INVARIANT GRAPH SUBSPACES OF A J -SELF-ADJOINT OPERATOR IN THE - - PowerPoint PPT Presentation

INVARIANT GRAPH SUBSPACES OF A J-SELF-ADJOINT OPERATOR IN THE FESHBACH CASE∗ Alexander K. Motovilov

Joint Institute for Nuclear Research and Dubna University, Dubna, Russia Workshop on Operator Theory and Indefinite Inner Product Spaces Vienna, 20 December 2016

∗Based on [S.Albeverio and A.K.Motovilov, “On invariant graph subspaces of

a J-self-adjoint operator in the Feshbach case”, Mathem. Notes 100 (2016), 761–773]

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The problem setup Assume that A is a self-adjoint operator on a Hilbert space H, block diagonal with respect to a decomposition H = A0 ⊕A1, that is, A0 and A1 are reducing subspaces and A may be written as a 2×2 block diagonal operator matrix, A = ( A0 A1 ) , A0 = A

• A0,

A1 = A

• A1.

In the following σ0 = spec(A0), σ1 = spec(A1) and J = ( I −I ) .

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We consider a perturbation of A by a bounded off-diagonal J-self-adjoint operator matrix V = ( B −B∗ )

• B ∈ B(A1,A0),

JV = (JV)∗. The perturbed operator L = A+V = ( A0 B −B∗ A1 ) . If B ̸= 0 then L is for sure a non-symmetric operator and, hence, it can have non-real spectrum. Nevertheless, there are known cases where spec(L) ⊂ R and L is similar to a self-adjoint operator. There are many contributors in this area (Adamyan, H.Langer, Shkalikov, Tretter, K.Veseli´ c . . . )

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It is known that L is similar to a self-adjoint operator if σ0 ∩σ1 = ∅, dist(σ0,σ1) = d > 0 and ∥B∥ < d π for both unbounded A0 and A1 (1) and generic σ0 and σ1, or ∥B∥ < d 2 for particular mutual positions of σ0 and σ1 (2)

• r if A0 and/or A1 is bounded

(see [K. Veseli´ c, 1972], [Albeverio, AM, Shkalikov, 2009]).

In these cases the corresponding (complementary) perturbed spectral subspaces L0 and L1 of L are maximal uniformly positive and maximal uniformly negative, respectively.

This is a reference to Krein space theory context.

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Introducing the indefinite inner product by formula [x,y] = (Jx,y), x,y ∈ H. turns the Hilbert space H into a Krein space which we denote by K, K = {H,J}; A, V, and L = A+V are s.a. on K. Recall that a (closed) subspace L ⊂ K is said to be uniformly positive if there exists a γ > 0 such that [x,x] ≥ γ ∥x∥2 for every x ∈ K, x ̸= 0, where ∥ · ∥ denotes the norm on H. The subspace L is called maximal uniformly positive if it is not a proper subset of another uniformly positive subspace of K. Uniformly negative and max- imal uniformly negative subspaces of K are defined in a similar way.

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In the spectral cases discussed (with d = dist(σ0,σ1) > 0) the per- turbed spectral subspaces L0 and L1 of L are written as a graph subspace of the form L0 = G (K) = { x0 ⊕Kx0

• x0 ∈ A0

} , L1 = G (K∗) = { K∗x1 ⊕x1

• x1 ∈ A1

} , where K is a uniform contraction K ∈ B(A0,A1), ∥K∥ < 1. In some spectral cases with d = dist(σ0,σ1) > 0 we have sharp (optimal) bounds for the angular operator K.

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In particular,

if conv(σ0)∩σ1 = ∅ or σ1 ∩conv(σ0) = ∅ then sin(2θ) ≤ 2∥B∥ d where θ = arctan∥K∥

[Albeverio, AM, Shkalikov, 2009], for more bounds on K see [Albeverio, AM, Tretter, 2010]. K is a solution to the operator Riccati equation KA0 −A1K +KBK = −K∗.

The present work: We suppose that σ0 ∩σ1 ̸= ∅ . More precisely, σ ac

0 ∩σ1 ̸= ∅.

In the talk, we adopt a simplified assumption that corresponds to the Feshbach spectral case:

• All the spectrum σ0 of A0 is absolutely continuous, σ0 = σ ac

0 , and

σ1 is completely embedded into σ0.

