Metric operators, generalized hermiticity and lattices of Hilbert spaces

Jean-Pierre Antoine

Institut de Recherche en Math´ematiqueet Physique (IRMP) Universit´ecatholique de Louvain, Louvain-la-Neuve, Belgium

(Joint work with Camillo Trapani)

15th International Workshop on Pseudo-Hermitian Hamiltonians in Quantum Physics (PHHQP 15)

Universit`adi Palermo, May 2015

Jean-Pierre Antoine Metric operators, generalized hermiticity and . . . 1/28 MOTIVATION

Non-self-adjoint operators with real spectrum appear in different contexts : PT-symmetric quantum mechanics (Bender et al.) Pseudo-Hermitian quantum mechanics (Mostafazadeh) Three-Hilbert-Space formulation of quantum mechanics (Znojil) Nonlinear pseudo-bosons (Bagarello) Nonlinear supersymmetry ... under various names : pseudo-Hermitian, quasi-Hermitian, cryptohermitian operators Generic structure : A∗G = GA, i.e., A∗ is similar to A via G : H → K, a metric operator, i.e. G > 0, thus invertible, with (possibly unbounded) inverse G −1 Aim of this talk : to study the problem of operator similarity under a metric operator in the framework of partial inner product spaces (pip-spaces)

Jean-Pierre Antoine Metric operators, generalized hermiticity and . . . 2/28 I. Metric operators and generalized hermiticity

Jean-Pierre Antoine Metric operators, generalized hermiticity and . . . 3/28 Metric operators - 1

Metric operator : strictly positive self-adjoint operator G > 0 : i.e., hGξ|ξi > 0, ∀ ξ ∈ D(G), and hGξ|ξi = 0 ⇔ ξ = 0 ⇒ G invertible, G −1 densely defined in H, not necessarily bounded G −1 bounded ⇒ G −1 is a metric operator

Let G, G1, G2 be metric operators. Then (1) if G1, G2 are bounded, G1 + G2 is a bounded metric operator (2) λG is a metric operator if λ > 0 1/2 t (3) G , G (t ∈ R) are metric operators Take G bounded and G −1 unbounded and define 1/2 1/2 1/2 hξ|ηiG := hG ξ|G ηi,ξ,η ∈ D(G ) : positive definite inner product 1/2 on H with corresponding norm kξkG = kG ξk Completion ⇒ new H(G) and H ⊆ H(G) Conjugate H(G)× of H(G) ' subspace of H and H(G)× ≡ H(G −1) = D(G −1/2) with inner product −1/2 −1/2 hξ|ηiG −1 = hG ξ|G ηi ⇒ Triplet (RHS) H(G −1) ,→ H ,→ H(G) (,→ = continuous embedding with dense range) G −1 bounded ⇒ H(G −1) = H(G) = H with norms equivalent (but different) to the norm of H

Jean-Pierre Antoine Metric operators, generalized hermiticity and . . . 4/28 Metric operators - 2

In the triplet H(G −1) ⊂ H ⊂ H(G) (*) G −1/2= H(G −1) → H and H → H(G) G 1/2= unitary operator H → H(G −1) and H(G) → H Triplet (*) = central part of the infinite scale of Hilbert spaces built on −1/2 −n/2 the powers of G , VI := {Hn, n ∈ Z}, where Hn = D(G ), n ∈ N, × with a norm equivalent to the graph norm, and H−n = Hn

... ⊂ H2 ⊂ H1 ⊂ H ⊂ H−1 ⊂ H−2 ⊂ ...

Question : what are the end spaces of the scale ?

