aa r X i v : . [ m a t h . SP ] A ug ON A MINIMAX PRINCIPLE IN SPECTRAL GAPS
ALBRECHT SEELMANN
Abstract.
The minimax principle for eigenvalues in gaps of the es-sential spectrum by Griesemer, Lewis, and Siedentop in [Doc. Math. (1999), 275–283] is adapted to cover certain abstract perturbative set-tings with bounded or unbounded perturbations, in particular ones thatare off-diagonal with respect to the spectral gap under consideration.This in part builds upon and extends the considerations in the author’sappendix to [arXiv:1804.07816]. Several monotonicity and continuityproperties 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 semiboundedoperator A on a Hilbert space H are given by λ k ( A ) = inf M ⊂ Dom( A )dim M = k sup x ∈ M k x k =1 h x, Ax i = inf M ⊂ Dom( | A | / )dim M = k sup x ∈ M k x k =1 a [ x, x ]for k ∈ N with k ≤ dim H , see, e.g., [14, Theorem 12.1] and also [20, Sec-tion 12.1 and Exercise 12.4.2]. Here, h· , ·i denotes the inner product of H ,and a with a [ x, x ] = h| A | / x, sign( A ) | A | / x i for x ∈ Dom( | A | / ) is theform associated with A .The above minimax values have proved to be a powerful description ofthe eigenvalues below the essential spectrum of A ; in fact, they agree withthese eigenvalues in nondecreasing order counting multiplicities. A standardapplication in this context is that these eigenvalues exhibit a monotonicitywith respect to the operator: for two 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., [20, Corollary 12.3].Matters get, however, much more complicated when eigenvalues in a gapof the essential spectrum are considered. If A + is the (lower semibounded)part of A with spectrum in an interval of the form ( γ, ∞ ), γ ∈ R , then theminimax values for A + still describe the eigenvalues of A + below its essentialspectrum and thus the eigenvalues of A in ( γ, ∞ ) below the essential spec-trum of A above γ . However, the subspaces over which the correspondinginfimum is taken are chosen within the spectral subspace for A associatedwith the interval ( γ, ∞ ) and therefore usually depend on the operator itselfrather than just its domain. This makes it difficult to compare minimax Mathematics Subject Classification.
Primary 49Rxx; Secondary 47A10, 47A75.
Key words and phrases.
Minimax values, eigenvalues in gap of the essential spectrum,block diagonalization, Stokes operator. values in spectral gaps of two different operators A and B , even if theirdomains agree.In [9], Griesemer, Lewis, and Siedentop devised an abstract minimax prin-ciple for eigenvalues in spectral gaps that allows to overcome these problems.However, the corresponding hypotheses seem to be hard to verify on an ab-stract level, cf. Remark 3.3 (2) below. In the particular situation of boundedadditive perturbations, the present author has adapted this abstract min-imax principle in the appendix to [19] with hypotheses that can in somecases be verified explicitly by means of the Davis-Kahan sin 2Θ theoremfrom [2] and variants thereof. This has been successfully applied in [19] tostudy lower bounds on the movement of eigenvalues in gaps of the essen-tial spectrum and of edges of the essential spectrum. In the present note,the considerations from [19, Appendix A] are supplemented and extendedto cover also certain unbounded perturbations, in particular ones that areoff-diagonal with respect to the spectral gap under consideration. It shouldbe mentioned that some of the results discussed here might also be obtainedwith the alternative approaches from [3,4,15,17]. However, the present workfocuses on [9] as a starting point since the techniques employed to apply thatabstract minimax principle promise to be of a broader interest. Main results.
In order to formulate our main results, it is convenient to fixthe following notational setup in the case where γ = 0; the case of general γ ∈ R can of course always be reduced to this situation by spectral shift,cf. Remark 3.2 and also the proofs of Proposition 2.1 below. Hypothesis 1.1.
Let A be a self-adjoint operator on a Hilbert space. De-note the spectral projectors for A associated with the intervals (0 , ∞ ) and( −∞ ,
0] by P + and P − , respectively, that is, P + := E A (cid:0) (0 , ∞ ) (cid:1) , P − := I − P + , and let D ± := Ran P ± ∩ Dom( A ) , D ± := Ran P ± ∩ Dom( | A | / ) . Moreover, let B be another self-adjoint operator on the same Hilbert spacewith analogously defined spectral projections Q + := E B (cid:0) (0 , ∞ ) (cid:1) , Q − := I − Q + , and denote by b the form associated with B , that is, b [ x, y ] = h| B | / x, sign( B ) | B | / y i for x, y ∈ Dom( b ) = Dom( | B | / ).Here, E A and E B stand for the projection-valued spectral measures forthe 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 thepositive part A | Ran P + of A can clearly be written as λ k ( A | Ran P + ) = inf M + ⊂D + dim M + = k sup x ∈ M + ⊕D − k x k =1 h x, Ax i = inf M + ⊂ D + dim M + = k sup x ∈ M + ⊕ D − k x k =1 a [ x, x ] N A MINIMAX PRINCIPLE IN SPECTRAL GAPS 3 for k ∈ N with k ≤ dim Ran P + . In our main results below we give conditionson B under which the minimax values for the positive part B | Ran Q + of B admit the same representations with h x, Ax i and a [ x, x ] replaced by h x, Bx i and b [ x, x ], respectively, but with the infima taken over the same respectivefamilies of subspaces as for A . It is natural to consider this in an perturbativeframework where B is obtained by an operator or form perturbation of A and, thus, one has Dom( A ) = Dom( B ) and/or Dom( | A | / ) = Dom( | B | / ).Four results in this direction are presented here, each addressing differentsituations. We first treat the case of operator perturbations and start withthe direct extension of [19, Theorem A.2] to infinitesimal perturbations.Recall that an operator V is called A -bounded with A -bound b ∗ ≥ V ) ⊃ Dom( A ) and for all b > b ∗ there is some a ≥ k V x k ≤ a k x k + b k Ax k for all x ∈ Dom( A ) . 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 ofthe form B = A + V with some symmetric operator V that is infinitesimalwith respect to A . Furthermore, suppose that k P + Q − k < and that h x, Bx i ≤ for all x ∈ D − . Then, λ k ( B | Ran Q + ) = inf M + ⊂D + dim M + = k sup x ∈ M + ⊕D − k x k =1 h x, Bx i = inf M + ⊂ D + dim M + = k sup x ∈ M + ⊕ D − k x k =1 b [ x, x ] for all k ∈ N with k ≤ dim Ran P + . Two remarks regarding Theorem 1.2 are in order: (1) also certain per-turbations V that are not infinitesimal with respect to A can be consideredhere, but at the cost of a stronger assumption on k P + Q − k , see Remark 4.2below; (2) the condition k P + Q − k < k P + − Q + k < P + andRan Q + have the same dimension, that is, dim Ran P + = dim Ran Q + , seeRemark 3.5 (a) below.The stronger condition k P + − Q + k < A is semibounded. Theorem 1.3.
Assume Hypothesis 1.1. Suppose, in addition, that A issemibounded and that k P + − Q + k < .If Dom( | A | / ) = Dom( | B | / ) and b [ x, x ] ≤ for all x ∈ D − , then (1.1) λ k ( B | Ran Q + ) = inf M + ⊂ D + dim M + = k sup x ∈ M + ⊕ D − k x k =1 b [ x, x ] for all k ≤ dim Ran P + = dim Ran Q + . If even Dom( A ) = Dom( B ) and h x, Bx i ≤ for all x ∈ D − , then also (1.2) λ k ( B | Ran Q + ) = inf M + ⊂D + dim M + = k sup x ∈ M + ⊕D − k x k =1 h x, Bx i A. SEELMANN for all k ≤ dim Ran P + = dim Ran Q + . It should be emphasized that the conditions Dom( A ) = Dom( B ) and h x, Bx i ≤ x ∈ D − in Theorem 1.3 indeed imply that one has alsoDom( | A | / ) = Dom( | B | / ) and b [ x, x ] ≤ x ∈ D − , see Lemma 3.4below. Note also that in contrast to Theorem 1.2, Theorem 1.3 makes noassumptions on how the operator B is obtained from A . The latter will,however, be relevant when the hypotheses of Theorem 1.3 are to be verifiedin concrete situations.The condition h x, Bx i ≤ x ∈ D − plays an important role in bothTheorems 1.2 and 1.3. In the case where B = A + V with some A -boundedsymmetric operator V as in Theorem 1.2, this condition is automaticallysatisfied if h x, V x i ≤ x ∈ D − since h x, Ax i ≤ x ∈ D − by definition. A particular instance of such perturbations V are so-called off-diagonal perturbations with respect to the decomposition Ran P + ⊕ Ran P − ,in which case also the condition k P + − Q + k < A -bound of V here. Theorem 1.4.
Assume Hypothesis 1.1. Suppose, in addition, that B hasthe form B = A + V with some symmetric A -bounded operator V with A -bound smaller than and which is off-diagonal on Dom( A ) with respect tothe decomposition Ran P + ⊕ Ran P − , that is, P − V P − x = 0 = P + V P + x for all x ∈ Dom( A ) . Then, one has dim Ran P + = dim Ran Q + and λ k ( B | Ran Q + ) = inf M + ⊂D + dim M + = k sup x ∈ M + ⊕D − k x k =1 h x, Bx i = inf M + ⊂ D + dim M + = k sup x ∈ M + ⊕ D − k x k =1 b [ x, x ] for all k ∈ N with k ≤ dim Ran Q + . The last theorem can to some extend be formulated also for off-diagonalform perturbations, at least in the semibounded setting. The latter restric-tion is commented on in Section 5 below.
Theorem 1.5.