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σ0 = ∆ = [α,β] ⊂ R We address the following question: Does L have invariant graph subspaces of the form L0 = G (K) = { x0 ⊕Kx0

• x0 ∈ A0

} , L1 = G (K∗) = { K∗x1 ⊕x1

• x1 ∈ A1

} ? May one associate them with certain parts of the spectrum of L as spectral subspaces? As for the location of the spectrum of L, by this moment we

• nly know the following:

[Tretter, 2008]: spec(L) ⊂ { z ∈ C

• |Imz| ≤ ∥B∥, α ≤ Re z ≤ β

} , Surely, spec(L) is symmetric with respect to R.

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Factorization of the Schur complement Frobenius-Schur representation of the difference L−z, z ̸∈ σ0: L−z = ( I −B∗(A0 −z)−1 I )(A0 −z M1(z) )(I (A0 −z)−1B I ) , (3) where M1(z) stands for the Schur complement of A0 −z, M1(z) = A1 −z+W1(z), with W1(z) = B∗(A0 −z)−1B. (4) (3) = ⇒ Spectral properties of L are determined by those of M1. Idea: To factorize M1(z) in the form M1(z) = F1(z)(Z −z) by having found its operator root(s) Z. (F1(z) should be bounded and boundedly invertible for any z in a neighborhood of the spec- trum of Z.) This is a Markus-Matsaev-type factorization of M1.

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In the case under consideration we follow an approach that was elaborated in [Mennicken, AM (1999)] for self-agoint block

• perator matrices (and also for some non-self-adjoint ones in

[Hardt, Mennicken, AM (2003)]). M1(z) = A1 −z+B∗(A0 −z)−1B = A1 −z+

∫

∆0(= spec(A0))

B∗EA0(dµ)B 1 µ −z. Suppose that M1 admits analytic continuation through the interval ∆0 to certain domains on the so called unphysical sheets of the z plane. From, e.g., [Greenstein 1960] it follows that the integral term

• f M1 admits (as a Cauchy type integral) the analytic continuation
• nto some D− (or D+) under the cut ∆ = (α,β) if and only if

KB(µ) := B∗EA0 ( (−∞,µ] ) B admits such a continuation. In this case we have D+ = (D−)∗, KB is holomorphic in D− ∪D+, and KB(µ) = KB(µ)∗.

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We then pass from M1 to M(z,Γ±) := A0−z+

∫

Γ± dµK′ B(µ)

µ −z , where K′

B(µ) = d dµKB(µ)

and Γ± ⊂ D±, with the end points α and β.

K′

B(µ) is allowed to be slightly sin-

gular at the end points µ0 = α and µ0 = β, ∥K′

B(µ)∥ ≤ C|µ − µ0|γ,

γ ∈ (0,1]

Introduce the operator transformation M1(Z,Γ) = A1 −Z +

∫

Γdµ K′ B(µ)(µ −Z)−1,

(∗) where Γ = Γ±. It is assumed that Z ∈ B(A1) and spec(Z)∩Γ = ∅.

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And then find Z as a solution to the equation M1(Z,Γ±) = 0. (5)

The solvability of (5) is proven by applying Banach’s Fixed Point Theorem under the assumption V0(B,Γ) < 1 4d0(Γ)2, (6) where V0(B,Γ) :=

∫

Γ|dµ|∥K′ B(µ)∥

and d0(Γ) = dist(σ1,Γ).

• Theorem. Assume that Γ is an admissible contour satisfying (6) Then equation

(5) has a solution Z of the form Z = A0 +X with ∥X∥ ≤ rmin(Γl) := d(Γ) 2 − √ d(Γ)2 4 −V0(B,Γ). With respect to X, this solution is unique in the closed ball of the bigger radius d(Γ)− √ V0(B,Γ). The solution X does not depend on a specific contour Γ ⊂ Dl satisfying (6). Moreover, the bound on the norm of X may be optimized with respect to the

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admissible contours Γl in the form ∥X∥ ≤ r0(B) with r0(B) := inf

Γl:ω(B,Γl)>0rmin(Γl),

(7) where ω(B,Γl) = d2(Γl)−4V0(B,Γl). Unlike r0(B), the solution X depends on l, and thus we will supply its notation with the index l writing X(l) and Z(l) = A1 +X(l), l = ±1.