−1/2 \ −1/2 [ H∞(G ) := Hn, H−∞(G ) := Hn. n∈Z n∈Z

By quadratic interpolation, build continuous scale Ht , 0 6 t 6 1, between −t/2 −t/2 H1 and H, where Ht = D(G ), with norm kξkt = kG ξk × Define H−t = Ht and iterate ⇒ full continuous scale V := {H , t ∈ } : -space eI t R pip

Jean-Pierre Antoine Metric operators, generalized hermiticity and . . . 5/28 Similar operators

Easy properties : Given (A, D(A))= linear operator in H : D(A) dense in H ⇒ D(A) is dense in H(G) A closed in H(G) ⇒ closed in H Definitions (1) Let H, K be Hilbert spaces, A, B densely defined linear operators in H, resp. K. A T : H → K is called a bounded intertwining operator for A and B if (io1) T : D(A) → D(B);

(io2) BT ξ = TAξ, ∀ξ ∈ D(A). (2) A and B are similar( A ∼ B) if there exists a bounded intertwining operator T for A and B with bounded inverse T −1 : K → H, intertwining for B and A (∼ is an equivalence relation) (3) A and B are unitarily equivalent( A ∼u B) if A ∼ B and T : H → K is unitary Properties Let A ∼ B. Then TD(A) = D(B) A closed ⇔ B closed A−1 exists ⇔ B−1 exists; then, B−1 ∼ A−1

Jean-Pierre Antoine Metric operators, generalized hermiticity and . . . 6/28 Similar operators : spectral considerations

Decomposition of spectrum of a closed operator A:

Resolvent set −1 ρ(A) := {λ ∈ C :(A − λI ) one-to-one and (A − λI ) is bounded} Spectrum σ(A) := C \ ρ(A) Point spectrum σp(A) := {λ ∈ C :(A − λI ) not one-to-one} = {eigenvalues of A} Continuous spectrum σc (A) := {λ ∈ C :(A − λI ) one-to-one, dense range 6= H} ⇒ (A − λI )−1 densely defined, unbounded Residual spectrum σr (A) := {λ ∈ C :(A − λI ) one-to-one, range not dense}, ⇒ (A − λI )−1 not densely defined ∗ Note σr (A) = σp(A∗) = {λ : λ ∈ σp(A )}

σ(A) = σp(A) ∪ σc (A) ∪ σr (A) (Dunford-Schwartz)

Jean-Pierre Antoine Metric operators, generalized hermiticity and . . . 7/28 Similar operators : spectral considerations

Properties of similar operators A, B closed and A ∼ B with the bounded intertwining operator T . Then ρ(A) = ρ(B)

σp(A) = σp(B) and eigenvectors of A, B are in bijection, with same eigenvalues and same multiplicity : ξ ∈ D(A) eigenvector of A ⇒ T ξ eigenvector of B η ∈ D(B) eigenvector of B ⇒ T −1η eigenvector of A

σc (A) = σc (B) and σr (A) = σr (B)

A self-adjoint ⇒ B has real spectrum and σr (B) = ∅ Similarity : often too strong! A is quasi-similar to B (A a B) if there exists a bounded intertwining operator T for A and B which is invertible, with inverse T −1 densely defined (but not necessarily bounded). Note : one can always suppose that T is a metric operator (⇒ ‘metrically (quasi-)similar operators’) NB. Great confusion about terminology in the literature !

Jean-Pierre Antoine Metric operators, generalized hermiticity and . . . 8/28 Quasi-similar operators : spectral considerations - 1

Comparison of spectra Let A, B be closed and densely defined, and assume A a B with the bounded intertwining operator T . Then:

(i) σp(A) ⊆ σp(B) and, for every λ ∈ σp(A), one has mA(λ) 6 mB (λ) (ii) σr (B) ⊆ σr (A)

(iii) If TD(A) = D(B), then σp(B) = σp(A) −1 (iv) If T is bounded and TD(A) is a core for B, then σp(B) ⊆ σ(A) (v) T −1 everywhere defined and bounded ⇒

ρ(A) \ σp(B) ⊆ ρ(B) and ρ(B) \ σr (A) ⊆ ρ(A) (vi) Assume that T −1 is everywhere defined and bounded and TD(A) is a core for B. Then σp(A) ⊆ σp(B) ⊆ σ(B) ⊆ σ(A). Remark :This situation is important for applications : it gives some information on σ(B) once σ(A) is known. For instance : A has a pure point spectrum ⇒ B is to A A self-adjoint and A a B via an intertwining operator T with bounded inverse T −1 ⇒ B has real spectrum

Jean-Pierre Antoine Metric operators, generalized hermiticity and . . . 9/28 Quasi-similar operators : spectral considerations - 2

