Generalized Riesz systems and orthonormal sequences in Krein spaces

We analyse special classes of biorthogonal sets of vectors in Hilbert and in Krein spaces, and their relations with $\mathcal{G}$- quasi bases. We also discuss their relevance in some concrete quantum mechanical system driven by manifestly non self-adjoint Hamiltonians.


Introduction
The employing of non self-adjoint operators for the description of experimentally observable data goes back to the early days of quantum mechanics. In the past twenty years, the steady interest in this subject grew considerably after it has been discovered [10,12] that the spectrum of the manifestly non self-adjoint Hamiltonian is real. It was conjectured [10] that the reality of eigenvalues of H is a consequence of its PT -symmetry: PT H = HPT , where the space parity operator P and the complex conjugation operator T are defined as follows: (Pf )(x) = f (−x) and (T f )(x) = f (x). This gave rise Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
to a consistent development of theory of PT -symmetric Hamiltonians in quantum physics, see [4,5,11] and references therein.
Usually, PT -symmetric Hamiltonians can be interpreted as self-adjoint ones for a suitable choice of indefinite inner product. For instance, the operator H in (1) is self-adjoint with respect to the indefinite inner product [ f , g] = ∞ −∞ f (−x)g(x)dx in the Krein space (L 2 (R), [·, ·]) (see section 3.1 for the definition of Krein spaces). The eigenstates of H lose the property of being Riesz basis in the original Hilbert space L 2 (R) but they still form a complete set in L 2 (R) [18,21]. Moreover, they form a sequence which is orthonormal with respect to the indefinite inner product [·, ·]. Such kind of phenomenon is typical for PT -symmetric Hamiltonians and it gives rise to a natural problem: What can we say about properties of vectors which form a complete set in a Hilbert space and, additionally, are orthonormal with respect to the indefinite inner product?
The main objective of the paper is to investigate this problem with the use of theory of generalized Riesz systems (GRS) and G -quasi bases. These two concepts were originally introduced in [15] and [2], respectively and then analyzed in a series of papers, see, e.g. [4, chapter 3], [6,7,14]. The motivation for introducing GRS and G -quasi bases was the need to put on a mathematically rigorous ground several physical models where the eigenstates of some non self-adjoint operator were usually claimed to be bases, while they were not.
The paper is structured as follows. Section 2 contains facts related to GRS and G -quasi bases. We slightly change the original definition of GRS given in [15] putting in evidence the role of a self-adjoint operator Q, since it is more convenient for our purpose: a sequence {φ n } in a Hilbert space H is called a generalized Riesz system (GRS) if there exists a self-adjoint operator Q in H and an orthonormal basis (ONB) {e n } such that e n ∈ D(e Q/2 ) ∩ D(e −Q/2 ) and φ n = e Q/2 e n .
A GRS {φ n } is a Riesz basis if and only if Q is a bounded operator. For each GRS {φ n }, the dual GRS is determined by the formula {ψ n = e −Q/2 e n }. The dual GRS {φ n } and {ψ n } are biorthogonal.
In section 2.3, the phenomenon of nonuniqueness of a self-adjoint operator Q in the formulas φ n = e Q/2 e n , ψ n = e −Q/2 e n (2) is investigated in the case of regular biorthogonal sequences {φ n } and {ψ n }. We firstly prove that the operator Q and ONB {e n } are determined uniquely for dual bases. In general, this property does not hold for regular biorthogonal sequences. To describe all possible selfadjoint operators Q in (2) we consider a positive densely defined operator G 0 which acts as G 0 φ n = ψ n and then is extended on D(G 0 ) = span{φ n } by the linearity. The proved statement is: for given {φ n } and {ψ n }, the set of admissible operators Q in (2) is in one-to-one correspondence with the set of extremal extensions G of G 0 , precisely, Q = − ln G. This fact allows one to characterize the important case where Q is determined uniquely. Section 3 contains the main results. We begin with the simple fact that each complete sequence {φ n } which is orthonormal in a Krein space (H, [·, ·]) (briefly, J-orthonormal) is a GRS and therefore, it can be expressed by (2) for some choice of Q. If Q is uniquely determined, then the anti-commutation relation JQ = −QJ holds. For this reason, the anti-commutation relation is reasonable to keep in the general case (if Q is not determined uniquely, then we cannot state, in general, that JQ = −QJ): we say that a complete J-orthonormal sequence {φ n } is of the first type if there exists Q in (2) such that JQ = −QJ. Otherwise, {φ n } is of the second type.
We proved that the formula (2) defines first type sequences if and only if Q anticommutes with J and elements of ONB {e n } are eigenvectors of J.
The first type sequences have a lot of useful properties. One of benefits is the fact that a first type sequence generates a C-symmetry operator C = e Q J where Q is the same operator as in (2). The latter allows one to construct the Hilbert space (H −Q , ·, · −Q ) involving {φ n } as ONB, directly as the completion of D(C) with respect to 'CPT -norm': For a second type sequence, the inner product ·, · −Q generated by an operator Q from (2) cannot be expressed via [·, ·] and one should apply more efforts for the determination of ·, · −Q , see section 3.2.
Assume that a complete J-orthonormal sequence {φ n } consists of eigenfunctions of a Hamiltonian H corresponding to real eigenvalues {λ n }. The sequence {φ n } generates C-symmetry operators (proposition 3.9). If {φ n } is of the first type, then there exists at least one C such that the operator H restricted on span{φ n } turns out to be essential self-adjoint in the Hilbert space H −Q with the inner product (3). The spectrum of the closure of H in H −Q coincides with the closure of {λ n }. Hence, we construct an isospectral realization of H in H −Q .
For a second type sequence, each operator C generated by {φ n } gives rise to the Hilbert space (H −Q , ·, · −Q ) with non-densely defined symmetric operator H. Its extensions to selfadjoint operators in H −Q lead to the appearance of new spectral points. Therefore, self-adjoint realizations constructed with the use of CPT -norm cannot be isospectral. The isospectrality of self-adjoint realizations of H in H −Q can be achieved via the construction of the Friedrichs extension G = e −Q of the symmetric operator G 0 φ n = [φ n , φ n ]Jφ n defined on span{φ n }, that is quite complicated problem.
Section 4 contains examples of J-orthonormal sequences. We show that the eigenstates of the shifted harmonic oscillator constitute a first type sequence. This example leads to the following conjecture: eigenstates of a PT -symmetric Hamiltonian H with unbroken PT -symmetry [11, p 41] form a first type sequence.
In what follows, H means a complex Hilbert space with inner product linear in the first argument. Sometimes, it is useful to specify the inner product ·, · associated with H. In that case the notation (H, ·, · ) has already been used, and will be used in the following. All operators in H are supposed to be linear, the identity operator is denoted by I. The symbols D(A) and R(A) denote the domain and the range of a linear operator A. An operator A is called

