Unifying the treatment of indefinite and semidefinite perturbations in the subspace perturbation problem
aa r X i v : . [ m a t h . SP ] J un UNIFYING THE TREATMENT OF INDEFINITE ANDSEMIDEFINITE PERTURBATIONS IN THE SUBSPACEPERTURBATION PROBLEM
ALBRECHT SEELMANN
Abstract.
The variation of spectral subspaces for linear self-adjointoperators under an additive bounded perturbation is considered. Theobjective is to estimate the norm of the difference of two spectral pro-jections associated with isolated parts of the spectrum of the perturbedand unperturbed operators. Recent results for semidefinite and general,not necessarily semidefinite, perturbations are unified to statements thatcover both types of perturbations and, at the same time, also allow forcertain perturbations that were not covered before. Introduction and main result
The present note continues the considerations on the subspace perturba-tion problem previously discussed in several recent works such as [1,5–7]; seealso the references cited therein. More specifically, the results for semidef-inite perturbations from [7] and those for general, not necessarily semidefi-nite, perturbations from [4, 6] are unified to general statements which coverboth types of perturbations and, at the same time, allow for certain pertur-bations that have not been covered before. This is achieved by decomposingthe perturbation into its nonnegative and nonpositive parts with respect toits spectral decomposition. Naturally, the corresponding results here barea great similarity to its predecessors. Since the proofs require only smallmodifications to the previous ones and the essential parts of the theory re-main the same, this note concentrates on giving the formal statements butotherwise skips on discussions as well as details of the proofs as much aspossible and refers to the previous works instead.Let A be a self-adjoint, not necessarily bounded, operator on a separableHilbert space such that the spectrum of A is separated into two disjointcomponents, that is,(1.1) spec( A ) = σ ∪ Σ with d := dist( σ, Σ) > . Moreover, given a bounded self-adjoint operator V on the same Hilbertspace, define bounded nonnegative operators V ± with V = V + − V − viafunctional calculus by V + := (cid:0) V ) (cid:1) V / , V − := (cid:0) sign( V ) − (cid:1) V / . We clearly have k V ± k ≤ k V k , and V − or V + vanish if V is nonnegative ornonpositive, respectively. Mathematics Subject Classification.
Primary 47A55; Secondary 47A15, 47B15.
Key words and phrases.
Subspace perturbation problem, spectral subspaces, maximalangle between closed subspaces.
If(1.2) k V + k + k V − k < d, then it can be shown (see Corollary 2.2 below and the discussion thereafter)that the spectrum of the perturbed operator A + V is likewise separated intotwo disjoint components,(1.3) spec( A + V ) = ω ∪ Ω with dist( ω, Ω) ≥ d − k V + k − k V − k , where(1.4) ω = spec( A + V ) ∩ (cid:0) σ + [ −k V − k , k V + k ] (cid:1) and analogously for Ω (with σ replaced by Σ); here we have used the short-hand notation σ + [ −k V − k , k V + k ] := { λ + t : λ ∈ σ, −k V − k ≤ t ≤ k V + k} .Clearly, the gap non-closing condition (1.2) is sharp. Also note that (1.2)covers semidefinite perturbations V with k V k < d , as well as general, notnecessarily semidefinite, perturbations satisfying k V k < d/
2. On the otherhand, condition (1.2) also includes certain indefinite perturbations with d/ ≤ k V k < d that were not covered in the previous works. It is in-teresting to observe that (1.2) formally differs from the condition k V k < d for semidefinite perturbations and k V k < d/ k V + k + k V − k instead of k V k and 2 k V k , respectively. Infact, this seemingly na¨ıve observation remains valid also when it comes tothe main results discussed here and, in this sense, represents the essence ofthe present note.The variation of the spectral subspaces associated with the componentsof the spectrum is studied in terms of the difference of the correspondingspectral projections E A ( σ ) and E A + V ( ω ), where E A and E A + V denote theprojection-valued spectral measures for the unperturbed and perturbed self-adjoint operators A and A + V , respectively.The first main result discussed here holds under a certain favourable spec-tral separation condition for the unperturbed operator A and unifies [7, The-orem 1.1] for semidefinite perturbations and its corresponding well-knownvariant for general perturbations (cf., e.g., [4, Remark 2.9]). Theorem 1.1.
