthogonal projections onto H + and H − , respectively. In particular, we have Thus, by the well-known KLMN theorem, the form b S := a + v with Dom[b S ] = Dom[a] = Dom(|A| 1/2 ) is associated to a unique lower semibounded self-adjoint operator B S on H with Dom(|B S | 1/2 ) = Dom(|A| 1/2 ), the so-called Stokes operator. It is a self-adjoint extension of the (non-closed) upper dominant block operator matrix In fact, the closure of the latter is a self-adjoint operator, see [8, Theorems 3.7 and 3.9], which yields another characterization of the Stokes operator B S .
By rescaling, one obtains from [8,Theorem 3.15] that the essential spectrum of B S is given by cf. [12,Remark 2.2]. In particular, the essential spectrum of B S is purely negative. In turn, the positive spectrum of B S , that is, spec The above shows that the hypotheses of Theorem 1.5 are satisfied in this situation, so that we obtain from Theorem 1.5 and Corollary 2.7 the following result. Proposition 2.9. Let B S be the Stokes operator as above. Then, the positive spectrum of B S , spec(B S ) ∩ (0, ∞), is discrete, and the positive eigenvalues λ k (B S | Ran E B S ((0,∞)) ), k ∈ N, of B S , enumerated in nondecreasing order and counting multiplicities, admit the representation The latter depend locally Lipschitz continuously on ν and v * and satisfy the two-sided estimate Proof. In view the above considerations, the representation of the eigenvalues follows from Theorem 1.5, and the lower bound on the eigenvalues follows from Corollary 2.7 (a). Moreover, by rescaling, the continuity statement is a consequence of Corollary 2.7 (b). It remains to show the upper bound on the eigenvalues. To this end, let and ν div u 2 Similarly as in (2.5), we now obtain by means of Young's inequality that Since a[f, f ] = µ u 2 L 2 (Ω) n , this gives and taking the infimum over subspaces M + ⊂ H 1 0 (Ω) n with dim M + = k proves the upper bound. This completes the proof.
Remark 2.11. Since B S is lower semibounded, Proposition 2.9 can alternatively be proved via [9], see Remark 1.6 (1). Moreover, since A = −ν∆ ⊕ 0 has a spectral gap to the right of 0, the same is true with [23], see Remark 1.6 (2). In fact, [26] gives for λ k = λ k (B S | Ran E B S ((0,∞)) ), k ∈ N, with the same reasoning also the representation Reducing to the off-diagonal framework. Apart from the Stokes operator from the previous subsection, the consideration of off-diagonal perturbations in Theorems 1.4 and 1.5 may seem a bit restrictive. However, such perturbations naturally appear when the perturbation is decomposed into its diagonal and off-diagonal part. If the diagonal perturbation can then be handled efficiently in a suitable way, the discussed off-diagonal framework can be applied to the remaining part of the perturbation. The following result makes this precise in the setting of operator perturbations.
with respect to the decomposition Ran P + ⊕ Ran P − . Suppose that the following hold: ∞)), one has dim Ran P + = dim Ran Q and x, Bx = inf for all k ∈ N with k ≤ dim Ran Q.
Proof. By (iii) and (iv) we obviously have Upon a spectral shift by µ, the claim now follows from Theorem 1.4 with A, V , and Q + replaced by A + V diag , V off , and Q, respectively.  ([ν, ∞)). This is the situation encountered in the alternative proof of Proposition 2.1 (b) and the proof of Corollary 2.15 below. However, the conclusions can then alternatively be obtained also via [5,23], cf. Remark 1.6 (2).
Since conditions (i) and (ii) in Proposition 2.12 are clearly satisfied if V is bounded, the above provides an alternative way to prove part (b) of Proposition 2.1: Alternative proof of Proposition 2.1 (b). By spectral shift we may assume without loss of generality that c < 0 < d. We then have P = P + with P + as in Hypothesis 1.1. Let A ± and V diag be defined as in Proposition 2.12, and for • ∈ {p, n} denote by V in the sense of quadratic forms. In light of c + V (p) < d − V (n) and Remark 2.13, applying Proposition 2.12 proves the claim.