Assume Hypothesis 1.1. Suppose, in addition, that B issemibounded and that its form b is given by b = a + v , where a is the formassociated with A and v is a symmetric sesquilinear form satisfying v [ P + x, P + y ] = 0 = v [ P − x, P − y ] for all x, y ∈ Dom[ a ] ⊂ Dom[ v ] and (1.3) | v [ x, x ] | ≤ a k x k + b | a [ x, x ] | for all x ∈ Dom( | A | / ) = Dom[ a ] with some constants a, b ≥ .Then, one has dim Ran P + = dim Ran Q + and λ k ( B | Ran Q + ) = inf M + ⊂ D + dim M + = k sup x ∈ M + ⊕ D − k x k =1 b [ x, x ] for all k ∈ N with k ≤ dim Ran Q + . N A MINIMAX PRINCIPLE IN SPECTRAL GAPS 5
The semiboundedness of B in Theorem 1.5 forces A to be semiboundedas well, see the proof of Theorem 1.5 below. In turn, it is natural to supposethat A is semibounded and then to guarantee the semiboundedness of B via the well-known KLMN theorem by ensuring (1.3) with b <
1. In thisregard, Theorem 1.5 can be interpreted as a particular case of Theorem 1.3with Dom( | A | / ) = Dom( | B | / ), in which the remaining hypotheses areautomatically satisfied due to the structure of the perturbation.The rest of this note is organized as follows. In Section 2 we discussapplications of the main theorems and revisit the Stokes operator as an ex-ample in the framework of Theorem 1.5. Section 3 is devoted to an abstractminimax principle based on [9]. Two approaches are then used to verify thehypotheses of this abstract minimax principle, the graph norm approach andthe block diagonalization approach , respectively, which are discussed sepa-rately in Sections 4 and 5 below. Theorem 1.2 is proved in Section 4, whichis based on the author’s appendix to [19] and extends the correspondingconsiderations to certain unbounded perturbations V . Theorems 1.3–1.5are proved in Section 5, which builds upon recent developments on block di-agonalization of operators and forms from [16] and [7], respectively. Finally,Appendix A provides some consequences of the well-known Heinz inequalitythat are used at various spots in this note and are probably folklore.2. Applications and examples
In this section, we use the main results from Section 1 to prove mono-tonicity and continuity properties of minimax values in gaps of the essentialspectrum in various situations and also revisit the well-known Stokes op-erator in the framework of Theorem 1.5 as an example. We first considerthe situation of indefinite or semidefinite bounded perturbations, which hasessentially been discussed in a slightly different form in [19].For a bounded self-adjoint operator V we define bounded nonnegativeoperators V ( p ) and V ( n ) with V = V ( p ) − V ( n ) via functional calculus by(2.1) V ( p ) := (1 + sign( V )) V / , V ( n ) := (sign( V ) − V / . We clearly have k V ( p ) k ≤ k V k and k V ( n ) k ≤ k V k . Proposition 2.1.
Let the finite interval ( c, d ) belong to the resolvent setof the self-adjoint operator A , and let V be a bounded self-adjoint operatoron the same Hilbert space. Define D + := Ran E A ([ d, ∞ )) ∩ Dom( A ) and D − := Ran E A (( −∞ , c ]) ∩ Dom( A ) .If k V ( p ) k + k V ( n ) k < d − c with V ( p ) and V ( n ) as in (2.1) , then the interval ( c + k V ( p ) k , d − k V ( n ) k ) belongs to the resolvent set of the operator A + V ,and one has dim Ran E A ([ d, ∞ )) = dim Ran E A + V ([ d − k V ( n ) k , ∞ )) and λ k (cid:0) ( A + V ) | Ran E A + V ([ d −k V ( n ) k , ∞ )) (cid:1) = inf M + ⊂D + dim M + = k sup x ∈ M + ⊕D − k x k =1 h x, ( A + V ) x i for all k ∈ N with k ≤ dim Ran E A + V ([ d − k V ( n ) k , ∞ )) .Remark . A corresponding representation of the minimax values in termsof the form associated with A + V as in Theorems 1.2–1.4 holds here as well. A. SEELMANN
However, for the sake of simplicity and since this is not needed in Corollar-ies 2.3 and 2.4 below, this has not been formulated in Proposition 2.1.The above proposition includes the particular cases where V satisfies k V k < ( d − c ) / V is semidefinite with k V k < d − c , whichessentially have been discussed in the proofs of Theorems 3.14 and 3.15in [19]; cf. also the discussion after Corollaries 2.3 and 2.4 below. How-ever, Proposition 2.1 allows also certain indefinite perturbations V with( d − c ) / ≤ k V k < d − c that were not covered before.We discuss here two proofs of Proposition 2.1, one based on Theorem 1.2that is close to the proofs of Theorems 3.14 and 3.15 in [19] and the otherone based on Theorem 1.4. Both emphasize different aspects on how to dealwith the perturbation V . Proof of Proposition 2.1 based on Theorem 1.2.
By Proposition 2.1 in [25],the interval ( c + k V ( p ) k , d − k V ( n ) k ) belongs to the resolvent set of the op-erator A + V .Pick γ ∈ ( c + k V ( p ) k , d −k V ( n ) k ). We then have E A − γ ((0 , ∞ )) = E A ([ d, ∞ ))as well as E A − γ (( −∞ , E A (( −∞ , c ]).For x ∈ D − we clearly have h x, ( A + V − γ ) x i = h x, ( A − γ ) x i + h x, V ( p ) x i − h x, V ( n ) x i≤ ( c − γ + k V ( p ) k ) k x k < . Moreover, with P + := E A ([ d, ∞ )) and Q + := E A + V ([ d − k V ( n ) k , ∞ )), thevariant of the Davis-Kahan sin 2Θ theorem in [25, Theorem 1.1] implies that k P + − Q + k ≤ sin (cid:16)
12 arcsin k V ( p ) k + k V ( n ) k d − c (cid:17) < √ < . Taking into account that E A + V − γ ((0 , ∞ )) = Q + and λ k (( A + V − γ ) | Ran Q + ) = λ k (( A + V ) | Ran Q + ) − γ for all k ≤ dim Ran Q + , the claim now follows by applying Theorem 1.2;cf. also Remark 3.5 (1) below. (cid:3) Proof of Proposition 2.1 based on Theorem 1.4.
As in the first proof, the in-terval ( c + k V ( p ) k , d − k V ( n ) k ) belongs to the resolvent set of the operator A + V . Pick again γ ∈ ( c + k V ( p ) k , d − k V ( n ) k ).Let A + := A | Ran E A ([ d, ∞ )) and A − := A | Ran E A (( −∞ ,c ]) denote the parts of A associated with Ran E A ([ d, ∞ )) and Ran E A (( −∞ , c ]), respectively. More-over, for • ∈ { p, n } , decompose V ( • ) as V ( • ) = V ( • )diag + V ( • )off , where V ( • )diag = V ( • )+ ⊕ V ( • ) − is the diagonal part of V ( • ) and V ( • )off is the off-diagonal part of V ( • ) with respect to Ran E A ([ d, ∞ )) ⊕ Ran E A (( −∞ , c ]). Weclearly have V ( • ) ± ≥ k V ( • ) ± k ≤ k V ( • ) k . Thus, A − + V ( p ) − − V ( n ) − ≤ c + k V ( p ) k < γ < d − k V ( n ) k ≤ A + + V ( p )+ − V ( n )+ , so that E A ([ d, ∞ )) = E A + V ( p )diag − V ( n )diag (( γ, ∞ )) N A MINIMAX PRINCIPLE IN SPECTRAL GAPS 7 and E A (( −∞ , c ]) = E A + V ( p )diag − V ( n )diag (( −∞ , γ ]) , cf. the proof of [24, Proposition 2.1]. Taking into account that V ( p )off − V ( n )off is off-diagonal with respect to Ran E A ([ d, ∞ )) ⊕ Ran E A (( −∞ , c ]) and that A + V = ( A + V ( p )diag − V ( n )diag ) + V ( p )off − V ( n )off , the claim now follows fromTheorem 1.4 via a spectral shift by γ as in the first proof. (cid:3) As corollaries to Proposition 2.1, we obtain the following monotonicityand continuity statements for the minimax values in gaps of the essentialspectrum.
Corollary 2.3 (cf. [19, Theorem 3.15 (2) and Theorem 3.14]) . Let A be asin Proposition 2.1, and let V and V be bounded self-adjoint operators on thesame Hilbert space satisfying max {k V ( p )0 k + k V ( n )0 k , k V ( p )1 k + k V ( n )1 k} < d − c .If, in addition, V ≤ V , then λ k (cid:0) ( A + V ) | Ran E A + V ([ d −k V ( n )0 k , ∞ )) (cid:1) ≤ λ k (cid:0) ( A + V ) | Ran E A + V ([ d −k V k ( n ) , ∞ )) (cid:1) for k ≤ dim Ran E A ([ d, ∞ )) = dim Ran E A + V j ([ d − k V ( n ) j k , ∞ )) , j ∈ { , } . Corollary 2.4.
Let A and V be as in Proposition 2.1. Then, the interval ( c + k V ( p ) k , d −k V ( n ) k ) belongs to the resolvent set of every A + tV , t ∈ [0 , ,and for each k ≤ dim Ran E A ([ d, ∞ )) = dim Ran E A + tV ([ d − t k V ( n ) k , ∞ )) , t ∈ [0 , , the mapping [0 , ∋ t λ k (( A + tV ) | Ran E A + tV ([ d − t k V ( n ) k , ∞ )) ) is Lipschitz continuous with Lipschitz constant k V k .Proof. Taking into account that h x, ( A + sV ) x i − | t − s |k V k ≤ h x, ( A + tV ) x i ≤ h x, ( A + sV ) x i + | t − s |k V k for all x ∈ Dom( A ), the claim follows immediately from Proposition 2.1. (cid:3) It should again be mentioned that the above statements include the par-ticular cases where the norm of the perturbations is less than ( d − c ) / d − c . Thesecases have essentially been discussed in [19]. There, especially lower boundson the movement of eigenvalues in gaps of the essential spectrum under cer-tain conditions and the behaviour of edges of the essential spectrum havebeen studied. However, since this is not the main focus of the present note,this is not pursued further here.The second proof of Proposition 2.1 discussed above is flexible enoughto handle also unbounded perturbations that are small enough in a certainsense, at least in the semibounded setting. This is demonstrated in thefollowing result for the case where A is lower semibounded. Proposition 2.5.