Generalized Riesz systems in Hilbert spaces
Let {φ n } be a Riesz basis in H. Then there exists a bounded and boundedly invertible operator R such that φ n = Re n , where {e n } is an orthonormal basis (ONB) of H. The operator RR * is positive and self-adjoint in H and it admits the presentation RR * = e Q , where Q is a bounded self-adjoint operator. The polar decomposition of R has the form R = √ RR * U = e Q/2 U, where U is a unitary operator in H. The unitarity of U means that {e n = Ue n } is an ONB of H and we can rewrite the definition of Riesz bases as follows: a sequence {φ n } is called a Riesz basis if there exists a bounded self-adjoint operator Q in H and an ONB {e n } such that φ n = e Q/2 e n . This simple observation leads to: Definition 2.1. A sequence {φ n } is called a generalized Riesz system (GRS) if there exists a self-adjoint operator Q in H and an ONB {e n } such that e n ∈ D(e Q/2 ) ∩ D(e −Q/2 ) and φ n = e Q/2 e n .
Let {φ n } be a GRS. In view of definition 2.1, the sequence {ψ n = e −Q/2 e n } is well defined and it is a biorthogonal sequence for {φ n = e Q/2 e n }. Obviously, {ψ n } is a GRS which we will call a dual GRS. Dual GRS are Riesz bases if and only if Q is a bounded operator.