It is worth to note that an analogous reasoning for part (c) of Proposition 2.1 suffers from similar obstacles as the alternative proof based on [5,23] mentioned in Remark 2.2 (2).
If V is not bounded, conditions (i) and (ii) in Proposition 2.12 can still be guaranteed via the well-known Kato-Rellich theorem by means of a sufficiently small A-bound of V , as the following lemma shows.
Lemma 2.14. In the situation of Proposition 2.12, let V have A-bound smaller than 1/2. Then V diag and V off both have A-bound smaller than 1/2, and V off has (A + V diag )-bound smaller than 1.
Proof. By hypothesis, there are constants a, b ≥ 0, b < 1/2 such that we have V x ≤ a x + b Ax for all x ∈ Dom(A). Using Young's inequality, this gives for every ε > 0 that for all x ∈ Dom(A); cf. [15,Section V.4.1]. Since AP ± x = P ± Ax for all x ∈ Dom(A) and the ranges of P ± are orthogonal, this implies that for all x ∈ Dom(A). We choose ε > 0 such that β := b(1 + ε) 1/2 < 1/2 and set α := a(1 + 1/ε) 1/2 . It then follows from the above that and analogously the same for V off . This shows that V diag and V off indeed have A-bound smaller than 1/2. Using standard arguments as, for instance, in [33, Lemma 2.1.6], we obtain and, in turn, for all x ∈ Dom(A + V diag ) = Dom(A), where β/(1 − β) < 1. This shows that V off has (A + V diag )-bound smaller than 1 and, hence, completes the proof.
A suitable smallness assumption on the perturbation may also be used to guarantee condition (iii) and (iv) in Proposition 2.12 in the sense of Remark 2.13 if the unperturbed operator has a gap in the spectrum. This is demonstrated in the following corollary to the above. A, (c, d), and D ± be as in Proposition 2.1, and let V be a symmetric operator that is A-bounded with A-bound smaller than 1/2. Suppose, in addition, that c < 0 < d, and define A ± and V ± as in Proposition 2.12. Suppose that there are constants a ± , b ± ≥ 0, b ± < 1, with

Corollary 2.15. Let
Then, the interval (a − +(1−b − )c, (1−b + )d−a + ) belongs to the resolvent set of A + V , and one has dim Ran Proof. By Lemma 2.14 and the Kato-Rellich theorem, conditions (i) and (ii) in Proposition 2.12 are satisfied. By (2.8) we have by (2.9), the claim now again follows from Proposition 2.12 and Remark 2.13.
Remark 2.16. (1) It is easy to see that the left-hand side of (2.9) is invariant under a spectral shift in A. In this respect, an analogous statement as in Corollary 2.15 holds for arbitrary spectral gaps (c, d), not just ones satisfying c < 0 < d.
(3) As indicated in Remark 2.13, Corollary 2.15 can also be proved via [5,23] by verifying the operator analogue to (1.1). In fact, that approach even allows to weaken the assumption on the A-bound of V from being smaller than 1/2 to merely being smaller than 1 since it is then not necessary to have that V off has (A + V diag )-bound smaller than 1.

An abstract minimax principle in spectral gaps
We rely on the following abstract minimax principle in spectral gaps, part (a) of which is extracted from [13] and part (b) of which is its natural adaptation to the operator framework; cf. also [25,Proposition A.3]. For the convenience of the reader, its proof is reproduced in Appendix A below.
x, Bx for all k ∈ N with k ≤ dim Ran P + .
Remark 3.2. The above proposition is tailored towards spectral gaps to the right of 0, but by a spectral shift we can of course handle also spectral gaps to the right of any point γ ∈ R. Indeed, we have E A−γ ((0, ∞)) = E A ((γ, ∞)) for γ ∈ R and analogously for B. Moreover, the form associated to the operator B − γ is known to agree with the form b − γ. The latter can be seen for instance with an analogous reasoning as in [27, Proposition 10.5 (a)]; cf. also Lemma B.6 in Appendix B below.