Examples: 2 (1) In H = L (R, dx), take Q of multiplication by x, on the dense domain  Z  2 2 2 D(Q) = f ∈ L (R): x |f (x)| dx < ∞ R 2 Pϕ = |ϕihϕ| ≡ ϕ ⊗ ϕ, for ϕ ∈ L (R), with kϕk = 1 2 2 −1 Aϕf = h(I + Q )f |ϕi(I + Q ) ϕ, ϕ ∈ D(Aϕ) ⇒ Then 2 −1 Pϕ a Aϕ with the bounded intertwining operator T = (I + Q )

Pϕ is everywhere defined and bounded, but the operator Aϕ is closable iff ϕ ∈ D(Q2). 2 If ϕ ∈ D(Q ), Aϕ is bounded and everywhere defined, and σ(Aϕ) = σ(Pϕ) = {0, 1} 2 (2) In H = L (R, dx), define (Af )(x) = f 0(x) − 2x f (x), f ∈ D(A) = W 1,2( ) 1+x2 R 0 1,2 (Bf )(x) = f (x), f ∈ D(B) = W (R) ⇒ A a B with the bounded intertwining operator T = (I + Q2)−1 σ(A) = σ(B) σp(A) = ∅, σr (A) = {0}, but σ(B) = σc (B) = iR

Jean-Pierre Antoine Metric operators, generalized hermiticity and . . . 10/28 Quasi-similarity with unbounded intertwining operators

A closed (densely defined) operator T : H → K is called an intertwining operator for A and B if

(io0) D(TA) = D(A) ⊂ D(T );

(io1) T : D(A) → D(B);

(io2) BT ξ = TAξ, ∀ ξ ∈ D(A).

First part of condition (io0): ξ ∈ D(A) implies Aξ ∈ D(T ). A is quasi-similar to B, A a B, if there exists a (possibly unbounded) intertwining operator T for A and B which is invertible, with inverse T −1 densely defined ∗ ∗ ∗ Note : A a B 6⇒ B a A , since (io0) may fail for B ⇒ This is a very singular situation ! Indeed if A a B w.r. to T and ρ(A) 6= ∅, then T is bounded A and B are mutually quasi-similar if we have both A a B and B a A, which we denote by A a` B a` is an equivalence relation A a` B ⇒ A∗ a` B∗

Jean-Pierre Antoine Metric operators, generalized hermiticity and . . . 11/28 Quasi-Hermitian operators - 1

Dieudonn´e’sdefinition :A bounded operator A is called quasi-Hermitian if there exists a metric operator G such that GA = A∗G New definitions : A closed operator A is called quasi-Hermitian if there exists a metric operator G such that D(A) ⊂ D(G) and hAξ|Gηi = hGξ|Aηi, ξ, η ∈ D(A) A is strictly quasi-Hermitian if it is quasi-Hermitian and AD(A) ⊂ D(G) ⇔ D(GA) = D(A) Then A strictly quasi-Hermitian ⇔ A a A∗ (1) Bounded quasi-Hermitian operators : If A is bounded, the following statements are equivalent (i) A is quasi-Hermitian (ii) There exists a bounded metric operator G, with bounded inverse, such that GA (= A∗G) is self-adjoint (iii) A is metrically similar to a self-adjoint operator K (2) Unbounded quasi-Hermitian operators : If G is bounded, then A is quasi-Hermitian ⇔ GA is symmetric in H Let G be bounded. If A is self-adjoint in H(G), then GA is symmetric in H. If G −1 is also bounded, A is self-adjoint in H(G) ⇔ GA is self-adjoint in H

Jean-Pierre Antoine Metric operators, generalized hermiticity and . . . 12/28 Quasi-Hermitian operators - 2

