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]

2 The problem setup Assume that A is a self-adjoint operator on a Hilbert space H , block diagonal with respect to a decomposition H = A 0 ⊕ A 1 , that is, A 0 and A 1 are reducing subspaces and A may be written as a 2 × 2 block diagonal operator matrix, ( A 0 ) � � 0 � � A = A 0 = A A 1 = A , A 0 , A 1 . 0 A 1 In the following σ 0 = spec ( A 0 ) , σ 1 = spec ( A 1 ) and ( I ) 0 J = . − I 0

3 We consider a perturbation of A by a bounded off-diagonal J -self-adjoint operator matrix � ( ) � 0 B � JV = ( JV ) ∗ . V = B ∈ B ( A 1 , A 0 ) , � − B ∗ 0 The perturbed operator ( A 0 ) B L = A + V = . − B ∗ A 1 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 . . . )

4 It is known that L is similar to a self-adjoint operator if σ 0 ∩ σ 1 = ∅ , dist ( σ 0 , σ 1 ) = d > 0 and ∥ B ∥ < d for both unbounded A 0 and A 1 (1) π and generic σ 0 and σ 1 , or ∥ B ∥ < d for particular mutual positions of σ 0 and σ 1 (2) 2 or if A 0 and/or A 1 is bounded c, 1972], [Albeverio, AM, Shkalikov, 2009]) . (see [K. Veseli´ In these cases the corresponding (complementary) perturbed spectral subspaces L 0 and L 1 of L are maximal uniformly positive and maximal uniformly negative, respectively. This is a reference to Krein space theory context .

5 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.

6 In the spectral cases discussed (with d = dist ( σ 0 , σ 1 ) > 0 ) the per- turbed spectral subspaces L 0 and L 1 of L are written as a graph subspace of the form � { } � x 0 ∈ A 0 L 0 = G ( K ) = x 0 ⊕ Kx 0 , � { } � x 1 ∈ A 1 L 1 = G ( K ∗ ) = K ∗ x 1 ⊕ x 1 , where K is a uniform contraction K ∈ B ( A 0 , A 1 ) , ∥ K ∥ < 1 . In some spectral cases with d = dist ( σ 0 , σ 1 ) > 0 we have sharp (optimal) bounds for the angular operator K .

7 In particular, if conv ( σ 0 ) ∩ σ 1 = ∅ or σ 1 ∩ conv ( σ 0 ) = ∅ then sin ( 2 θ ) ≤ 2 ∥ B ∥ where θ = arctan ∥ K ∥ d [Albeverio, AM, Shkalikov, 2009], for more bounds on K see [Albeverio, AM, Tretter, 2010]. K is a solution to the operator Riccati equation KA 0 − A 1 K + KBK = − K ∗ . We suppose that σ 0 ∩ σ 1 ̸ = ∅ . The present work: More precisely, σ ac 0 ∩ σ 1 ̸ = ∅ . In the talk, we adopt a simplified assumption that corresponds to the Feshbach spectral case: • All the spectrum σ 0 of A 0 is absolutely continuous, σ 0 = σ ac 0 , and σ 1 is completely embedded into σ 0 .

8 σ 0 = ∆ = [ α , β ] ⊂ R We address the following question: Does L have invariant graph subspaces of the form � { } � x 0 ∈ A 0 L 0 = G ( K ) = x 0 ⊕ Kx 0 , � { } � x 1 ∈ A 1 L 1 = G ( K ∗ ) = K ∗ x 1 ⊕ x 1 ? 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 only know the following: � { } � | Im z | ≤ ∥ B ∥ , α ≤ Re z ≤ β [Tretter, 2008]: spec ( L ) ⊂ z ∈ C , Surely, spec ( L ) is symmetric with respect to R .

9 Factorization of the Schur complement Frobenius-Schur representation of the difference L − z , z ̸∈ σ 0 : ( )( A 0 − z )( I ( A 0 − z ) − 1 B ) I 0 0 L − z = (3) , − B ∗ ( A 0 − z ) − 1 I M 1 ( z ) 0 0 I where M 1 ( z ) stands for the Schur complement of A 0 − z , with W 1 ( z ) = B ∗ ( A 0 − z ) − 1 B . M 1 ( z ) = A 1 − z + W 1 ( z ) , (4) (3) = ⇒ Spectral properties of L are determined by those of M 1 . Idea: To factorize M 1 ( z ) in the form M 1 ( z ) = F 1 ( z )( Z − z ) by having found its operator root(s) Z . ( F 1 ( 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 M 1 .

10 In the case under consideration we follow an approach that was elaborated in [Mennicken, AM (1999)] for self-agoint block operator matrices (and also for some non-self-adjoint ones in [Hardt, Mennicken, AM (2003)]). M 1 ( z ) = A 1 − z + B ∗ ( A 0 − z ) − 1 B ∫ 1 B ∗ E A 0 ( d µ ) B = A 1 − z + µ − z . ∆ 0 (= spec ( A 0 )) Suppose that M 1 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 of M 1 admits (as a Cauchy type integral) the analytic continuation onto some D − (or D + ) under the cut ∆ = ( α , β ) if and only if ( ) K B ( µ ) : = B ∗ E A 0 ( − ∞ , µ ] B admits such a continuation. In this case we have D + = ( D − ) ∗ , K B is holomorphic in D − ∪ D + , and K B ( µ ) = K B ( µ ) ∗ .