Let A , ( c, d ) , and D ± be as in Proposition 2.1, and sup-pose, in addition, that A is lower semibounded. Let V be a symmetric op-erator that is A -bounded with A -bound smaller than . Moreover, supposethat there are constants a, b ≥ , b < , with |h x, V x i| ≤ a k x k + b h x, Ax i for all x ∈ Dom( A ) A. SEELMANN and (2.2) 2 a + b ( c + d ) < d − c. Then, the interval ( a + (1 + b ) c, (1 − b ) d − a ) belongs to the resolvent set of A + V , and one has dim Ran E A ([ d, ∞ )) = dim E A + V ([(1 − b ) d − a, ∞ )) and λ k (( A + V ) | Ran E A + V ([(1 − b ) d − a, ∞ )) ) = inf M + ⊂D + dim M + = k sup x ∈ M + ⊕D − k x k =1 h x, ( A + V ) x i for all k ∈ N with k ≤ dim Ran E A + V ([(1 − b ) d − a, ∞ )) .Proof. For x ∈ Dom( A ) = Dom( A + V ), we clearly have(1 − b ) h x, Ax i − a k x k ≤ h x, ( A + V ) x i ≤ (1 + b ) h x, Ax i + a k x k . According to (2.2), we may pick γ ∈ R satisfying the two-sided inequality a + (1 + b ) c < γ < (1 − b ) d − a . Let A ± be as in the second proof ofProposition 2.1, and again decompose the perturbation V as V = V diag + V off with diagonal part V diag = V + ⊕ V − and off-diagonal part V off . The abovethen gives A − + V − ≤ a + (1 + b ) c < γ < (1 − b ) d − a ≤ A + + V + , so that E A ([ d, ∞ )) = E A + V diag (( γ, ∞ )) = E A + V diag ([(1 − b ) d − a, ∞ )) and E A (( −∞ , c ]) = E A + V diag (( −∞ , γ ]), and the interval ( a + (1 + b ) c, (1 − b ) d − a )belongs to the resolvent set of A + V diag . By [18, Theorem 1] (cf. also [1,Theorem 2.1]), the interval ( a + (1 + b ) c, (1 − b ) d − a ) then belongs also tothe resolvent set of A + V = ( A + V diag ) + V off . The rest of the claim isnow proved as in the second proof of Proposition 2.1 via Theorem 1.4 anda spectral shift by γ . (cid:3) Remark . (1) If A is lower semibounded and the symmetric operator V is A -bounded with A -bound smaller than 1, then constants a, b ≥ b < |h x, V x i| ≤ a k x k + b h x, Ax i for all x ∈ Dom( A )exist by [11, Theorem VI.1.38]. Condition (2.2) can then be guaranteed for tV instead of V for t ∈ R with sufficiently small modulus.(2) A similar result as in Proposition 2.5 holds also if instead A is uppersemibounded. In this case, one requires constants a, b ≥ b <
1, satisfying |h x, V x i| ≤ a k x k − b h x, Ax i for all x ∈ Dom( A ) and 2 a − b ( c + d ) < d − c .We then get in a completely analogous way a representation for the minimaxvalues of ( A + V ) | Ran E A + V ([(1+ b ) d − a )) .As another consequence of Theorem 1.4, we obtain the following lowerbound for the minimax values in the setting of off-diagonal operator pertur-bations. Corollary 2.7.
In the situation of Theorem 1.4, we have λ k ( A | Ran P + ) ≤ λ k ( B | Ran Q + ) for all k ≤ dim Ran P + = dim Ran Q + . N A MINIMAX PRINCIPLE IN SPECTRAL GAPS 9
Proof.
Let M + ⊂ D + with dim M + = k . Since h x, V x i = 0 for all x ∈ D + by hypothesis, we havesup x ∈ M + k x k =1 h x, Ax i = sup x ∈ M + k x k =1 h x, ( A + V ) x i ≤ sup x ∈ M + ⊕D − k x k =1 h x, ( A + V ) x i . Taking the infimum over all such subspaces M + proves the claim by Theo-rem 1.4. (cid:3) As in Corollary 2.4, we also obtain a continuity statement in the situationof Theorem 1.4 with bounded off-diagonal perturbations. Here, however, wedo not have to impose any condition on the norm of the perturbation.
Corollary 2.8.
Let A and V be as in Theorem 1.4, and suppose that V isbounded. Then, for each k ≤ dim Ran E A ((0 , ∞ )) = dim Ran E A + tV ((0 , ∞ )) , t ∈ R , the mapping R ∋ t λ k (cid:0) ( A + tV ) | Ran E A + tV ((0 , ∞ )) (cid:1) is Lipschitz continuous with Lipschitz constant k V k . In the particular case where B is semibounded, Theorem 1.5 allows usto extend Corollaries 2.7 and 2.8 to some degree to off-diagonal form per-turbations. Recall here, that semiboundedness of B implies that also A issemibounded, see the proof of Theorem 1.5 below. Corollary 2.9.
Assume the hypotheses of Theorem 1.5. (a)
For each k ∈ N with k ≤ dim Ran P + = dim Ran Q + one has λ k ( A | Ran P + ) ≤ λ k ( B | Ran Q + ) . (b) Denote for t ∈ ( − /b, /b ) by B t the self-adjoint operator associatedwith the form b t := a + t v with form domain Dom[ b t ] := Dom[ a ] .Then, for each k ≤ dim Ran E A ((0 , ∞ )) = dim Ran E B t ((0 , ∞ )) , themapping ( − /b, /b ) ∋ t λ k ( B t | Ran E Bt ((0 , ∞ )) ) is locally Lipschitz continuous.Proof. (a). Taking into account that v [ x, x ] = 0 for all x ∈ D + by hy-pothesis, the inequality λ k ( A | Ran P + ) ≤ λ k ( B | Ran Q + ) is proved by means ofTheorem 1.5 in a way analogous to Corollary 2.7.(b). Upon a rescaling, we may assume without loss of generality that b <
1. Also recall that each B t is indeed a semibounded self-adjoint operatorwith Dom[ b t ] = Dom( | B t | / ) by the well-known KLMN theorem, and notethat each t v satisfies the hypotheses of Theorem 1.5. Pick t, s ∈ ( − /b, /b )with b | t − s | ≤ − b | s | .Consider first the case where A (and hence a ) is lower semibounded withlower bound m ∈ R . We then have | a [ x, x ] | ≤ a [ x, x ] + ( | m | − m ) k x k for all x ∈ Dom[ a ]. With ˜ a := a + | m | − m , this gives | v [ x, x ] | ≤ ˜ a k x k + b a [ x, x ] ≤ ˜ a k x k + b b s [ x, x ] + b | s || v [ x, x ] | and, hence, | v [ x, x ] | ≤ ˜ a − b | s | k x k + b − b | s | b s [ x, x ] for all x ∈ Dom[ a ] = Dom[ b s ]. Since b t = b s + ( t − s ) v , we thus obtain − ˜ a | t − s | − b | s | + (cid:16) − b | t − s | − b | s | (cid:17) b s ≤ b t ≤ ˜ a | t − s | − b | s | + (cid:16) b | t − s | − b | s | (cid:17) b s . Abbreviating λ k ( t ) := λ k ( B t | Ran E Bt ((0 , ∞ )) ), Theorem 1.5 then implies that − ˜ a | t − s | − b | s | + (cid:16) − b | t − s | − b | s | (cid:17) λ k ( s ) ≤ λ k ( t ) ≤ ˜ a | t − s | − b | s | + (cid:16) b | t − s | − b | s | (cid:17) λ k ( s )and, therefore,(2.3) | λ k ( t ) − λ k ( s ) | ≤ ˜ a | t − s | − b | s | + b | t − s | − b | s | | λ k ( s ) | . This proves that t λ k ( t ) is continuous on ( − /b, /b ) and, in particular,bounded on every compact subinterval of ( − /b, /b ). In turn, it then easilyfollows from (2.3) 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.We then have | a [ x, x ] | ≤ − a [ x, x ] + ( m + | m | ) k x k for all x ∈ Dom[ a ]. With˜ a := a + m + | m | , this leads to | v [ x, x ] | ≤ ˜ a − b | s | k x k − b − b | s | b s [ x, x ]for all x ∈ Dom[ a ] = Dom[ b s ]. Analogously as above, we then eventuallyobtain again (2.3), which proves the claim in the case where A is uppersemibounded. This completes the proof. (cid:3) Remark . (1) In part (a) of Corollary 2.9, one can also give an upperbound for λ k ( B | Ran Q + ) in terms of the form bounds of v : If A is lowersemibounded with lower bound m ∈ R , then | v [ x, x ] | ≤ ( a + b | m | ) k x k + b ( a − m )[ x, x ]= ( a + b | m | − bm ) k x k + b a [ x, x ]for all x ∈ Dom[ a ], leading to λ k ( B | Ran Q + ) ≤ (1 + b ) λ k ( A | Ran P + ) + ( a + b | m | − bm )for all k ≤ dim Ran P + = dim Ran Q + . Similarly, if A is upper semiboundedwith upper bound m ∈ R , we have | v [ x, x ] | ≤ ( a + b | m | ) k x k + b ( m − a )[ x, x ]= ( a + b | m | + bm ) k x k − b a [ x, x ]for all x ∈ Dom[ a ]. If, in addition, b ≤
1, this then leads to λ k ( B | Ran Q + ) ≤ (1 − b ) λ k ( A | Ran P + ) + ( a + b | m | + bm )for all k ≤ dim Ran P + = dim Ran Q + .(2) A similar continuity result as in part (b) of Corollary 2.9 can beformulated also in the framework of Proposition 2.5: in addition to thehypotheses of Proposition 2.5, let I ⊂ ( − /b, /b ) be an interval such thatfor all t ∈ I we have 2 a | t | + b | t | ( c + d ) < d − c . Then, for all k ∈ N satisfying k ≤ dim Ran E A ([ d, ∞ )) = dim E A + tV ([(1 − b | t | ) d − a | t | , ∞ )), the mapping I ∋ t λ k (( A + tV ) | Ran E A + tV ([(1 − b | t | ) d − a | t | , ∞ )) ) N A MINIMAX PRINCIPLE IN SPECTRAL GAPS 11 is locally Lipschitz continuous. The proof is analogous to the one of part (b)of Corollary 2.9.A corresponding result can be formulated also in the framework of Theo-rem 1.3, provided that the interval I ⊂ ( − /b, /b ) is then chosen such thatfor all t ∈ I we have k E A ((0 , ∞ )) − E A + tV ((0 , ∞ )) k < h x, ( A + tV ) x i ≤ x ∈ D − . An example. The Stokes operator.