Let A be a self-adjoint operator on a separable Hilbert spacesuch that the spectrum of A is separated as in (1.1) . Let V be a boundedself-adjoint operator on the same Hilbert space satisfying (1.2) , and choose ω ⊂ spec( A + V ) as in (1.4) .If, addition, the convex hull of one of the components σ and Σ is disjointfrom the other component, that is, conv( σ ) ∩ Σ = ∅ or σ ∩ conv(Σ) = ∅ ,then (1.5) arcsin (cid:0) k E A ( σ ) − E A + V ( ω ) k (cid:1) ≤
12 arcsin (cid:16) k V + k + k V − k d (cid:17) < π , and this estimate is sharp. As indicated above, Theorem 1.1 differs from its predecessors by the ap-pearance of k V + k + k V − k instead of k V k and 2 k V k , respectively. The samedifference shows up when considering the generic result where no additionalspectral separation condition other than (1.1) is assumed. In order to dis-cuss this here, it is necessary to recall from [6] (cf. also [7]) the function NIFYING PERTURBATIONS IN THE SUBSPACE PERTURBATION PROBLEM 3 that has played the crucial role in the corresponding results for general andsemidefinite perturbations:Set c crit := 12 − (cid:16) − √ π (cid:17) = 0 . . . . and define N : [0 , c crit ] → [0 , π/
2] by(1.6) N ( x ) = arcsin( πx ) for 0 ≤ x ≤ π +4 , arcsin (cid:16)q π x − π − (cid:17) for π +4 < x < π − π , arcsin (cid:0) π (1 − √ − x ) (cid:1) for 4 π − π ≤ x ≤ κ, arcsin (cid:0) π (1 − √ − x ) (cid:1) for κ < x ≤ c crit . Here, κ ∈ (4 π − π , π − π ) is the unique solution to the equationarcsin (cid:16) π (cid:0) − √ − κ (cid:1)(cid:17) = 32 arcsin (cid:16) π (cid:0) − √ − κ (cid:1)(cid:17) in the interval (0 , π − π ].The second principal result of this note now unifies [7, Theorem 1.2] forsemidefinite perturbations and [6, Theorem 1] for general perturbations. Theorem 1.2.
Let A be a self-adjoint operator on a separable Hilbert spacesuch that the spectrum of A is separated as in (1.1) , and let V and ω as inTheorem 1.1. If, in addition, V satisfies k V + k + k V − k < c crit · d, then arcsin (cid:0) k E A ( σ ) − E A + V ( ω )) k (cid:1) ≤ N (cid:16) k V + k + k V − k d (cid:17) < π , where N is given by (1.6) . A more detailed discussion on the function N can be found in [6].The rest of this note is organized as follows:In Section 2.1, the perturbation of the spectrum with respect to the de-composition V = V + − V − of the perturbation and, in particular, the spectralseparation (1.3), (1.4) is discussed. Section 2.2 is devoted to a correspondingvariant of the Davis-Kahan sin 2Θ theorem, which is the core of the proofsof the main theorems. The latter are finally presented in Section 2.3.2. Proofs
Perturbation of the spectrum.
The following result extends thestatement of [7, Proposition 2.1] to not necessarily semidefinite perturba-tions; cf. also [8, Theorem 3.2] and [2, Eq. (9.4.4)].
Proposition 2.1.
Let the finite interval ( a, b ) ⊂ R , a < b , be contained inthe resolvent set of the self-adjoint operator A . Moreover, let V be a boundedself-adjoint operator on the same Hilbert space with k V + k + k V − k < b − a .Then, the interval ( a + k V + k , b − k V − k ) belongs to the resolvent set of theperturbed operator A + V . A. SEELMANN
Proof.
Since V + is nonnegative with k V + k < b − a , it follows from [7, Propo-sition 2.1] that the interval ( a + k V + k , b ) belongs to the resolvent set of A + V + . In turn, since − V − is nonpositive and k V − k < b − a − k V + k , itfollows from the same result that the interval ( a + k V + k , b − k V − k ) belongsto the resolvent set of the operator A + V = ( A + V + ) − V − . (cid:3) We have the following corollary to Proposition 2.1. The proof worksexactly as the one of [7, Corollary 2.2] and is hence omitted.
Corollary 2.2.