Remark 3.3. Since P + and Q + are spectral projections for the respective operators, we have Ran(P + Q + | D + ) ⊂ D + and Ran(P + Q + | D + ) ⊂ D + . In this respect, the condition Ran(P + Q + | D + ) ⊃ D + in part (a) of Proposition 3.1 actually means that the restriction P + Q + | D + : D + → D + is surjective. This has not been formulated explicitly in the statement of [13, Theorem 1] but has instead been guaranteed by the stronger condition (|A| + I) 1/2 P + Q − (|A| + I) −1/2 < 1.
In fact, taking into account that D + = Ran((|A| + I) −1/2 | Ran P + ), a standard Neumann series argument in the Hilbert space Ran P + then even gives bijectivity of the restriction P + Q + | D + , see Step 2 of the proof of [13, Theorem 1]. In this reasoning, the operators (|A| + I) ±1/2 can be replaced by (|A| + αI) ±1/2 for any α > 0; if |A| has a bounded inverse, also α = 0 can be considered here.
In the context of our main theorems, the restriction P + Q + | Ran P + , understood as an endomorphism of Ran P + , will always be bijective, cf. Remark 3.5 (1) below. It turns out that then the hypotheses of part (b) in Proposition 3.1 imply those of part (a), in which case both representations for the minimax values in Proposition 3.1 are valid. More precisely, we have the following lemma, essentially based on the well-known Heinz inequality, cf. Appendix B below. (c) If the restriction P + Q + | Ran P + : Ran P + → Ran P + is bijective and Ran(P + Q + | D + ) ⊃ D + , then also Ran(P + Q + | D + ) ⊃ D + .
Proof. (a). This is a consequence of the well-known Heinz inequality, see, e.g., Corollary B.3 below. Alternatively, this follows by classical considerations regarding operator and form boundedness, see Remark B.4 below. (b). It follows from part (a) that the operator |B| 1/2 (|A| 1/2 + I) −1 is closed and everywhere defined, hence bounded by the closed graph theorem. Thus, x] ≤ 0 for x ∈ D − now follows from the hypothesis x, Bx ≤ 0 for all x ∈ D − by approximation.
Remark 3.5. (1) In light of the identity P + Q + = P + − P + Q − , the bijectivity of P + Q + | Ran P + : Ran P + → Ran P + can be guaranteed, for instance, by the condition P + Q − < 1 via a standard Neumann series argument. Since P + − Q + = P + Q − − P − Q + and, in particular, P + Q − ≤ P + − Q + , this condition holds if the stronger inequality P + − Q + < 1 is satisfied. In the latter case, there also is a unitary operator U with Q + U = U P + , see, e.g., [15,Theorem I.6.32], so that automatically dim Ran P + = dim Ran Q + . It is this situation we encounter in Theorems 1.3-1.5.
(2) In the case where B is an infinitesimal operator perturbation of A, the inequality P + Q − < 1 already implies that Ran(P + Q + | D + ) ⊃ D + , see the following section; the particular case where B is a bounded perturbation of A has previously been considered in [25,Lemma A.6]. For more general, not necessarily infinitesimal, perturbations, this remains so far an open problem.

Proof of Theorem 1.2: The graph norm approach
In this section we show that the inequality P + Q − < 1 in the context of Theorem 1.2 implies that Ran(P + Q + | D + ) ⊃ D + , which is essentially what is needed to deduce Theorem 1.2 from Proposition 3.1 and Lemma 3.4. The main technique used to accomplish this can in fact be formulated in a much more general framework: Recall that for a closed operator Λ on a Banach space with norm · , its domain Dom(Λ) can be equipped with the graph norm which makes (Dom(Λ), · Λ ) a Banach space. Also recall that a linear operator K with Dom(K) ⊃ Dom(Λ) is called Λ-bounded with Λ-bound β * ≥ 0 if for all β > β * there is an α ≥ 0 with and if there is no such α for 0 < β < β * .