Example 1.
A first simple example of GRS can be extracted from [6]: if we take Q = − x 2 2 , x being the position operator, it is clear that D(e Q/2 ) = L 2 (R), while This set is dense in L 2 (R), since contains each eigenfunction of the quantum harmonic oscillator The Hermite functions {e n (x)} form an orthonormal basis of L 2 (R). In general, the inner product is not equivalent to ·, · and the linear space Generally, the Hilbert spaces H −Q and H Q differ from H (as the sets of elements). We can just say, in view of (5) and (6)  Proof. In view of (5), Therefore, By construction, the vectors φ n belong to Proof. Due to (5), the sequence {φ n = e Q/2 e n } is orthonormal in H −Q . Its completeness follows from lemma 2.2. The case {ψ n } is considered similarly with the use of (6). □ Dual GRS could be used to define manifestly non self-adjoint Hamiltonians with known complex eigenvalues {λ n } and eigenvectors {φ n } and {ψ n }, respectively. We refer to [6,7] for the connection between H φ,ψ and the adjoint of H ψ,φ and for the analysis of ladder operators associated to similar bi-orthogonal sets, and how these ladder operators can be used to factorize the Hamiltonians above.

Dual GRS and G -quasi bases
Dual GRS {φ n } and {ψ n } can be considered as examples of more general object: G -quasi bases. These are biorthogonal sets originally introduced in [2], and then analyzed in a series of papers (see [4] for a relatively recent review).
the following holds: The last relations yield To complete the proof, it suffices notice that G is dense in H since each vector of ONB {e n } belongs to D(e Q/2 ) ∩ D(e −Q/2 ). □ Remark 2.6. Proposition 2.5 implies that example 1 above of GRS provides also an example of G -quasi bases, with G = D(e x 2 /4 ), in agreement with what was found in [6].

Regular biorthogonal sequences and dual GRS
We say that biorthogonal sequences {φ n } and {ψ n } are regular if {φ n } and {ψ n } are complete sets in H. In other words, a biorthogonal sequence {ψ n } is defined uniquely by {φ n } and vice versa. Proof. Let {φ n } and {ψ n } be regular biorthogonal sequences. Then an operator G 0 defined initially as and extended on D(G 0 ) = span{φ n } by the linearity is densely defined and positive. The later follows from the fact that (10), e n = e Q/2 ψ n . Therefore, e n ∈ D(e Q/2 ) ∩ D(e −Q/2 ) and φ n = e Q/2 e n , ψ n = e −Q/2 e n .
The sequence {e n } is orthonormal in H since e n , e m = e −Q/2 φ n , e Q/2 ψ m = φ n , ψ m = δ nm .
Let us assume that γ ∈ H is orthogonal to {e n }. Then there exists a sequence {f m } ( f m ∈ D(e −Q )) such that e −Q/2 f m → γ in H (because e −Q/2 D(e −Q ) is a dense set in H). In this case, due to (7), {f m } is a Cauchy sequence in H −Q and therefore, f m tends to some f ∈ H −Q . This means that We note that the set D(G 0 ) = span{φ n } is dense in the Hilbert space  Proof. Let γ be orthogonal to R(G 0 + I). Then, in view of (10), γ, φ n = − γ, ψ n and the basis property of {φ n } and {ψ n } leads to the relation: By virtue of (13), the sequence γ m = m n=1 γ, ψ n φ n tends to γ, while G 0 γ m = m n=1 γ, ψ n ψ n tends to −γ . Therefore, that is possible when γ = 0. Hence, R(G 0 + I) is a dense set in H and, as a result, G 0 is an essentially self-adjoint operator in H. Its closure G 0 gives a unique positive self-adjoint extension G which determines a unique self-adjoint operator Q = − ln G (i.e. G = e −Q ). Moreover, because of the equality e n = e −Q/2 φ n , the ONB {e n } is also determined uniquely. □ In view of proposition 2.9 a natural question arise: is the operator Q determined uniquely for a given GRS {φ n }?
The choice of the Friedrichs extension G = e −Q of G 0 in the proof of theorem 2.7 was inspired by the fact that the sequence {φ n } must be complete in the Hilbert space H −Q (that, in view of (12) and lemma 2.2, is equivalent to the completeness of orthonormal system {e n } in H). Generally, there are many self-adjoint extensions G of G 0 which preserve this property and each of them can be used instead of the Friedrichs extension.
We recall [1] that a nonnegative self-adjoint extension G of G 0 is called extremal if −Q due to (7). For this reason, the definition of extremal extensions can be rewritten as follows: let G 0 be determined by (10), where {φ n } and {ψ n } are regular biorthogonal sequences. A self- Therefore, the extremality of a self-adjoint extension e −Q of G 0 is equivalent to the completeness of the sequence {φ n } in H −Q . This means that for each extremal self-adjoint extension e −Q one can repeat the proof of theorem 2.7 and establish the relations (11) where G * 0 means the adjoint operator of G 0 with respect to ·, · .
Proof. It suffices to establish the equivalence (ii) and (iii). Indeed, the set of extremal extensions involves the Friedrichs G F and the Krein-von Neumann G K extensions of G 0 and it contains only one element when G F = G K [1]. The last equality is equivalent to (14) due to [17, theorem 9]. □