(2) Unbounded quasi-Hermitian operators (continued) : Conversely, given A, find a metric operator G that makes A quasi-Hermitian If A is closed and densely defined, the following statements are equivalent (i) ∃G bounded with bounded inverse, s.t. A is self-adjoint in H(G) (ii) ∃G bounded with bounded inverse, s.t. GA = A∗G, i.e., A ∼ A∗ (iii) ∃G bounded with bounded inverse, s.t. G 1/2AG −1/2 is self-adjoint (iv) A is a spectral operator of scalar type with real spectrum : A = R λ dX (λ), where {X (λ)} = spectral family R If A is closed and densely defined, consider the statements (i) ∃G bounded s.t. GD(A) = D(A∗), A∗Gξ = GAξ, ∀ ξ ∈ D(A); in particular, A a A∗ (ii) ∃G bounded s.t G 1/2AG −1/2 is self-adjoint. (iii) ∃G bounded s.t. A is self-adjoint in H(G) (iv) ∃G bounded s.t. GD(A) = D(G −1A∗), A∗Gξ = GAξ, ∀ ξ ∈ D(A); in particular, A a A∗ Then, the following implications hold : (i) ⇒ (ii) ⇒ (iii) ⇒ (iv) If R(A∗) ⊂ D(G −1), then the four conditions (i)-(iv) are equivalent More results if one uses pip-space formalism

Jean-Pierre Antoine Metric operators, generalized hermiticity and . . . 13/28 Pseudo-Hermitian Hamiltonians - 1

Non-self-adjoint Hamiltonians appear in Pseudo-Hermitian quantum mechanics: In general PT -symmetric Hamiltonians, that is, invariant under the joint action of space reflection (P) and complex conjugation (T ) Typical examples : H = p2 + ix 3 and H = p2 − x 4 : PT -symmetric, but non-self-adjoint purely point spectrum, real and positive Usual assumption : H Dieudonn´e-pseudo-Hermitian ⇔ ∃ unbounded metric operator G satisfying the relation H∗G = GH Our assumption : H quasi-Hermitian ⇔ D(H) ⊂ D(G) and hHξ|Gηi = hGξ|Hηi, ∀ ξ, η ∈ D(H)

G bounded ⇒ H a H∗ and G 1/2HG −1/2 is self-adjoint G unbounded and H is strictly quasi-Hermitian ⇒ H a H∗ if, in addition, G −1 is bounded ⇒ G −1H∗Gη = Hη, ∀ η ∈ D(H): restrictive form of similarity

Jean-Pierre Antoine Metric operators, generalized hermiticity and . . . 14/28 Pseudo-Hermitian Hamiltonians - 2

Assume H quasi-Hermitian operator and possesses a (large) set of vectors, ω φ ∈ DG (H), analytic in the norm k·kG and contained in D(G):

∞ n X kH φk n G t < ∞, for some t ∈ n! R n=0

ω 1/2 1/2  ⇒ DG (H) ⊂ D(H) ⊂ D(G) ⊂ D(G ) ⊂ H, with H(G) = D(G ), k·kG Construction of physical system ω HG := completion of (DG (H), k·kG ) : closed subspace HG of H(G) and ω hφ|HψiG = hHφ|ψiG , ∀ φ, ψ ∈ DG (H)

H = densely defined symmetric operator in HG , with a dense set of analytic vectors ⇒ H essentially self-adjoint (Nelson)

⇒ closure H self-adjoint in HG ⇒ (HG , H) = physical system 1/2 ω ω WD = G  DG (H) is isometric from DG (H) into H extends to isometry W = WD : HG → H RanW = Hphys : closed subspace of H W : HG → Hphys is unitary −1 h = W HW is self-adjoint in Hphys : genuine Hamiltonian of the system, acting in the physical Hilbert space Hphys

Jean-Pierre Antoine Metric operators, generalized hermiticity and . . . 15/28 Pseudo-Hermitian Hamiltonians - 3

ω Simplified case: DG (H) dense in H ω W (DG (H)) also dense HG = H(G)

Hphys = H W = G 1/2 unitary from H(G) onto H 1 2 1 2 2 2 Example: Hα = 2 (p − iα) + 2 ω x in H = L (R), for any α ∈ R

Hα has an orthonormal basis of eigenvectors eigenvectors are automatically analytic ⇒ construction applicable

Jean-Pierre Antoine Metric operators, generalized hermiticity and . . . 16/28 II. Metric operators and lattices of Hilbert spaces

Jean-Pierre Antoine Metric operators, generalized hermiticity and . . . 17/28 The lattice generated by a single metric operator - 1