We now briefly revisit the Stokesoperator in the framework of Theorem 1.5. Here, we mainly rely on [8],but the reader is referred also to [7, Section 7], [21, Chapter 5], [5], and thereferences cited therein.Let Ω ⊂ R n , n ≥
2, be a bounded domain with C -boundary, and let ν > v ∗ ≥
0. On the Hilbert space H = H + ⊕ H − with H + = L (Ω) n and H − = L (Ω), we consider the closed, densely defined, and nonnegativeform a with Dom[ a ] := H (Ω) n ⊕ L (Ω) and a [ v ⊕ q, u ⊕ p ] := ν n X j =1 Z Ω h ∂ j v ( x ) , ∂ j u ( x ) i C n d x for u ⊕ p, v ⊕ q ∈ Dom[ a ]. Clearly, a is the form associated to the nonnegativeself-adjoint operator A := − ν ∆ ⊕ H = H + ⊕ H − with Dom( A ) := ( H (Ω) ∩ H (Ω)) n ⊕ L (Ω) and Dom( | A | / ) = Dom[ a ],where ∆ = ∆ · I C n is the vector-valued Dirichlet Laplacian on Ω. Moreover, P + := E A ((0 , ∞ )) and P − := E A (( −∞ , E A ( { } ) are the orthogonalprojections onto H + and H − , respectively. In particular, we have D + := Ran P + ∩ Dom( | A | / ) = H (Ω) n ⊕ D − := Ran P − ∩ Dom( | A | / ) = 0 ⊕ L (Ω) . Define the symmetric sesquilinear form v on H = H + ⊕ H − with domainDom[ v ] := Dom[ a ] by v [ v ⊕ q, u ⊕ p ] := − v ∗ h div v, p i L (Ω) − v ∗ h q, div u i L (Ω) for u ⊕ p, v ⊕ q ∈ Dom[ a ]. One can show that ν k div u k L (Ω) ≤ a [ u ⊕ , u ⊕ u ∈ D + = H (Ω) n , see, e.g., [21, Proof of Theorem 5.12]. UsingYoung’s inequality, this then implies that v is infinitesimally form boundedwith respect to a , see [21, Remark 5.1.3]; cf. also [8, Section 2]. Indeed, for ε > f = u ⊕ p ∈ Dom[ a ] we obtain(2.4) | v [ f, f ] | ≤ v ∗ |h p, div u i L (Ω) | ≤ v ∗ k p k L (Ω) k div u k L (Ω) ≤ εν k div u k L (Ω) + ε − ν − v ∗ k p k L (Ω) ≤ ε a [ u ⊕ , u ⊕
0] + ε − ν − v ∗ k f k H = ε a [ f, f ] + ε − ν − v ∗ k f k H . Thus, by the well-known KLMN theorem, the form b S := a + v withDom[ b S ] = Dom[ a ] = Dom( | A | / ) is associated to a unique lower semi-bounded self-adjoint operator B S on H with Dom( | B S | / ) = Dom( | A | / ), the so-called Stokes operator . It is a self-adjoint extension of the (non-closed)upper dominant block operator matrix (cid:18) − ν ∆ v ∗ grad − v ∗ div 0 (cid:19) defined on ( H (Ω) ∩ H (Ω)) n ⊕ H (Ω). In fact, the closure of the latter isa self-adjoint operator, see [5, Theorems 3.7 and 3.9], which yields anothercharacterization of the Stokes operator B S .By rescaling, one obtains from [5, Theorem 3.15] that the essential spec-trum of B S is given by spec ess ( B S ) = n − v ∗ ν , − v ∗ ν o , see [8, Remark 2.2]. In particular, the essential spectrum of B S is purelynegative. In turn, the positive spectrum of B S , that is, spec( B S ) ∩ (0 , ∞ ),is discrete [8, Theorem 2.1 (i)].The above shows that the hypotheses of Theorem 1.5 are satisfied inthis situation, so that we obtain from Theorem 1.5 and Corollary 2.9 thefollowing result. Proposition 2.11.
Let B S be the Stokes operator as above. Then, the posi-tive spectrum of B S , spec( B S ) ∩ (0 , ∞ ) , is discrete, and the positive eigenval-ues λ k ( B S | Ran E BS ((0 , ∞ )) ) , k ∈ N , of B S , enumerated in nondecreasing orderand counting multiplicities, admit the representation λ k ( B S | Ran E BS ((0 , ∞ )) ) = inf M + ⊂ H (Ω) n dim M + = k sup u ⊕ p ∈ M + ⊕ L (Ω) k u k L n + k p k L =1 b S [ u ⊕ p, u ⊕ p ] . The latter depend locally Lipschitz continuously on ν and v ∗ and satisfy thetwo-sided estimate νλ k ( − ∆ ) ≤ λ k ( B S | Ran E BS ((0 , ∞ )) ) ≤ νλ k ( − ∆ ) + v ∗ ν . Proof.
In view the above considerations, the representation of the eigenval-ues, the continuity statement, and the lower bound on the eigenvalues followfrom Theorem 1.5 and Corollary 2.9. It remains to show the upper bound onthe eigenvalues. To this end, let M + ⊂ H (Ω) n with dim M + = k ∈ N , andlet f = u ⊕ p ∈ H (Ω) n ⊕ L (Ω) be a normalized vector with u = 0. Then, µ := a [ u ⊕ , u ⊕ / k u k L (Ω) n = a [ f, f ] / k u k L (Ω) n is positive and satisfies(2.5) µ ≤ sup v ∈ M + k v k L n =1 a [ v ⊕ , v ⊕ ν k div u k L (Ω) µ = k u k L (Ω) n ν k div u k L (Ω) a [ u ⊕ , u ⊕ ≤ k u k L (Ω) n ≤ . N A MINIMAX PRINCIPLE IN SPECTRAL GAPS 13
Similarly as in (2.4), we now obtain by means of Young’s inequality that | v [ f, f ] | ≤ v ∗ k p k L (Ω) k div u k L (Ω) ≤ µ k p k L (Ω) + v ∗ k div u k L (Ω) µ ≤ µ k p k L (Ω) + v ∗ ν . Since a [ f, f ] = µ k u k L (Ω) n , this gives(2.6) b S [ f, f ] ≤ a [ f, f ] + µ k p k L (Ω) + v ∗ ν = µ + v ∗ ν . In light of b S [0 ⊕ p, ⊕ p ] = a [0 ⊕ p, ⊕ p ] = 0, we conclude from (2.5)and (2.6) thatsup u ⊕ p ∈ M + ⊕ L (Ω) k u k L + k p k L =1 b S [ u ⊕ p, u ⊕ p ] ≤ sup v ∈ M + k v k L n =1 a [ v ⊕ , v ⊕
0] + v ∗ ν , and taking the infimum over subspaces M + ⊂ H (Ω) n with dim M + = k proves the upper bound. This completes the proof. (cid:3) Remark . (1) Choosing ε = 1 in (2.4), the upper bound from Re-mark 2.10 (1) reads λ k ( B S | Ran E BS ((0 , ∞ )) ) ≤ νλ k ( − ∆ ) + v ∗ ν for all k ∈ N , while the choice ε = v ∗ in (2.4) leads to λ k ( B S | Ran E BS ((0 , ∞ )) ) ≤ (1 + v ∗ ) νλ k ( − ∆ ) + v ∗ ν for all k ∈ N .(2) For the particular case of k = 1, a similar upper bound has beenestablished in the proof of [8, Theorem 2.1 (i)]: νλ ( − ∆) ≤ λ ( B S | Ran E BS ((0 , ∞ )) ) ≤ νλ ( − ∆) + v ∗ k div u k L (Ω) , where u ∈ ( H (Ω) ∩ H (Ω)) n is a normalized eigenfunction for − ∆ corre-sponding to the first positive eigenvalue λ ( − ∆ ) = λ ( − ∆).3. 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 [9] and part (b) of which is its naturaladaptation to the operator framework; cf. also [19, Proposition A.3].