The following lemma extends part (a) of [25,Proposition A.5], taken from Lemma 3.9 in the author's Ph.D. thesis [30], to relatively bounded commutators.
We are now in position to prove Theorem 1.2.
Proof of Theorem 1.2. We mainly follow the line of reasoning in the proof of [25,Lemma A.6]. Only a few additional considerations are necessary in order to accommodate unbounded perturbations V by means of Lemma 4.1.
For convenience of the reader, we nevertheless reproduce the whole argument here. Define S, T : Ran P + → Ran P + by S := P + Q − | Ran P + , T := P + Q + | Ran P + = I Ran P + − S.
By hypothesis, we have S ≤ P + Q − < 1, so that T is bijective. In light of Proposition 3.1 and Lemma 3.4, it now remains to show the inclusion Ran(P + Q + | D + ) ⊃ D + , that is, Ran(T −1 | D + ) ⊂ D + . To this end, we rewrite T −1 as a Neumann series, Clearly, S maps the domain D + = Dom(A| Ran P + ) into itself, so that the inclusion Ran(T −1 | D + ) ⊂ D + holds if the above series converges also with respect to the graph norm for the closed operator Λ := A| Ran P + . This, in turn, is the case if the corresponding spectral radius r Λ (S) of S is smaller than 1. For x ∈ D + ⊂ Ran P + we compute We show that the operator K is Λ-bounded with Λ-bound 0. Indeed, let b > 0, and choose a ≥ 0 with V x ≤ a x + b Ax for all x ∈ Dom(A); recall that V is infinitesimal with respect to A by hypothesis. Then, Thus, for x ∈ Dom(Λ) = D + . Since b > 0 was chosen arbitrarily, this implies that K is Λ-bounded with Λ-bound 0. It therefore follows from Lemma 4.1 that r Λ (S) ≤ S < 1, which completes the proof.
and the right-hand side of the latter is smaller than 1 if and only if This is a reasonable condition on the norm P + Q − only for b * < √ 2 − 1.
and, in turn, for all x ∈ Dom(A). Plugging these into (4.2) gives for all x ∈ Dom(Λ) = D + , which eventually leads to The right-hand side of the latter is smaller than 1 if and only if which is a reasonable condition on P + Q − only forb < 1/2.

The block diagonalization approach
In this section, we discuss an approach to verify the hypotheses of Proposition 3.1 and Lemma 3.4 which relies on techniques previously discussed in the context of block diagonalizations of operators and forms, for instance in [22] and [11], respectively; cf. also Remark 5.4 below.
Recall that for the two orthogonal projections P + and Q + from Hypothesis 1.1 the inequality P + − Q + < 1 holds if and only if Ran Q + can be represented as with some bounded linear operator X : Ran P + → Ran P − ; in this case, one has see, e.g., [16,Corollary 3.4 (i)]. The orthogonal projection Q + can then be represented as the 2 × 2 block operator matrices (5.3) Q + = (I Ran P + + X * X) −1 (I Ran P + + X * X) −1 X * X(I Ran P + + X * X) −1 X(I Ran P + + X * X) −1 X * = (I Ran P + + X * X) −1 X * (I Ran P − + XX * ) −1 (I Ran P − + XX * ) −1 X XX * (I Ran P − + XX * ) −1 with respect to Ran P + ⊕ Ran P − , see, e.g., [16,Remark 3.6]. In particular, we have (5.4) P + Q + | Ran P + = (I Ran P + + X * X) −1 , which is in fact the starting point for the current approach: With regard to the desired relations Ran(P + Q + | D + ) ⊃ D + and Ran(P + Q + | D + ) ⊃ D + , we need to establish that the operator I Ran P + + X * X maps D + and D + into D + and D + , respectively. Define the skew-symmetric operator Y via the 2×2 block operator matrix with respect to Ran P + ⊕ Ran P − . Then, the operators I ± Y are bijective with The following lemma is extracted from various sources. We comment on this afterwards in Remark 5.2 below. Lemma 5.1. Suppose that the projections P + and Q + from Hypothesis 1.1 satisfy P + − Q + < 1, and let the operators X and Y be as in (5.1) and (5.5), respectively. Moreover, let C be an invariant subspace for both P + and Q + such that C = (C ∩ Ran P + ) ⊕ (C ∩ Ran P − ) =: Then, the following are equivalent: Proof. Clearly, the hypotheses imply that P + Q + maps C into C + and P − Q + maps C into C − . (i)⇒(ii). Let g ∈ C − . Using the first representation in (5.3), we then have (I Ran P + + X * X) −1 X * g = (P + Q + | Ran P − )g ∈ C + . Hence, X * g ∈ C + by (i) and, in turn, h := (I Ran P + + X * X)X * g ∈ C + . Using again (5.3), this yields As a byproduct, we have also shown that X * maps C − into C + .