General case : G and G −1 both possibly unbounded

1/2 Define H(RG ) as D(G ) equipped with the following norm, equivalent to the graph norm 2 2 kξk2 = (I + G)1/2ξ  kξk2 := kξk2 + G 1/2ξ RG gr

where RG = 1 + G 2 Define H(G) = completion of H(R ) in the norm kξk2 := G 1/2ξ G G ⇒ H(RG ) = H ∩ H(G), with projective norm × −1 Conjugate dual H(RG ) = H(RG ) −1 −1 ⇒ H(RG ) ⊂ H ⊂ H(RG ) = H + H(G ) ⇒ Full lattice −1 1 H(G )  PP Pq −1 H(R −1 ) H(R ) G P 1 G PPq  1 H P  PPq H(R ) H(R−1 ) G P 1 G −1 PPq  H(G)

Jean-Pierre Antoine Metric operators, generalized hermiticity and . . . 18/28 The lattice generated by a single metric operator - 2

2 2 Example : G = x in L (R, dx)

Corresponding lattice : 2 −2 1 L (R, x dx)  PPPq 2 −2 2 2 −1 L (R, (1 + x ) dx) P 1 L (R, (1 + x ) ) dx) PPq  2 1 L (R, dx) P  PPq L2( , (1 + x2) dx) L2( , (1 + x−2)−1 dx) R P 1 R PPq  2 2 L (R, x dx)

−1 G unbounded ⇒ RG = 1 + G > 1 and RG bounded −1 ⇒ triplet H(RG ) ⊂ H ⊂ H(RG ) 1/2 Iteration ⇒ infinite Hilbert scale on powers of RG , n/2 × Hn = D(RG ), n ∈ N, and H−n = Hn

... ⊂ H2 ⊂ H1 ⊂ H ⊂ H−1 ⊂ H−2 ⊂ ... 2 Examples: H = L (R, dx) 2 1/2 1/2 2 Gx = (1 + x ) , H∞(Gx ) : fast decreasing L functions 2 2 1/2 1/2 Gp = (1 − d /dx ) = FGx F−1, H∞(Gp ) : Sobolev spaces

Jean-Pierre Antoine Metric operators, generalized hermiticity and . . . 19/28 The LHS generated by metric operators - 1

Define M(H) = {all metric operators}

Mb(H) = {bounded metric operators}

Natural order on M(H): G1  G2 ⇔ ∃ γ > 0 such that G2 6 γG1 ⇔ H(G1) ⊂ H(G2) and the embedding is continuous with dense range. Consequences :

−1 −1 G2  G1 ⇔ G1  G2 if G1, G2 ∈ M(H)

−1 G  I  G, ∀ G ∈ Mb(H) ⇒ Let X , Y ∈ M(H). Then, X  Y ⇔ H(X ) ,→ H(Y ) We will show that the spaces {H(X ); X ∈ M(H)} constitute a lattice of Hilbert spaces (LHS)

Jean-Pierre Antoine Metric operators, generalized hermiticity and . . . 20/28 The LHS generated by metric operators - 2

Let O ⊂ M(H) be a family of metric operators, containing I , and assume that \ 1/2 D := D(G ) G∈O is a dense subspace of H Every operator G ∈ O is self-adjoint and invertible

⇒ on D, define the graph topology tO by means of the norms

ξ ∈ D 7→ kG 1/2ξk, G ∈ O

× × D := conjugate dual of D[tO], with strong dual topology tO

× × ⇒ D[tO] ,→ H ,→ D [tO] the Rigged Hilbert Space associated to O Then O generates a canonical lattice of Hilbert spaces (LHS) interpolating between D and D×

Jean-Pierre Antoine Metric operators, generalized hermiticity and . . . 21/28 The LHS generated by metric operators - 3

On the family {H(X ); X ∈ O} define the lattice operations as

H(X ∧ Y ) := H(X ) ∩ H(Y ) H(X ∨ Y ) := H(X ) + H(Y )

The corresponding operators read as

X ∧ Y := X u Y −1 −1 −1 X ∨ Y := (X u Y )