• Theorem. Let Γl be an admissible contour satisfying V0(B,Γ) < 1

4d0(Γ)2, and let

Z(l) is the corresponding unique solution (of the main equation) mentioned in the previous theorem. Then, for z ∈ C\Γl, M1(z,Γl) = F1(z,Γl)(Z(l) −z), (8) where F1(z,Γl) = I +

∫

Γl dµ K′ B(µ)(Z(l) − µ)−1(µ −z)−1

(9) is a bounded operator on A1. Moreover, if dist ( z,σ1 ) ≤ 1

2d(Γl) then for sure

F1(z,Γl) has a bounded inverse.

• Lemma. The spectrum of Z(l) lies in the closed r0(B)-neighborhood of σ1 =

spec(A1) in C.

• Lemma. The operators Z(−)∗ and Z(+) are similar to each other and, thus, the

spectrum of Z(−)∗ coincides with that of Z(+).

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The spectrum of Z(−) on the real axis and in the upper half-plane represents (a part of) the spectrum of L. The (complex) spectrum of Z(−) in C− = {z ∈ C | Imz < 0}: the discrete eigenvalues are called resonances. (In the case of a self-adjoint L these were the Feshbach resonances.)

The no-resonance case Hypothesis (NR). Assume dist ( spec(Z(l)),Dl) > 0. Lemma. Assume Hypothesis (NR) for some l = ±1 and set Y (l) =

∫

∆0

EA0(dµ)B(Z(l) − µ)−1. The operator K(l) := Y (l)∗ belongs to B(A0,A1 and satisfies the Riccati equation KA0 −A1K +KBK = −B∗.

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Thus, under Hypothesis (NR) we will have two pairs of J-

• rthogonal invariant graph subspaces for L:

L(l)

0 = G (K(l)) and L(l) 1 = G (K(l)∗),

l = ±1. Furthermore, the root Z(l) is given by Z(l) = A1 −B∗K(l)∗, and there is an analogous factorizer Z = A0+BK(l) for the Schur complement M0(z).

• Remark. Under Hypothesis (NR) necessarily ∥K(±)∥ ≥ 1 (!)

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Purely Feshbach spectral case Recall that KB(µ) = B∗EA0 ( (−∞,µ] ) B is a non-decreasing operator- valued function of µ ∈ R. Then notice that ⟨K′

B(µ)x.x⟩ ≥ 0

∀µ ∈ ∆0,∀x ∈ A1. Hypothesis (F). Assume dim(A1) < ∞ and there is c0 > 0 s.t. ⟨K′

B(µ)x,x⟩ ≥ c0∥x∥2

for any µ ∈ ∆0 ∩Or0(σ1) and any x ∈ A1.

• Lemma. Under Hypothesis (F) the operator root Z(l) = A1 + Z(l),

l = ±1, has no spectrum in the corresponding complex domain Dl and, thus, all the eigenvalues of Z(l) are simultaneously eigen- values of the original (not yet continued) Schur complement M1(·) and, hence, the eigenvalues of L.

None of these eigenvalues is real and none of them is a resonance! Hy- pothesis (NR) holds and then one may talk on the invariant graph subspaces for L.

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The simplest example

∆0 = [−α,α], α > 0, A0 = L2(−α,α), A1 = C, a1 ∈ ∆0 L = ( Λ0 b −⟨·,b⟩ a1 ) , where Λ0 (= A0) is multiplication by the independent variable, (Λ0u0)(µ) = µ u0(µ), u0 ∈ A0. The coupling operator B : A1 → A0 is assumed to be the multiplication by a constant b ≥ 0, namely (Bu1)(µ) := bu1, µ ∈ [−α,α] (that is, Bu1 is constant function on [−α,α]). Obviously, the adjoint operator B∗ is given by B∗u0 = b

∫ α

−α u0(µ)dµ = ⟨u0,b⟩.

In this example for −α ≤ µ ≤ α one finds KB(µ) = b2

∫ µ

−α dν = b2(µ +α).

Hence, K′

B(µ) = b2 and Hypotheses (F) and (NR) hold true. In case of a1 = 0,

almost everything is computed explicitly. For K we find ∥K∥ = 1.

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Summary

• In the case of disjoint σ0 and σ1 we have recalled some known conditions

ensuring that the perturbed spectral subspaces of a J-s.a. operator L are maximal uniformly definite.

• In the case of overlapping σ (ac)

and σ1 we have found conditions guaran- teeing the Markus-Matsaev-type factorization of the analytically continued Schur complement M.

• Furthermore, we pointed out conditions (in particular, the absence of reso-

nances) ensuring the graph representation of the perturbed reducing sub- spaces of L.