Proposition 3.1 (cf. [9, Theorem 1], [19, Proposition A.3]) . Assume Hy-pothesis 1.1. (a)
If we have
Dom( | B | / ) = Dom( | A | / ) , b [ x, x ] ≤ for all x ∈ D − ,and Ran( P + Q + | D + ) ⊃ D + , then λ k ( B | Ran Q + ) = inf M + ⊂ D + dim( M + )= k sup x ∈ M + ⊕ D − k x k =1 b [ x, x ] for all k ∈ N with k ≤ dim Ran P + . (b) If we have
Dom( B ) = Dom( A ) , h x, Bx i ≤ for all x ∈ D − , and Ran( P + Q + | D + ) ⊃ D + , then λ k ( B | Ran Q + ) = inf M + ⊂D + dim( M + )= k sup x ∈ M + ⊕D − k x k =1 h x, Bx i for all k ∈ N with k ≤ dim Ran P + .Proof. For part (a), we first recall that the spectral projections P + and Q + map D := Dom( | A | / ) = Dom( | B | / ) into itself, so that P + mapsRan Q + ∩ D into D + . Next, we observe that under the hypotheses of part (a)the restriction(3.1) P + | Ran Q + ∩ D : Ran Q + ∩ D → D + is bijective. Indeed, its surjectivity follows directly from the hypothesisRan( P + Q + | D + ) ⊃ D + . For the injectivity, we follow Step 2 of the proofof [9, Theorem 1]: assume to the contrary that P + x = 0 for some non-zero x ∈ Ran Q + ∩ D . Then, on the one hand we have b [ x, x ] >
0, and on theother hand x ∈ Ran P − , that is, x ∈ D − . The latter gives b [ x, x ] ≤ D by Dom( A ) = Dom( B )and D ± by D ± , part (b) can be proved in the same manner. (cid:3) Remark . The above proposition is tailored towards spectral gaps aroundzero, but by a spectral shift we can of course handle also spectral gaps aroundany point γ ∈ R . Indeed, we have E A − γ ((0 , ∞ )) = E A (( γ, ∞ )) for γ ∈ R andanalogously for B . Moreover, the form associated to the operator B − γ isknown to agree with the form b − γ . The latter can be seen for instance withan analogous reasoning as in [20, Proposition 10.5 (a)]; cf. also Lemma A.6in Appendix A below. Remark . (1) The hypothesis b [ x, x ] ≤ x ∈ D − in part (a)of Proposition 3.1 is used not only to verify the injectivity of the restric-tion (3.1) but is also a crucial ingredient in the cited Step 1 of the proofof [9, Theorem 1]. The same applies for the hypothesis h x, Bx i ≤ x ∈ D − in part (b).(2) Since P + and Q + are spectral projections for the respective operators,we always have Ran( P + Q + | D + ) ⊂ D + and Ran( P + Q + | D + ) ⊂ D + . In thisrespect, the condition Ran( P + Q + | D + ) ⊃ D + in part (a) of Proposition 3.1actually means that the restriction P + Q + | D + : D + → D + is surjective. Thishas not been formulated explicitly in the statement of [9, Theorem 1] buthas instead been guaranteed by the stronger condition(3.2) k ( | A | + I ) / P + Q − ( | A | + I ) − / k < . Since D + = Ran(( | A | + I ) − / | Ran P + ), a standard Neumann series argumentthen even gives bijectivity of the restriction P + Q + | D + , see Step 2 of theproof of [9, Theorem 1]. In this reasoning, the operators ( | A | + I ) ± / canbe replaced by ( | A | + αI ) ± / for any α >
0; if | A | has a bounded inverse,also α = 0 can be considered here. N A MINIMAX PRINCIPLE IN SPECTRAL GAPS 15
Of course, the above reasoning also applies in the situation of part (b) ofProposition 3.1, but with ( | A | + αI ) ± / replaced by ( | A | + αI ) ± .Condition (3.2) has been considered in [9] in the setting of Dirac operators,but it seems to be hard to verify it on an abstract level. This approach istherefore not pursued further here.In the context of our main theorems, the restriction P + Q + | Ran P + , un-derstood as an endomorphism of Ran P + , will always be bijective, cf. Re-mark 3.5 (1) below. It turns out that then the hypotheses of part (b) inProposition 3.1 imply those of part (a), in which case both representationsfor the minimax values in Proposition 3.1 are valid. More precisely, we havethe following lemma, essentially based on the well-known Heinz inequality,cf. Appendix A below. Lemma 3.4.
Assume Hypothesis 1.1 with
Dom( A ) = Dom( B ) . (a) One has
Dom( | A | / ) = Dom( | B | / ) . (b) If h x, Bx i ≤ for all x ∈ D − , then b [ x, x ] ≤ for all x ∈ D − . (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 A.3 below. Alternatively, this follows by classical considera-tions regarding operator and form boundedness, see Remark A.4 below.(b). It follows from part (a) that the operator | B | / ( | A | / + I ) − isclosed and everywhere defined, hence bounded by the closed graph theorem.Thus, k| B | / x k ≤ k| B | / ( | A | / + I ) − k · k ( | A | / + I ) x k for all x ∈ Dom( | A | / ) = Dom( | B | / ). Since D − is a core for the operator | A | Ran P − | / = | A | / | Ran P − with Dom( | A | / | Ran P − ) = D − , the inequality b [ x, x ] ≤ x ∈ D − now follows from the hypothesis h x, Bx i ≤ x ∈ D − by approximation.(c). We clearly have Ran( P + Q + | D + ) = D + , D + = Dom( A | Ran P + ),and D + = Dom( | A | Ran P + | / ). Applying Corollary A.5 below with thechoices Λ = Λ = A | Ran P + and S = P + Q + | Ran P + therefore implies thatRan( P + Q + | D + ) = D + , which proves the claim. (cid:3) Remark . (1) In light of the identity P + Q + = P + − P + Q − , the bijectivityof P + Q + | Ran P + : Ran P + → Ran P + can be guaranteed, for instance, by thecondition k P + Q − k < P + − Q + = P + Q − − P − Q + and, in particular, k P + Q − k ≤ k P + − Q + k ,this condition holds if the stronger inequality k P + − Q + k < U with Q + U = U P + , see,e.g., [11, Theorem I.6.32], which implies that dim Ran P + = dim Ran Q + . Itis this situation we encounter in Theorems 1.3–1.5.(2) In the case where B is an infinitesimal operator perturbation of A , theinequality k P + Q − k < P + Q + | D + ) ⊃ D + , see thefollowing section; the particular case where B is a bounded perturbation of A has previously been considered in [19, Lemma A.6]. For more general, notnecessarily 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 k P + Q − k < P + Q + | D + ) ⊃ D + , which is essentially whatis needed to deduce Theorem 1.2 from Proposition 3.1 and Lemma 3.4. Themain technique used to accomplish this can in fact be formulated in a muchmore general framework:Recall that for a closed operator Λ on a Banach space with norm k · k , itsdomain Dom(Λ) can be equipped with the graph norm k x k Λ := k x k + k Λ x k , x ∈ Dom(Λ) , which makes (Dom(Λ) , k · k Λ ) a Banach space. Also recall that a linearoperator K with Dom( K ) ⊃ Dom(Λ) is called Λ -bounded with Λ -bound β ∗ ≥ β > β ∗ there is an α ≥ k Kx k ≤ α k x k + β k Λ x k for all x ∈ Dom(Λ) . The following lemma extends part (a) of [19, Proposition A.5], takenfrom Lemma 3.9 in the author’s Ph.D. thesis [23], to relatively boundedcommutators.
Lemma 4.1.
Let Λ be a closed operator on a Banach space, K be Λ -boundedwith Λ -bound β ∗ ≥ , and let S be bounded with Ran( S | Dom(Λ) ) ⊂ Dom(Λ) and Λ Sx − S Λ x = Kx for all x ∈ Dom(Λ) . Then, S | Dom(Λ) is bounded on
Dom(Λ) with respect to the graph norm for Λ , and the corresponding spectral radius satisfies r Λ ( S ) := lim k →∞ k ( S | Dom(Λ) ) k k /k Λ ≤ k S k + β ∗ . Proof.
Only small modifications to the reasoning from [23, Lemma 3.9], [19,Proposition A.5] are necessary. For the sake of completeness, we reproducethe full argument here:Let β > β ∗ and α ≥ x ∈ Dom(Λ) onehas k Λ Sx k ≤ k S kk Λ x k + k Kx k ≤ ( k S k + β ) k Λ x k + α k x k , so that k Sx k Λ = k Sx k + k Λ Sx k ≤ (cid:0) k S k + β (cid:1) k x k Λ + α k x k . In particular, S | Dom(Λ) is bounded with respect to the graph norm k · k Λ with k S k Λ ≤ k S k + β + α .Now, a straightforward induction yields k S k x k Λ ≤ (cid:0) k S k + β (cid:1) k k x k Λ + kα (cid:0) k S k + β (cid:1) k − k x k , x ∈ Dom(Λ) , for k ∈ N . Hence, k ( S | Dom(Λ) ) k k Λ ≤ ( k S k + β ) k + kα ( k S k + β ) k − , so that r Λ ( S ) = lim k →∞ k ( S | Dom(Λ) ) k k /k Λ ≤ lim k →∞ (cid:0) ( k S k + β ) k + kα ( k S k + β ) k − (cid:1) /k = k S k + β. Since β > β ∗ was chosen arbitrarily, this proves the claim. (cid:3) We are now in position to prove Theorem 1.2.
N A MINIMAX PRINCIPLE IN SPECTRAL GAPS 17
Proof of Theorem 1.2.