Here u stands for the form sum and X , Y ∈ O : given two positive operators, T := T1 u T2 is the positive operator associated to the quadratic form t = t1 + t2 Both X ∨ Y and X ∧ Y are metric operators, but they need not belong to O However, for O = M(H), the corresponding family M(H) is a lattice by \ 1/2 itself, but D := D(G ) could be {0} G∈M(H)

Jean-Pierre Antoine Metric operators, generalized hermiticity and . . . 22/28 The LHS generated by metric operators - 4

±1/2 T Define R = {G , G ∈ O} and the domain DR := X ∈R D(X ) Let Σ = minimal set of self-adjoint operators containing O, stable under inversion and form sums, s.t. DR is dense in H(Z), for every Z ∈ Σ

⇒O generates a lattice of Hilbert spaces J := JΣ = {H(X ), X ∈ Σ} and a pip-space VΣ with central Hilbert space H = H(I ): P # V = G∈Σ H(G), “smallest” space V = DR Compatibility : ξ#η ⇔ ∃ G ∈ Σ s.t. ξ ∈ H(G), η ∈ H(G −1) 1/2 −1/2 Partial inner product : hξ|ηi ≡ hξ|ηiΣ := hG ξ|G ηiH, ξ#η Example : for O = {I , G}, the lattice Σ consists of nine operators :

−1 1 G  PPPq I ∧ G −1 I ∨ G −1 1 PP 1 Pq  PPPq −1 I −1 G ∧ G 1 P 1 G ∨ G PPPq  PPq I ∧ G I ∨ G P 1 PPq  G

Jean-Pierre Antoine Metric operators, generalized hermiticity and . . . 23/28 The LHS generated by metric operators - 5

Operators on VΣ

A ∈ Op(VΣ) ⇔ j(A) = {(X , Y ) ∈ Σ × Σ} such that A : H(X ) → H(Y ), continuously (i.e. bounded)

A '{AYX :(X , Y ) ∈ j(A)} : (maximal) coherent family of representatives (E.. ' identity):

H(W ) ⊂ H(X ), H(Y ) ⊂ H(Z) ⇒ AZW = EZY AYX EXW More generally, (X , Y ) ∈ j(A) ⇔ Y 1/2AX −1/2 is bounded in H Adjoint operator A× :(X , Y ) ∈ j(A) ⇒ (Y −1, X −1) ∈ j(A×) and

hA×η|ξi = hη|Aξi, for ξ ∈ H(X ), η ∈ H(Y −1)

In particular, (X , X ) ∈ j(A) ⇒ (X −1, X −1) ∈ j(A×) Symmetric operator: A = A× ⇒ (X , X ) ∈ j(A) ⇒ (X −1, X −1) ∈ j(A) ⇒ by interpolation, (I , I ) ∈ j(A), that is, A has a bounded representative AII : H → H

Jean-Pierre Antoine Metric operators, generalized hermiticity and . . . 24/28 (Quasi)-similarity of pip-space operators

(1) Let (G, G) ∈ j(A), G ∈ M(H) 1/2 −1/2 Then B = G AGG G is bounded on H and AGG a B (2) Let (G, G) ∈ j(A), G bounded, with G −1 unbounded, so that H(G −1) ⊂ H ⊂ H(G)

Consider the restriction A of AGG to H and assume D(A) = {ξ ∈ H : Aξ ∈ H} is dense in H ⇒ G 1/2 : D(A) → D(B) and B G 1/2η = G 1/2A η, ∀ η ∈ D(A), i.e.A a B (both acting in H) B G 1/2η = G 1/2A η, ∀ η ∈ H(G) and G 1/2 : H(G) → H unitary operator ⇒ A and B are unitarily equivalent (but acting in different Hilbert spaces) (3) Let (G, G) ∈ j(A), G unbounded, with G −1 bounded, so that H(G) ⊂ H ⊂ H(G −1) ⇒ A : H(G) → H(G) = densely defined operator in H 1/2 −1/2 B = G AGG G is bounded and everywhere defined on H −1/2 −1/2 G Bξ = AGG G ξ, ∀ ξ ∈ H, i.e.,B a AGG u ±1/2 By (1), AGG a` B; in addition AGG ∼ B, since G are unitary between H and H(G)

Jean-Pierre Antoine Metric operators, generalized hermiticity and . . . 25/28 The case of symmetric pip-space operators - 1

× A ∈ Op(VJ ) symmetric if A = A Possibility of self-adjoint restrictions to H = candidates for quantum observables If A = A×, then (G, G) ∈ j(A) ⇔ (G −1, G −1) ∈ j(A) ⇒ (I , I ) ∈ j(A), thus ⇒ Every symmetric operator A ∈ Op(VJ ) such that( G, G) ∈ j(A), with G ∈ M(H), has a bounded restriction AII to H Assumption (G, G) ∈ j(A) too strong for applications !