Let A be a self-adjoint operator, and let V be a boundedself-adjoint operator on the same Hilbert space. Then, spec( A + V ) ⊂ spec( A ) + [ −k V − k , k V + k ] . It is easy to see from Corollary 2.2 that in the situation of Theorems 1.1and 1.2 the spectrum of the perturbed operator is indeed separated asin (1.3) and (1.4). Moreover, for each t ∈ [0 ,
1] we have ( tV ) ± = tV ± ,so that the spectrum of A + tV is likewise separated into two disjoint com-ponents ω t and Ω t , defined analogously to ω and Ω in (1.4), respectively.Namely,(2.1) ω t = spec( A + tV ) ∩ (cid:0) σ + [ − t k V − k , t k V + k ] (cid:1) and analogously for Ω t (with σ replaced by Σ).We need the following variant of [7, Lemma 2.3] (see also [1, Theorem 3.5])for future reference. Lemma 2.3.
Let A be as in Theorem 1.2, and let V be a bounded self-adjoint operator on the same Hilbert space satisfying k V + k + k V − k < d .Then, the projection-valued path [0 , ∋ t E A + tV ( ω t ) with ω t as in (2.1) is continuous in norm.Proof. Let 0 ≤ s ≤ t ≤
1. Since dist( ω s , Ω t ) ≥ d − t k V + k − t k V − k as well asdist(Ω s , ω t ) ≥ d − t k V + k − t k V − k , we obtain as in the proof of [7, Lemma 2.3](see also [1, Theorem 3.5]) that k E A + sV ( ω s ) − E A + tV ( ω t ) k ≤ π | t − s |k V k d − t k V + k − t k V − k , which proves the claim. (cid:3) The sin 2Θ theorem.
The following variant of the Davis-Kahan sin 2Θtheorem from [3] unifies [4, Theorem 1] and [7, Proposition 2.4].
Proposition 2.4.
Let A be as in Theorem 1.2. Moreover, let V be abounded self-adjoint operator on the same Hilbert space and Q be an or-thogonal projection onto a reducing subspace for A + V . Then, the operatorangle Θ = arcsin | E A ( σ ) − Q | associated with E A ( σ ) and Q satisfies (2.2) k sin 2Θ k ≤ π k V + k + k V − k d . If, in addition, conv( σ ) ∩ Σ = ∅ or σ ∩ conv(Σ) = ∅ , then π/ in (2.2) canbe replaced by . NIFYING PERTURBATIONS IN THE SUBSPACE PERTURBATION PROBLEM 5
Proof.
Recall from the proof of [4, Theorem 1] that k sin 2Θ k ≤ π k V − KV K k d , where K = Q − Q ⊥ is self-adjoint and unitary. Also recall from [4, Re-mark 2.5] that π/ σ ) ∩ Σ = ∅ or σ ∩ conv(Σ) = ∅ . It only remains to show that k V − KV K k ≤ k V + k + k V − k .As in the proof of [7, Proposition 2.4], the nonnegative operators V ± satisfy k V ± − KV ± K k ≤ k V ± k . Hence, using V = V + − V − , we have k V − KV K k ≤ k V + − KV + K k + k V − − KV − K k ≤ k V + k + k V − k , which completes the proof. (cid:3) Remark . With the maximal angle θ := arcsin( k E A ( σ ) − Q k ) between thesubspaces Ran E A ( σ ) and Ran Q and the inequality sin 2 θ ≤ k sin 2Θ k , weobtain from (2.2) the sin 2 θ estimate (2.3) sin 2 θ ≤ π k V + k + k V − k d , where π/ θ = π/
2, and for θ < π/ θ ≤ π k E A (Σ) U ∗ V U E A ( σ ) k d with a certain unitary operator U satisfying U ∗ QU = E A ( σ ). Also recall(see, e.g., [4, Remark 3.2]) that the constant π here can be replaced by 2if conv( σ ) ∩ Σ = ∅ or σ ∩ conv(Σ) = ∅ . Since U ∗ V ± U are nonnegative,by [7, Lemma A.2] we have 2 k E A (Σ) U ∗ V ± U E A ( σ ) k ≤ k U ∗ V ± U k = k V ± k .Thus,2 k E A (Σ) U ∗ V U E A ( σ ) k ≤ k E A (Σ) U ∗ V + U E A ( σ ) k + 2 k E A (Σ) U ∗ V − U E A ( σ ) k≤ k V + k + k V − k , which together with (2.4) proves (2.3).2.3. Proof of the main results.
Proof of Theorem 1.1.