Elements of the Krein spaces theory
Here The principal difference between the initial inner product ·, · and the indefinite inner product [·, ·] is that there exist nonzero elements respectively. A closed subspace L of the Hilbert space (H, ·, · ) is called positive or negative if all nonzero elements f ∈ L are, respectively, positive or negative. A positive (negative) subspace L is called uniformly positive (uniformly negative) if there exists α > 0 such that In each of these classes we can define maximal subspaces. For instance, a positive subspace L is called maximal positive if L is not a subspace of another positive subspace in H. The maximality of a (negative, uniformly positive, uniformly negative) closed subspace is defined similarly.
Let a subspace L be a maximal positive (negative). Then its orthogonal complement with respect to the indefinite inner product [·, ·] is a maximal negative (positive) subspace and the J-orthogonal sum is dense in the Hilbert space (H, ·, · ) (the symbol [+] in (15) indicates that the subspaces L and L [⊥] are orthogonal with respect to [·, ·], i.e. J-orthogonal).
if and only if L is a maximal uniformly positive (uniformly negative) subspace (in this case, L [⊥] is maximal uniformly negative (uniformly positive)).  (16) is uniquely determined by a bounded operator C which coincides with the identity operator on the positive subspace L + := L and with the minus identity operator on the negative subspace L − := L [⊥] . By the construction, L ± = (I ± C)H and C 2 = I. Moreover, the operator JC is positive self-adjoint since Summing up: the fundamental decompositions (16)

of a Krein space are in one-to-one correspondence with the set of bounded operators
The J-orthogonal sums (15) of maximal positive/maximal negative subspaces are in oneto-one correspondence with the set of unbounded operators C = Je −Q = e Q J . In both cases, Q anticommutes with J.
The operator C is called a C-symmetry operator and this notion is widely used in PT -symmetric approach in quantum mechanics [11]. (17) is unbounded, then we understood (17) as the identity JQf = −QJf, where f ∈ D(Q) and J leaves D(Q) invariant. From now on, we will adopt this simplifying notation.

Remark 3.2. If Q in
A C-symmetry operator allows one to define a new inner product via the indefinite inner product [·, ·]: The corresponding norm · −Q is equivalent to the original norm of H when C is bounded. If C is unbounded, then the completion of D(C) with respect to · −Q leads to the Hilbert space (H −Q , ·, · −Q ) defined in section 2.

Remark 3.3.
It is maybe worth mentioning that the unboundedness of the C operator, and of the related metric, is a serious issue in PT quantum mechanics. For example, [4], it may happen that the basis property of the eigenvectors of a PT -symmetric Hamiltonian, obtained by considering a suitable deformation of a self-adjoint operator, is lost. This is the case, for instance, of the Swanson model and of the shifted harmonic oscillator, [3,8,9]. We meet similar difficulties also when working with Krein spaces, as it will appear clear in the remaining part of the paper.