We mainly follow the line of reasoning in the proofof [19, Lemma A.6]. Only a few additional considerations are necessary inorder to accommodate unbounded perturbations V by means of Lemma 4.1.For convenience of the reader, we nevertheless reproduce the whole argumenthere.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 k S k ≤ k P + Q − k <
1, so that T is bijective. In lightof Proposition 3.1 and Lemma 3.4, it now remains to show the inclusionRan( P + Q + | D + ) ⊃ D + , that is, Ran( T − | D + ) ⊂ D + . To this end, we rewrite T − as a Neumann series, T − = ( I Ran P + − S ) − = ∞ X k =0 S k . Clearly, S maps the domain D + = Dom( A | Ran P + ) into itself, so that theinclusion Ran( T − | D + ) ⊂ D + holds if the above series converges also withrespect to the graph norm for the closed operator Λ := A | Ran P + . This, inturn, is the case if the corresponding spectral radius r Λ ( S ) of S is smallerthan 1.For x ∈ D + ⊂ Ran P + we computeΛ Sx = AP + Q − x = P + ( A + V ) Q − x − P + V Q − x = P + Q − ( A + V ) x − P + V Q − x = S Λ x + Kx with K := ( P + Q − V − P + V Q − ) | Ran P + . We show that the operator K is Λ-bounded with Λ-bound 0. Indeed, let b >
0, and choose a ≥ k V x k ≤ a k x k + b k Ax k for all x ∈ Dom( A );recall that V is infinitesimal with respect to A by hypothesis. Then, k V Q − x k ≤ a k Q − x k + b k AQ − x k ≤ a k x k + b k ( A + V ) x k + b k V Q − x k , so that k V Q − x k ≤ a − b k x k + b − b (cid:0) k Ax k + k V x k (cid:1) ≤ a (1 + b )1 − b k x k + b (1 + b )1 − b k Ax k . Thus,(4.2) k Kx k ≤ k P + Q − kk V x k + k V Q − x k≤ a (cid:16) k P + Q − k + 1 + b − b (cid:17) k x k + b (cid:16) k P + Q − k + 1 + b − b (cid:17) k Λ x k for x ∈ Dom(Λ) = D + . Since b > K is Λ-bounded with Λ-bound 0. It therefore follows from Lemma 4.1 that r Λ ( S ) ≤ k S k <
1, which completes the proof. (cid:3)
Remark . (1) Estimate (4.2) suggests that also relatively bounded per-turbations V that are not necessarily infinitesimal with respect to A can beconsidered here. In fact, if b ∗ ∈ [0 ,
1) is the A -bound of V , then by (4.2)and Lemma 4.1 we have r Λ ( S ) ≤ k P + Q − k + b ∗ (cid:16) k P + Q − k + 1 + b ∗ − b ∗ (cid:17) , and the right-hand side of the latter is smaller than 1 if and only if k P + Q − k < − b ∗ − b ∗ − b ∗ . This is a reasonable condition on the norm k P + Q − k only for b ∗ < √ − A + V )-boundof V : If for some ˜ b ∈ [0 ,
1) and ˜ a ≥ k V x k ≤ ˜ a k x k +˜ b k ( A + V ) x k forall x ∈ Dom( A ) = Dom( A + V ), then standard arguments as in the aboveproof of Theorem 1.2 show that k V x k ≤ ˜ a − ˜ b k x k + ˜ b − ˜ b k Ax k and, in turn, k V Q − x k ≤ ˜ a k x k + ˜ b k ( A + V ) x k ≤ ˜ a k x k + ˜ b k Ax k + ˜ b k V x k≤ ˜ a (cid:16) b − ˜ b (cid:17) k x k + ˜ b (cid:16) b − ˜ b (cid:17) k Ax k = ˜ a − ˜ b k x k + ˜ b − ˜ b k Ax k for all x ∈ Dom( A ). Plugging these into (4.2) gives k Kx k ≤ k P + Q − kk V x k + k V Q − x k≤ (1 + k P + Q − k ) (cid:16) ˜ a − ˜ b k x k + ˜ b − ˜ b k Λ x k (cid:17) for all x ∈ Dom(Λ) = D + , which eventually leads to r Λ ( S ) ≤ k P + Q − k + ˜ b (1 + k P + Q − k )1 − ˜ b = k P + Q − k + ˜ b − ˜ b . The right-hand side of the latter is smaller than 1 if and only if k P + Q − k < − b. This is a reasonable condition on k P + Q − k only for ˜ b < / The block diagonalization approach
In this section, we discuss an approach to verify the hypotheses of Propo-sition 3.1 and Lemma 3.4 which relies on techniques previously discussed inthe context of block diagonalizations of operators and forms, for instancein [16] and [7], respectively; see also Remark 5.4 below.
N A MINIMAX PRINCIPLE IN SPECTRAL GAPS 19
Recall that for the two orthogonal projections P + and Q + from Hypoth-esis 1.1 the inequality k P + − Q + k < Q + can berepresented as(5.1) Ran Q + = { f ⊕ Xf | f ∈ Ran P + } with some bounded linear operator X : Ran P + → Ran P − ; in this case, onehas(5.2) k P + − Q + k = k X k q k X k , see, e.g., [12, Corollary 3.4 (i)]. The orthogonal projection Q + can then berepresented as the 2 × Q + = (cid:18) ( I Ran P + + X ∗ X ) − ( I Ran P + + X ∗ X ) − X ∗ X ( I Ran P + + X ∗ X ) − X ( I Ran P + + X ∗ X ) − X ∗ (cid:19) = (cid:18) ( I Ran P + + X ∗ X ) − X ∗ ( I Ran P − + XX ∗ ) − ( I Ran P − + XX ∗ ) − X XX ∗ ( I Ran P − + XX ∗ ) − (cid:19) with respect to Ran P + ⊕ Ran P − , see, e.g., [12, Remark 3.6]. In particular,we have(5.4) P + Q + | Ran P + = ( I Ran P + + X ∗ X ) − , which is in fact the starting point for the current approach. With regard tothe desired relations Ran( P + Q + | D + ) ⊃ D + and Ran( P + Q + | D + ) ⊃ D + , weneed 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 × Y = (cid:18) − X ∗ X (cid:19) with respect to Ran P + ⊕ Ran P − . Then, the operators I ± Y are bijectivewith(5.6) ( I − Y )( I + Y ) = (cid:18) I Ran P + + X ∗ X I Ran P − + XX ∗ (cid:19) . The following lemma is extracted from various sources. We comment onthis afterwards in Remark 5.2 below.
Lemma 5.1.
Suppose that the projections P + and Q + from Hypothesis 1.1satisfy k P + − Q + k < , and let the operators X and Y be as in (5.1) and (5.5) , respectively. Moreover, let C be an invariant subspace for P + and Q + such that C = ( C ∩
Ran P + ) ⊕ ( C ∩
Ran P − ) =: C + ⊕ C − .Then, the following are equivalent: (i) I Ran P + + X ∗ X maps C + into itself; (ii) I Ran P − + XX ∗ maps C − into itself; (iii) Y maps C into itself; (iv) ( I + Y ) maps C into itself; (v) ( I − Y ) maps C into itself. 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 ) − 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( I Ran P − + XX ∗ ) g = g + XX ∗ g = g + X ( I Ran P + + X ∗ X ) − h = g + ( P − Q + | Ran P + ) h ∈ C − . As a byproduct, we have also shown that X ∗ maps C − into C + .(ii) ⇒ (i). Using the identities ( I Ran P − + XX ∗ ) − X = P − Q + | Ran P + and X ∗ ( I Ran P − + XX ∗ ) − = P + Q + | Ran P − taken from the second representationin (5.3), the proof is completely analogous to the implication (i) ⇒ (ii). Inparticular, 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.(iii) ⇔ (iv),(v). This is clear.(iv),(v) ⇒ (i),(ii). This follows immediately from identity (5.6). (cid:3) Remark . The proof of the equivalence (i) ⇔ (ii) and the one of the impli-cation (i),(ii) ⇒ (iii) in Lemma 5.1 are extracted from the proof of [7, Theo-rem 5.1]; see also [21, Theorem 6.3.1 and Lemma 6.3.3].The equivalence (iv) ⇔ (v) can alternatively be directly obtained from theidentity (cid:18) I Ran P + − I Ran P − (cid:19) ( I + Y ) (cid:18) I Ran P + − I Ran P − (cid:19) = I − Y. Such an argument has been used in the proof of [16, Proposition 3.3].The implication (iv),(v) ⇒ (i) can essentially be found in the proof of [7,Theorem 5.1] and [21, Remark 6.3.2].Below, we apply Lemma 5.1 with C = Dom( A ) = Dom( B ) = D + ⊕ D − or C = Dom( | A | / ) = Dom( | B | / ) = D + ⊕ D − , depending on the situation.The easiest case is encountered in Theorem 1.3: Proof of Theorem 1.3.
Let Dom( | A | / ) = Dom( | B | / ) and b [ x, x ] ≤ x ∈ D − . We then have D − = Ran P − if A is bounded from belowand D + = Ran P + if A is bounded from above. Hence, item (ii) or (i) inLemma 5.1, respectively, with C = D + ⊕ D − is automatically satisfied. Inany case, we have by Lemma 5.1 that I Ran P + + X ∗ X maps D + into D + ,which by identity (5.4) means that Ran( P + Q + | D + ) ⊃ D + . The represen-tation (1.1) now follows from Proposition 3.1 (a) and Remark 3.5 (1). Ifeven Dom( A ) = Dom( B ) and h x, Bx i ≤ x ∈ D − , we use the samereasoning as above with D + and D − replaced by D + and D − , respectively,and obtain representation (1.2) from Proposition 3.1 (b) and Remark 3.5 (1).The representation (1.1) is then still valid by Lemma 3.4 and the first partof the proof. (cid:3) While certain conditions for Proposition 3.1 and Lemma 3.4 are part ofthe hypotheses of Theorems 1.2 and 1.3, in the situations of Theorems 1.4and 1.5 these need to be verified explicitly from the specific hypotheses at
N A MINIMAX PRINCIPLE IN SPECTRAL GAPS 21 hand. Here, we rely on previous considerations on block diagonalizationsfor block operator matrices and forms. In case of Theorem 1.4, the crucialingredient is presented in the following result, extracted from [16]. An earlierresult in this direction is commented on in Remark 5.4 (2) below.