Following the proof of [7, Theorem 1.1], we see thatestimate (1.5) is a direct consequence of Proposition 2.4 with Q = E A + V ( w )and Lemma 2.3.It remains to show the sharpness of estimate (1.5). This can be seen fromthe following example of 2 × ≤ v ± < v := v + + v − < A := (cid:18) − (cid:19) and V := v + − v − − v v √ − v v √ − v v + v + − v − ! with σ := { / } , Σ := {− / } , and d := dist( σ, Σ) = 1.
A. SEELMANN
It is easy to verify that spec( V ) = {− v − , v + } and that the spectrumof A + V is given by spec( A + V ) = { ( v + − v − ± √ − v ) / } . Denote ω := { ( v + − v − + √ − v ) / } ⊂ [1 / − v − , / v + ] and θ := arcsin( v ) / − √ − v v = tan θ = v √ − v , it is then straightforward to show that( A + V ) (cid:18) cos θ sin θ (cid:19) = v + − v − + √ − v (cid:18) cos θ sin θ (cid:19) and, therefore,arcsin( k E A ( σ ) − E A + V ( ω ) k ) = θ = 12 arcsin (cid:16) v + + v − d (cid:17) . Thus, estimate (1.5) is sharp, which completes the proof. (cid:3)
Just as Theorem 1.1, we also obtain from Proposition 2.4 and Lemma 2.3the follow result, which applies in the generic situation where no additionalspectral separation condition other than (1.1) is assumed. It unifies [4,Corollary 2] and [7, Corollary 2.5] and plays a crucial role in the proof ofTheorem 1.2, see below.
Corollary 2.6.
In the situation of Theorem 1.2 one has arcsin( k E A ( σ ) − E A + V ( ω ) k ) ≤
12 arcsin (cid:16) π k V + k + k V − k d (cid:17) ≤ π whenever k V + k + k V − k ≤ d/π .Proof of Theorem 1.2. Let 0 = t ≤ · · · ≤ t n = 1, n ∈ N , be a finite partitionof the interval [0 , λ j := ( t j +1 − t j )( k V + k + k V − k ) d − t j ( k V + k + k V − k ) < , j = 0 , . . . , n − . Considering A + t j +1 V = ( A + t j V ) + ( t j +1 − t j ) V , we obtain with Corol-lary 2.6 as in [5–7] (cf. also [1]) that(2.5) arcsin( k E A ( σ ) − E A + V ( ω ) k ) ≤ n − X j =0 arcsin (cid:16) πλ j (cid:17) whenever λ j ≤ π . Moreover, with 1 − t j +1 ( k V + k + k V − k ) /d = (1 − λ j )(1 − t j ( k V + k + k V − k ) /d )for j = 0 , . . . , n − t = 0, and t n = 1, we have(2.6) 1 − k V + k + k V − k d = n − Y j =0 (1 − λ j ) . Recalling from the proof of [7, Theorem 1.2] that the function N satisfies N (cid:16) x (cid:17) = inf (cid:26) n − X j =0 arcsin (cid:16) πλ j (cid:17) : n ∈ N , ≤ λ j ≤ π , n − Y j =0 (1 − λ j ) = 1 − x (cid:27) , for 0 ≤ x ≤ c crit , the claim now follows from (2.5) and (2.6). (cid:3) NIFYING PERTURBATIONS IN THE SUBSPACE PERTURBATION PROBLEM 7
Remark . Considering partitions of the interval [0 ,
1] with arbitrarilysmall mesh size, we obtain from (2.5) analogously to [6, Remark 2.2], [5,Remark 2.1] and [7, Remark 2.6] thatarcsin( k E A ( σ ) − E A + V ( ω ) k ) ≤ π Z k V + k + k V − k d − t k V + k − t k V − k d t = π dd − k V + k − k V − k , and the latter is strictly less than π/ k V + k + k V − k d ≤ < c crit . References [1] S. Albeverio, A. K. Motovilov,
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The rotation of eigenvectors by a perturbation. III , SIAMJ. Numer. Anal. (1970), 1–46.[4] A. Seelmann, Notes on the sin 2Θ theorem , Integral Equations Operator Theory (2014), 579–597.[5] A. Seelmann, Notes on the subspace perturbation problem for off-diagonal perturba-tions , Proc. Amer. Math. Soc. (2016), 3825–3832.[6] A. Seelmann,
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