11 We then pass from M 1 to ∫ Γ ± d µ K ′ B ( µ ) M ( z , Γ ± ) : = A 0 − z + µ − z , K ′ d B ( µ ) = d µ K B ( µ ) where and Γ ± ⊂ D ± , with the end points α and β . K ′ B ( µ ) is allowed to be slightly sin- gular at the end points µ 0 = α and µ 0 = β , ∥ K ′ B ( µ ) ∥ ≤ C | µ − µ 0 | γ , γ ∈ ( 0 , 1 ] Introduce the operator transformation ∫ Γ d µ K ′ B ( µ )( µ − Z ) − 1 , M 1 ( Z , Γ ) = A 1 − Z + ( ∗ ) where Γ = Γ ± . It is assumed that Z ∈ B ( A 1 ) and spec ( Z ) ∩ Γ = ∅ .

12 And then find Z as a solution to the equation M 1 ( Z , Γ ± ) = 0 . (5) The solvability of (5) is proven by applying Banach’s Fixed Point Theorem under the assumption V 0 ( B , Γ ) < 1 4 d 0 ( Γ ) 2 , (6) where ∫ Γ | d µ |∥ K ′ V 0 ( B , Γ ) : = B ( µ ) ∥ d 0 ( Γ ) = dist ( σ 1 , Γ ) . and Theorem. Assume that Γ is an admissible contour satisfying (6) Then equation (5) has a solution Z of the form Z = A 0 + X with √ ∥ X ∥ ≤ r min ( Γ l ) : = d ( Γ ) d ( Γ ) 2 − − V 0 ( B , Γ ) . 2 4 With respect to X , this solution is unique in the closed ball of the bigger radius √ d ( Γ ) − V 0 ( B , Γ ) . The solution X does not depend on a specific contour Γ ⊂ D l satisfying (6). Moreover, the bound on the norm of X may be optimized with respect to the

13 admissible contours Γ l in the form ∥ X ∥ ≤ r 0 ( B ) with Γ l : ω ( B , Γ l ) > 0 r min ( Γ l ) , r 0 ( B ) : = (7) inf where ω ( B , Γ l ) = d 2 ( Γ l ) − 4 V 0 ( B , Γ l ) . Unlike r 0 ( B ) , the solution X depends on l , and thus we will supply its notation with the index l writing X ( l ) and Z ( l ) = A 1 + X ( l ) , l = ± 1 . Theorem. Let Γ l be an admissible contour satisfying V 0 ( B , Γ ) < 1 4 d 0 ( Γ ) 2 , and let Z ( l ) is the corresponding unique solution (of the main equation) mentioned in the previous theorem. Then, for z ∈ C \ Γ l , M 1 ( z , Γ l ) = F 1 ( z , Γ l )( Z ( l ) − z ) , (8) where ∫ B ( µ )( Z ( l ) − µ ) − 1 ( µ − z ) − 1 Γ l d µ K ′ F 1 ( z , Γ l ) = I + (9) ( ) ≤ 1 2 d ( Γ l ) then for sure is a bounded operator on A 1 . Moreover, if dist z , σ 1 F 1 ( z , Γ l ) has a bounded inverse. Lemma . The spectrum of Z ( l ) lies in the closed r 0 ( B ) -neighborhood of σ 1 = spec ( A 1 ) in C . Lemma . The operators Z ( − ) ∗ and Z (+) are similar to each other and, thus, the spectrum of Z ( − ) ∗ coincides with that of Z (+) .

14 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 | Im z < 0 } : the discrete eigenvalues are called resonances. (In the case of a self-adjoint L these were the Feshbach resonances.) The no-resonance case ( spec ( Z ( l ) ) , D l ) Hypothesis (NR) . Assume dist > 0 . Lemma . Assume Hypothesis (NR) for some l = ± 1 and set ∫ Y ( l ) = E A 0 ( d µ ) B ( Z ( l ) − µ ) − 1 . ∆ 0 The operator K ( l ) : = Y ( l ) ∗ belongs to B ( A 0 , A 1 and satisfies the Riccati equation KA 0 − A 1 K + KBK = − B ∗ .

15 Thus, under Hypothesis (NR) we will have two pairs of J - orthogonal invariant graph subspaces for L : 1 = G ( K ( l ) ∗ ) , L ( l ) 0 = G ( K ( l ) ) and L ( l ) l = ± 1 . Furthermore, the root Z ( l ) is given by Z ( l ) = A 1 − B ∗ K ( l ) ∗ , and there Z = A 0 + BK ( l ) for the Schur complement is an analogous factorizer � M 0 ( z ) . Remark . Under Hypothesis (NR) necessarily ∥ K ( ± ) ∥ ≥ 1 (!)