−1 Assume instead that (G , G) ∈ j(A), G ∈ Mb(H) with unbounded inverse, so that H(G −1) ⊂ H ⊂ H(G) ⇒ Can apply the KLMN theorem: Given a symmetric operator A = A×, assume there is a metric operator G ∈ Mb(H) with an unbounded inverse, for which there exists a λ ∈ R such that A − λI has a boundedly invertible −1 representative (A − λI )GG −1 : H(G ) → H(G). Then AGG −1 has a unique restriction to a self-adjoint operator A in the Hilbert space H, with dense domain D(A) = {ξ ∈ H : Aξ ∈ H}. In addition, λ ∈ ρ(A).

Jean-Pierre Antoine Metric operators, generalized hermiticity and . . . 26/28 The case of symmetric pip-space operators - 2

If there is no bounded G as before, i.e. (G −1, G) ∈ j(A), one can use the KLMN theorem in the Hilbert scale VG built on the powers of −1/2 −1/2 G or (RG ) : Let VG = {Hn, n ∈ Z} be the Hilbert scale built on the powers of the ±1/2 −1/2 operator G or (RG ) , depending on the (un)boundedness of ±1 × G ∈ M(H) and let A = A be a symmetric operator in VG . (i) Assume there is a λ ∈ R such that A − λI has a boundedly invertible representative (A − λI )nm : Hm → Hn, with Hm ⊂ Hn. Then Anm has a unique restriction to a self-adjoint operator A in the Hilbert space H, with dense domain D(A) = {ξ ∈ H : Aξ ∈ H}. In addition, λ ∈ ρ(A).

(ii) If the natural embedding Hm → Hn is compact, the operator A has a purely point spectrum of finite multiplicity, thus σ(A) = σp(A), mA(λj ) < ∞ for every λj ∈ σp(A) and σc (A) = ∅.

No known (quasi-)similarity relation between AGG −1 or A and another operator ! On the contrary, under previous assumption A : H(G −1) → H(G), 1/2 1/2 B = G AGG −1 G is bounded on H, but AGG −1 6a B: −1/2 (io1) imposes T = G , hence unbounded, but then conditions (io0) and (io2) cannot be satisfied

Jean-Pierre Antoine Metric operators, generalized hermiticity and . . . 27/28 References

J-P. Antoine and C. Trapani, Partial Inner Product Spaces – Theory and Applications, Lecture Notes in Mathematics, vol. 1986, Springer, Berlin-Heidelberg, 2009

J-P. Antoine and C. Trapani, Partial inner product spaces, metric operators and generalized hermiticity, J. Phys. A: Math. Theor. 46 (2013) 025204; Corrigendum, Ibid. 46 (2013) 329501

J-P. Antoine and C. Trapani, Some remarks on quasi-Hermitian operators, J. Math. Phys. 55 (2014) 013503

J-P. Antoine and C. Trapani, Metric operators, generalized hermiticity and lattices of Hilbert spaces, Chap. 7 in Mathematical Aspects of Non-Selfadjoint Operators in Quantum Physics, pp. 345–402; F. Bagarello, J-P. Gazeau, F.H. Szafraniec and M. Znojil (eds.), J. Wiley, Hoboken, NJ, 2015

C.M. Bender, A. Fring, U. G¨untherand H. Jones, Quantum physics with non-Hermitian operators, J. Phys. A: Math. Theor. 45 (2012) 440301

A. Mostafazadeh, Pseudo–Hermitian quantum mechanics with unbounded metric operators, Phil. Trans. R. Soc. Lond. 371 (2013) 20120050

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