Proposition 5.3 (see [16, Theorem 6.1]) . In the situation of Theorem 1.4one has k P + − Q + k ≤ √ / < , and the operator identity (5.7) ( I − Y )( A + V )( I − Y ) − = A − Y V holds with Y as in (5.5) .Proof. Set V off := V | Dom( A ) , so that we have B = A + V = A + V off aswell as A − Y V = A − Y V off . Clearly, the hypotheses on V ensure that V off is A -bounded with A -bound b ∗ < P + ⊕ Ran P − . By [16, Lemma 6.3] we now haveKer( A + V off ) ⊂ Ker A ⊂ Ran P − . In light of (5.2), the claim therefore is just an instance of [16, Theorem 6.1]. (cid:3)
Remark . (1) Let A ± := A | Ran P ± be the parts of A associated with thesubspaces Ran P ± , and write V | Dom( A ) = (cid:18) WW ∗ (cid:19) , where W : Ran P − ⊃ D − → Ran P + is given by W x := P + V x , x ∈ D − .Then, A − Y V = (cid:18) A + − X ∗ W ∗ A − + XW (cid:19) . In this sense, identity (5.7) can be viewed as a block diagonalization of theoperator A + V . For a more detailed discussion of block diagonalizationsand operator Riccati equations in the operator setting, the reader is referredto [16] and the references cited therein.(2) In the particular case where 0 belongs to the resolvent set of A , theconclusion of Proposition 5.3 can be inferred also from [26, Theorems 2.7.21and 2.8.5]. Proof of Theorem 1.4.
For x ∈ D − , we have h x, V x i = h P − x, V P − x i = h x, P − V P − x i = 0and, thus, h x, ( A + V ) x i = h x, Ax i ≤ . Moreover, by Proposition 5.3 the inequality k P + − Q + k < Y be as in (5.5). Since Dom( A + V ) = Dom( A ) = Dom( A − Y V ), it thenfollows from identity (5.7) that I − Y maps C := Dom( A ) = D + ⊕ D − intoitself. In turn, Lemma 5.1 implies that I Ran P + + X ∗ X maps D + into itself,which by identity (5.4) means that Ran( P + Q + | D + ) ⊃ D + . The claim nowfollows from Proposition 3.1, Lemma 3.4, and Remark 3.5 (1). (cid:3) To the best of the author’s knowledge, no direct analogue of Proposi-tion 5.3 is known so far in the setting of form rather than operator pertur-bations. Although the inequality k P + − Q + k ≤ √ / I ± Y connected with a corresponding diag-onalization related to (5.7) are much harder to verify. The situation is evenmore subtle there since also the domain equality Dom( | A | / ) = Dom( | B | / )needs careful treatment. The latter is conjectured to hold in a general off-diagonal form perturbation framework [6, Remark 2.7]. Some characteriza-tions have been discussed in [22, Theorem 3.8], but they all are hard to verifyin a general abstract setting. A compromise in this direction is to requirethat the form b is semibounded, see [22, Lemma 3.9] and [7, Lemma 2.7],which forces the diagonal form a to be semibounded as well, see below. Asin the proof of Theorem 1.3 above, this simplifies the situation immensely: Proof of Theorem 1.5.
Set v off := v | Dom[ a ] , so that b = a + v = a + v off . For x ∈ D − = Ran P − ∩ Dom[ a ] we have v off [ x, x ] = v [ P − x, P − x ] = 0and, thus, b [ x, x ] = a [ x, x ] ≤ . In the same way, we see that b [ x, x ] = a [ x, x ] for x ∈ D + , which by theidentity a [ x, x ] = a [ P + x, P + x ] + a [ P − x, P − x ] for all x ∈ Dom[ a ] impliesthat along with b the form a is semibounded as well; cf. also the proofof [7, Lemma 2.7]. In particular, we have D − = Ran P − if a is boundedfrom below and D + = Ran P + if a is bounded from above.Let m ∈ R be the lower (resp. upper) bound of a . We then have | ( a − m )[ x, x ] | = k| A − m | / x k ≤ k| A − m | / ( | A | / + I ) − kk ( | A | / + I ) x k for all x ∈ Dom[ a ], where | A − m | / ( | A | / + I ) − is closed and everywheredefined, hence bounded by the closed graph theorem. From this and thehypothesis on v we see that | v [ x, x ] | ≤ β (cid:0) k| A | / x k + k x k (cid:1) for some β ≥ x ∈ Dom[ a ], which means that b = a + v is asemibounded saddle-point form in the sense of [7, Section 2].Since Dom( | B | / ) = Dom[ b ] = Dom[ a ] = Dom( | A | / ) by hypothesis, C = Dom[ a ] is invariant for both P + and Q + . Moreover, by [7, Theorem 3.3](cf. also [22, Theorem 2.13]) we haveKer B ⊂ Ker A ⊂ Ran P − and k P + − Q + k ≤ √ / <
1. Taking into account that a is semibounded asobserved above, Lemma 5.1 with C = Dom[ a ] = D + ⊕ D − and identity (5.4)then imply as in the proof of Theorem 1.3 that Ran( P + Q + | D + ) ⊃ D + . Theclaim now follows from Proposition 3.1 (a) and Remark 3.5 (1). (cid:3) N A MINIMAX PRINCIPLE IN SPECTRAL GAPS 23
Appendix A. Heinz inequality
In this appendix we discuss some consequences of the well-known Heinzinequality. These consequences or particular cases thereof are used at variousspots of the main part of the paper, but they may also be of independentinterest. Although probably folklore, in lack of a suitable reference they arenevertheless presented here in full detail.Throughout this appendix, we denote the norm associated with the innerproduct of a Hilbert space H by k · k H .The following variant of the Heinz inequality is taken from [13]. Proposition A.1 ([13, Theorem I.7.1]) . Let Λ and Λ be strictly positiveself-adjoint operators on Hilbert spaces H and H , respectively. Moreover,let S : H → H be a bounded operator mapping Dom(Λ ) into Dom(Λ ) ,and suppose that there is a constant C ≥ such that k Λ Sx k H ≤ C · k Λ x k H for all x ∈ Dom(Λ ) . Then, for all ν ∈ [0 , , the operator S maps Dom(Λ ν ) into Dom(Λ ν ) , andfor all x ∈ Dom(Λ ν ) one has k Λ ν Sx k H ≤ C ν k S k − ν H →H k Λ ν x k H . The above result admits the following extension to closed densely definedoperators between Hilbert spaces. For a generalization of Proposition A.1to maximal accretive operators, see [10].
Proposition A.2.
Let H , H , K , and K be Hilbert spaces, and let Λ : H ⊃ Dom(Λ ) → K and Λ : H ⊃ Dom(Λ ) → K be closed denselydefined operators. Moreover, let S : H → H be a bounded operator map-ping Dom(Λ ) into Dom(Λ ) , and suppose that there is a constant C ≥ such that k Λ Sx k K ≤ C · k Λ x k K for all x ∈ Dom(Λ ) . Then, for all ν ∈ [0 , , the operator S maps Dom( | Λ | ν ) into Dom( | Λ | ν ) .Proof. Recall that k Λ j y k K j = k| Λ j | y k H j for all y ∈ Dom(Λ j ) = Dom( | Λ j | ), j = 1 ,
2. Moreover, the operator S maps Dom( | Λ | + I H ) = Dom(Λ ) intoDom( | Λ | + I H ) = Dom(Λ ) by hypothesis. We estimate k ( | Λ | + I H ) Sx k H ≤ k Λ Sx k K + k Sx k H ≤ C k Λ x k K + k Sx k H ≤ e C k ( | Λ | + I H ) x k H for all x ∈ Dom(Λ ) with e C := C k Λ ( | Λ | + I H ) − k H →K + k S ( | Λ | + I H ) − k H →H . Here, we have taken into account that Λ ( | Λ | + I H ) − is a closed andeverywhere defined operator from H to K , hence bounded by the closedgraph theorem.Applying Proposition A.1 now yields that S maps Dom(( | Λ | + I H ) ν )into Dom(( | Λ | + I H ) ν ) for all ν ∈ [0 , | Λ j | + I H j ) ν ) = Dom( | Λ j | ν ) for j ∈ { , } ,which completes the proof. (cid:3) We now obtain several easy corollaries.
Corollary A.3 (cf. [21, Corollary 2.1.3]) . Let H , K , and K be Hilbertspaces, and let Λ : H ⊃
Dom(Λ ) → K and Λ : H ⊃
Dom(Λ ) → K beclosed densely defined operators.If Dom(Λ ) ⊂ Dom(Λ ) , then Dom( | Λ | ν ) ⊂ Dom( | Λ | ν ) for all ν ∈ [0 , .If even Dom(Λ ) = Dom(Λ ) , then also Dom( | Λ | ν ) = Dom( | Λ | ν ) for all ν ∈ [0 , .Proof. Suppose that Dom(Λ ) ⊂ Dom(Λ ). Since Dom( | Λ | ) = Dom(Λ ),we have as in the proof of the preceding proposition that Λ ( | Λ | + I H ) − isa closed everywhere defined, hence bounded, operator from H to K . Thus, k Λ x k K ≤ k Λ ( | Λ | + I H ) − k H→K · k ( | Λ | + I H ) x k H for all x ∈ Dom(Λ ), and applying Proposition A.2 with S = I H yields thatDom( | Λ | ν ) = Dom(( | Λ | + I H ) ν ) ⊂ Dom( | Λ | ν ) . If also Dom(Λ ) ⊃ Dom(Λ ), the above with switched roles of Λ and Λ yields that also Dom( | Λ | ν ) ⊂ Dom( | Λ | ν ), which completes the proof. (cid:3) Remark
A.4 . For the particular case of ν = 1 /
2, Corollary A.3 can alter-natively also be proved with classical considerations regarding operator andform boundedness:If Dom(Λ ) ⊂ Dom(Λ ), then also Dom( | Λ | ) ⊂ Dom( | Λ | ), so that | Λ | is relatively operator bounded with respect to | Λ | , see, e.g., [11, Re-mark IV.1.5]. In turn, by [11, Theorem VI.1.38], | Λ | is also form boundedwith respect to | Λ | , which extends to the closure of the forms. The latterincludes that Dom( | Λ | / ) ⊂ Dom( | Λ | / ). Corollary A.5.