(ii)⇒(i). Using the identities (I Ran P − + XX * ) −1 X = P − Q + | Ran P + and X * (I Ran P − + XX * ) −1 = P + Q + | Ran P − taken from the second representation in (5.3), the proof is completely analogous to the implication (i)⇒(ii). In particular, we likewise obtain as a byproduct that X maps C + into C − .
(i),(ii)⇒(iii). We have already seen that X maps C + into C − and that X * maps C − into C + . Taking into account that C = C + ⊕ C − , this means that Y maps C into itself.
(iv),(v)⇒(i),(ii). This follows immediately from identity (5.6). The equivalence (iv)⇔(v) can alternatively be directly obtained from the identity Such an argument has been used in the proof of [ 5 these need to be verified explicitly from the specific hypotheses at hand. Here, we rely on previous considerations on block diagonalizations for block operator matrices and forms. In case of Theorem 1.4, the crucial ingredient is presented in the following result, extracted from [22]. An earlier result in this direction is commented on in Remark 5.4 (2) below. Proposition 5.3 (see [22,Theorem 6.1]). In the situation of Theorem 1.4 one has P + − Q + ≤ √ 2/2 < 1, and the operator identity holds with Y as in (5.5). Proof.
Clearly, the hypotheses on V ensure that V 0 is A-bounded with A-bound b * < 1 and off-diagonal with respect to the decomposition Ran P + ⊕ Ran P − . By [22,Lemma 6.3] we now have In light of (5.2), the claim therefore is just an instance of [22, Theorem 6.1].
Remark 5.4. (1) Let A ± := A| Ran P ± be the parts of A associated with the subspaces Ran P ± , and write In this sense, identity (5.7) can be viewed as a block diagonalization of the operator A + V . For a more detailed discussion of block diagonalizations and operator Riccati equations in the operator setting, the reader is referred to [22] and the references cited therein. To the best of the author's knowledge, no direct analogue of Proposition 5.3 is known so far in the setting of form rather than operator perturbations. Although the inequality P + −Q + ≤ √ 2/2 can be established here as well under fairly reasonable assumptions, see [11,Theorem 3.3], the mapping properties of the operators I ± Y connected with a corresponding diagonalization related to (5.7) are much harder to verify. The situation is even more subtle there since also the domain equality Dom(|A| 1/2 ) = Dom(|B| 1/2 ) needs careful treatment. The latter is conjectured to hold in a general offdiagonal form perturbation framework [10,Remark 2.7]. Some characterizations have been discussed in [29,Theorem 3.8], but they all are hard to verify in a general abstract setting. A compromise in this direction is to require that the form b is semibounded, see [29,Lemma 3.9] and [11,Lemma 2.7], which forces the diagonal form a to be semibounded as well, see below. As in the situation of Theorem 1.3 above, this simplifies matters immensely: To this end, we first show that b is a semibounded saddle-point form in the sense of [11, Section 2]: Let m ∈ R be the lower (resp. upper) bound of a. We then have for all x ∈ Dom[a], where |A − m| 1/2 (|A| 1/2 + I) −1 is closed and everywhere defined, hence bounded by the closed graph theorem. From this and the hypothesis on v we see that with some β ≥ 0, which means that The following abstract minimax principle is extracted from [13, Theorem 1] and its proof follows the one in [13] almost literally. However, the result below is formulated in such a way that the operator and form settings can be handled simultaneously. To this end, we use a suitable subset C of Dom(|B| 1/2 ) satisfying Dom(B) ⊂ C. Part (a) of Proposition 3.1 then agrees with the choice C = Dom(|B| 1/2 ), whereas Part (b) of Proposition 3.1 corresponds to C = Dom(B).