J-orthonormal sequences of the first and of the second type
Obviously, {ψ n } is J-orthonormal and [φ n , φ n ] = [ψ n , ψ n ]. In view of (19), the positive symmetric operator G 0 in (10) acts as In what follows we assume that {φ n } is complete in the Hilbert space H. Then {ψ n } in (19) is complete too. Therefore, {φ n } and {ψ n } are regular biorthogonal sequences and, by theorem 2.7, they are dual GRS. Thus, each complete J-orthonormal sequence is a GRS. The corresponding operator Q = − ln G in (11) can be determined by every extremal extension G = e −Q of G 0 . Such kind of freedom allows us to select an appropriative operator Q which fits well with the J-orthonormality of {φ n }. Theorem 3.4. Let {φ n } be a complete J-orthonormal sequence. If a self-adjoint operator Q in (11) is determined uniquely, then the relation (17) holds.
Proof. Separating the sequence {φ n } by the signs of [φ n , φ n ]: By proposition 2.10, the uniqueness of Q means that the symmetric operator G 0 has a unique extremal extension G = e −Q . This is possible when G coincides with the Friedrichs extension of G 0 as well as with the Krein-von Neumann extension of G 0 . This fact, by virtue of [19, theorem 4.3], means that Je −Q f = e Q Jf for f ∈ D(e −Q ). The last relation and [19, theorem 2.1] justify (17). □ 4 In [19], the notation G 0 is used for G 0 .
In view of theorem 3.4, it seems natural to consider the anti-commutation relation (17) in the case where Q is not determined uniquely. Taking into account that (17) is equivalent to the relation where G = e −Q is an extremal extension of G 0 , we reduce the choice of Q which satisfies (17) to the choice of an extremal extension G satisfying (22). If extremal extensions G of G 0 are not determined uniquely, then not each Q = − ln G will anticommute necessarily with J. In particular, the operator Q that corresponds to the Friedrichs extension G = e −Q of G 0 does not satisfy (17) [16].  (11) is the first type. In particular, every J-orthonormal basis is a first type sequence. The example of a second type sequence can be found in [16, section 6.2]. In what follows, considering a first type sequence, we assume that Q anti-commutes with J. Proof. (i)→(ii). By virtue of (11) and (19), Comparing the third and the fifth terms in the equality above we get Je n = [φ n , φ n ]e n that implies (ii). The studies of the first type sequences began in [16], where they were called 'quasi bases'. Proposition 3.6 is a part of [16, theorem 6.3]. We present here a simpler proof.
For the first type sequence, the inner product in (H −Q , ·, · −Q ) is directly determined by the known indefinite inner product [·, ·], see (24) below. Let us briefly explain this important fact (see [16] for details).
Since Q anticommutes with J, the J-orthogonal sum L 0  (5) imply that for f = (I + Je −Q )u and g = (I + Je −Q )v from L +: Therefore, the indefinite inner product [·, ·] coincides with ·, · −Q on L +. Similar calculations show that [·, ·] coincides with − ·, · −Q on L −. Moreover, the subspaces L ± are orthogonal with respect to ·, · −Q since where γ = (I − Je −Q )w and u, w ∈ D(e −Q ). This leads to the conclusion that where L ± are the completion of the pre-Hilbert spaces (L ± , ±[·, ·]) and [⊕ −Q ] indicates the orthogonality with respect to ·, · −Q and with respect to [·, ·]. Keeping the same notation for the extension of [·, ·] onto H −Q we obtain the new Krein space (H −Q , [·, ·]) with the fundamental decomposition (23).
For the second type sequences, there are no operators Q in (11) which anticommute with J. The space H −Q cannot be presented as in (23). This implies that ·, · −Q cannot be directly expressed via [·, ·] and one should apply much more efforts for calculation of ·, · −Q .

J-orthonormal sequences and operators of C-symmetry
We say that an J-orthonormal sequence {φ n } generates a C-symmetry operator C = Je −Q = e Q J (the operator Q anti-commutes with J) if n are defined in (21). With each operator C one can associate a Hilbert space (H −Q , ·, · −Q ) (see section 3.1). In view of (18), Therefore, {φ n } is an orthonormal system in H −Q . Proposition 3.9 and corollary 3.10 follow from [16, sections 5 and 6]. Obviously, if {φ n } is the first type, then its bi-orthogonal sequence {ψ n } is also the first type. Remark 3.11. Corollary 3.10 can be easy extended for the general case of GRS {φ n = e Q/2 e n }. Indeed, in view of proposition 2.3, {φ n } and {ψ n } are ONB of the Hilbert spaces H −Q and H Q , respectively. Therefore, the operators H φ,ψ and H ψ,φ defined on span{φ n } and span{ψ n } have to be essentially self-adjoint in H −Q and H Q . This approach can be used for J-orthonormal sequences of the second type. The principal difference is: for the first type sequence, the new scalar product in H −Q is directly determined by the known indefinite inner product [·, ·], see (24). For the second type sequence, the inner product ·, · −Q cannot be expressed via [·, ·] and it becomes more complicated to determine ·, · −Q . respectively. The functions {φ n } and {ψ n } are determined by (11) with the operator of multiplication Q = 2p(x) in L 2 (R) which anticommutes with P . Proposition 3.6 implies that {φ n } and {ψ n } are P -orhonormal and they are sequences of the first type.