Let Λ and Λ be as in Proposition A.2, and suppose that S : H → H is bounded and bijective with Ran( S | Dom(Λ ) ) = Dom(Λ ) .Then, one has Ran( S | Dom( | Λ | ν ) ) = Dom( | Λ | ν ) for all ν ∈ [0 , .Proof. Consider the closed densely defined operator Λ := Λ S − with do-main Dom(Λ ) = Ran( S | Dom(Λ ) ) = Dom(Λ ). By definition, S mapsDom(Λ ) onto Dom(Λ ). Moreover, we have Λ Sx = Λ x and, in particular, k Λ Sx k K = k Λ x k K for all x ∈ Dom(Λ ). Proposition A.2 now implies that S maps Dom( | Λ | ν )into Dom( | Λ | ν ). Since Dom( | Λ | ν ) = Dom( | Λ | ν ) for all ν ∈ [0 ,
1] in lightof Corollary A.3, this proves the inclusion Ran( S | Dom( | Λ | ν ) ) ⊂ Dom( | Λ | ν ).Since S is bijective and S − maps Dom(Λ ) onto Dom(Λ ) by hypothesis,one verifies in an analogous way that S − maps Dom( | Λ | ν ) into Dom( | Λ | ν ).This shows the converse inclusion and, hence, completes the proof. (cid:3) The last corollary discussed here is related to the question whether anoperator sum agrees with the operator associated to the sum of the corre-sponding forms, at least in the semibounded setting. Part (b) of this corol-lary can in some sense also be regarded as an extension of [21, Lemma 2.2.7]to not necessarily off-diagonal perturbations.
N A MINIMAX PRINCIPLE IN SPECTRAL GAPS 25
Corollary A.6 (cf. [21, Lemma 2.2.7]) . Let Λ be a self-adjoint operator ona Hilbert space with inner product h· , ·i , and let K be an operator on thesame Hilbert space. (a) If K is symmetric with Dom( K ) ⊃ Dom( | Λ | / ) , then the operatorsum Λ + K defines a self-adjoint operator with h| Λ + K | / x, sign(Λ + K ) | Λ + K | / y i = h| Λ | / x, sign(Λ) | Λ | / y i + h x, Ky i for all x, y ∈ Dom( | Λ | / ) = Dom( | Λ + K | / ) . (b) If K is self-adjoint and Λ -bounded with Λ -bound smaller than , then Λ + K is self-adjoint with h| Λ + K | / x, sign(Λ + K ) | Λ + K | / y i = h| Λ | / x, sign(Λ) | Λ | / y i + h| K | / x, sign( K ) | K | / y i for all x, y ∈ Dom( | Λ | / ) = Dom( | Λ + K | / ) .Proof. (a). Since Dom( | Λ | / ) ⊂ Dom( K ) by hypothesis, Corollary 2.1.20in [26] yields that K is operator infinitesimal with respect to Λ. In partic-ular, the operator sum Λ + K is self-adjoint on Dom(Λ + K ) = Dom(Λ)by the well-known Kato-Rellich theorem. In turn, Corollary A.3 impliesthat Dom( | Λ + K | / ) = Dom( | Λ | / ). In particular, | Λ + K | / is rela-tively operator bounded with respect to | Λ | / , see, e.g., [11, Remark IV.1.5and Section V.3.3]. Since the symmetric operator K satisfies the inclusionDom( | Λ | / ) ⊂ Dom( K ) by hypothesis, K is likewise relatively boundedwith respect to | Λ | / .Now, for x, y ∈ Dom(Λ) = Dom(Λ+ K ), both sides of the claimed identityclearly agree. For x, y ∈ Dom( | Λ | / ) = Dom( | Λ + K | / ), this identity thenfollows by approximation, taking into account that Dom(Λ) is an operatorcore for | Λ | / .(b). The operator sum Λ + K with Dom(Λ + K ) = Dom(Λ) is self-adjoint by the well-known Kato-Rellich theorem, and Corollary A.3 impliesthat Dom( | Λ + K | / ) = Dom( | Λ | / ) and Dom( | Λ | / ) ⊂ Dom( | K | / ). Inturn, as in part (a), both | Λ + K | / and | K | / are relatively bounded withrespect to | Λ | / . The claimed identity now follows just as in part (a) byapproximation upon observing that it certainly holds for x, y ∈ Dom(Λ). (cid:3)
Acknowledgements
The author is grateful to Ivan Veseli´c, Matthias T¨aufer and StephanSchmitz for fruitful and inspiring discussions. He is especially indebted toStephan Schmitz for also commenting on an earlier version of this manu-script.
References [1] V. M. Adamjan, H. Langer,
Spectral properties of a class of rational operator valuedfunctions , J. Operator Theory (1995), 259–277.[2] C. Davis, W. M. Kahan, The rotation of eigenvectors by a perturbation. III , SIAMJ. Numer. Anal. (1970), 1–46.[3] J. Dolbeault, M. J. Esteban, E. S´er´e, On the eigenvalues of operators with gaps.Application to Dirac Operators , J. Funct. Anal. (2000), 208–226. [4] J. Dolbeault, M. J. Esteban, E. S´er´e,
General results on the eigenvalues of opera-tors with gaps, arising from both ends of the gaps. Application to Dirac operators ,J. Eur. Math. Soc. (JEMS) (2006), 243–251.[5] M. Faierman, R. J. Fries, R. Mennicken, M. M¨oller, On the essential spectrum of thelinearized Navier-Stokes operator , Integral Equations Operator Theory (2000),9–27.[6] L. Grubiˇsi´c, V. Kostrykin, K. A. Makarov, K. Veseli´c, The
Tan 2Θ
Theorem forindefinite quadratic forms , J. Spectr. Theory (2013), 83–100.[7] L. Grubiˇsi´c, V. Kostrykin, K. A. Makarov, S. Schmitz, K. Veseli´c, Diagonalizationof indefinite saddle point forms , In: Analysis as a Tool in Mathematical Physics:in Memory of Boris Pavlov, Oper. Theory Adv. Appl., vol. 276, Birkh¨auser, Basel,2020, pp. 373–400.[8] L. Grubiˇsi´c, V. Kostrykin, K. A. Makarov, S. Schmitz, K. Veseli´c,
The
Tan 2Θ
Theorem in fluid dynamics , J. Spectr. Theory (2019), 1431–1457.[9] M. Griesemer, R. T. Lewis, H. Siedentop, A minimax principle for eigenvalues inspectral gaps: Dirac operators with Coulomb potentials , Doc. Math. (1999), 275–283.[10] T. Kato, A generalization of the Heinz inequality , Proc. Japan Acad. (1961),305–308.[11] T. Kato, Perturbation Theory for Linear Operators , Classics Math., Springer, Berlin,1995.[12] V. Kostrykin, K. A. Makarov, A. K. Motovilov,
Existence and uniqueness of so-lutions to the operator Riccati equation. A geometric approach , In: Advances inDifferential Equations and Mathematical Physics (Birmingham, AL, 2002), Con-temp. Math., vol. 327, Amer. Math. Soc., Providence, RI, 2003, pp. 181–198.[13] S. G. Kre˘ın,
Linear Differential Equations in Banach Space , Transl. Math. Monogr.,vol. 29, Amer. Math. Soc., Providence, RI, 1969.[14] E. H. Lieb, M. Loss,
Analysis , second edition, Grad. Stud. Math., vol. 14,Amer. Math. Soc., Providence, RI, 2001.[15] M. Langer, M. Strauss,
Triple variational principles for self-adjoint operator func-tions , J. Funct. Anal. (2016), 2019–2047.[16] K. A. Makarov, S. Schmitz, A. Seelmann,
On invariant graph subspaces , IntegralEquations Operator Theory (2016), 399–425.[17] S. Morozov, D. M¨uller, On the minimax principle for Coulomb-Dirac operators ,Math. Z. (2015), 733–747.[18] A. K. Motovilov, A. V. Selin,
Some sharp norm estimates in the subspace perturba-tion problem , Integral Equations Operator Theory (2006), 511–542.[19] I. Naki´c, M. T¨aufer, M. Tautenhahn, I. Veseli´c, Unique continuation and lifting ofspectral band edges of Schr¨odinger operators on unbounded domains , with an appen-dix by Albrecht Seelmann, to appear in J. Spectr. Theory. E-print arXiv:1804.07816[math.SP] (2018).[20] K. Schm¨udgen,
Unbounded Self-Adjoint Operators on Hilbert Space , Grad. Texts inMath., vol. 265, Springer, Dordrecht, 2012.[21] S. Schmitz,
Representation theorems for indefinite quadratic forms and applications ,Dissertation, Johannes Gutenberg-Universit¨at Mainz, 2014.[22] S. Schmitz,
Representation theorems for indefinite quadratic forms without spectralgap , Integral Equations Operator Theory (2015), 73–94.[23] A. Seelmann, Perturbation theory for spectral subspaces , Dissertation, JohannesGutenberg-Universit¨at Mainz, 2014.[24] A. Seelmann,
Semidefinite perturbations in the subspace perturbation problem , J. Op-erator Theory (2019), 321–333.[25] A. Seelmann, Unifying the treatment of indefinite and semidefinite perturbationsin the subspace perturbation problem , e-print arXiv:2006.16102 [math.SP] (2020).(submitted)[26] C. Tretter,
Spectral Theory of Block Operator Matrices and Applications , ImperialCollege Press, London, 2008.
N A MINIMAX PRINCIPLE IN SPECTRAL GAPS 27
A. Seelmann, Technische Universit¨at Dortmund, Fakult¨at f¨ur Mathematik,D-44221 Dortmund, Germany
E-mail address ::