(a) For all k ∈ N with k ≤ dim Ran Q + we have k ≤ dim Ran P + and If, in addition, Ran(P + Q + | C + ) ⊃ C + holds, then for all k ∈ N with k ≤ dim Ran P + we have k ≤ dim Ran Q + and Proof. We first show that the restriction (A.1) P + | C∩Ran Q + : C ∩ Ran Q + → C + is injective. Indeed, assume to the contrary that P + x = 0 for some non-zero x ∈ C ∩ Ran Q + . Then, on the one hand we have b[x, x] > 0, and on the other hand x ∈ Ran P − , that is, x ∈ C − . The latter gives b[x, x] ≤ 0 by hypothesis, a contradiction. (a). Let k ∈ N with k ≤ dim Ran Q + . Let ε > 0 be arbitrary, and abbreviate λ k = λ k (B| Ran Q + ). Consider the subspace M := Ran E B ((0, λ k + ε)) ⊂ C ∩ Ran Q + , and denote by P M the orthogonal projection onto M. We clearly have dim M ≥ k. Choose any subspace N ⊂ M with dim N = k. The injectivity of (A.1) then gives dim Ran P + ≥ dim Ran P + | N = dim N = k.
Let x = P + u ⊕ w, x = 1, with u ∈ N and w ∈ C − . Then, x = u + v with v = w − P − u ∈ C − . Moreover, P M v ∈ M and, taking into account that M is reducing for B, we have Set In turn, taking into account that x = u + v = x 1 ⊕ x 2 and that M and M ⊥ are reducing for B, we obtain In light of dim Ran P + | N = k as observed above, we thus conclude that Since ε was chosen arbitrarily, this proves (a).
(b). The additional assumption Ran(P + Q + | C + ) ⊃ C + guarantees that the restriction (A.1) is also surjective, hence bijective. Let k ∈ N with k ≤ dim Ran P + , and let M + be any subspace of C + with dim M + = k; note that C + is dense in Ran P + , so that such a subspace indeed exists. By bijectivity of (A.1), there is a subspace M ⊂ C∩Ran Q + with dim M = k and M + = Ran P + | M . In particular, we have k ≤ dim Ran Q + . Since M ⊂ C, we have M ⊂ M + ⊕ C − and, therefore, Since M + was chosen arbitrarily, together with part (a) this proves (b) and, thus, completes the proof.

Appendix B. Heinz inequality
In this appendix we discuss some consequences of the well-known Heinz inequality. These consequences or particular cases thereof are used at various spots of the main part of the paper, but they may also be of independent interest. They are probably folklore, but in lack of a suitable reference they are nevertheless presented here in full detail.
Throughout this appendix, we denote the norm associated with the inner product of a Hilbert space H by · H .
The following variant of the Heinz inequality is taken from [18].
Then, for all ν ∈ [0, 1], the operator S maps Dom(Λ ν 1 ) into Dom(Λ ν 2 ), and for all x ∈ Dom(Λ ν 1 ) one has The above result admits the following extension to closed densely defined operators between Hilbert spaces. For a generalization of Proposition B.1 to maximal accretive operators, see [14].
The last corollary discussed here is related to the question whether an operator sum represents the sum of the corresponding forms. Part (b) of this corollary can in some sense also be regarded as an extension of [28, Lemma 2.2.7] to not necessarily off-diagonal perturbations.
Corollary B.6 (cf. [28,Lemma 2.2.7]). Let Λ be a self-adjoint operator on a Hilbert space with inner product ·, · , and let K be an operator on the same Hilbert space.