On Dual Definite Subspaces in Krein Space

Extensions of dual definite subspaces to dual maximal definite ones are described. The obtained results are applied to the classification of C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {C}$$\end{document}-symmetries. The concepts of dual quasi maximal subspaces and quasi bases are introduced and studied. It is shown that complex shift g(·)→g(·+ia)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g(\cdot )\rightarrow {g}(\cdot +ia)$$\end{document} of Hermite functions gn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_n$$\end{document} is an example of quasi bases in L2(R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_2({\mathbb {R}})$$\end{document}.

A (closed) subspace L of the Hilbert space H is called nonnegative, positive, uniformly positive with respect to the indefinite inner product [·, ·] if, respectively, Nonpositive, negative and uniformly negative subspaces are introduced similarly. In each of the above mentioned classes we can define maximal subspaces. For instance, a closed positive subspace L is called maximal positive if L is not a proper subspace of a positive subspace in H. The concept of maximality for other classes of closed subspaces is defined similarly. A subspace L of H is called definite if it is either positive or negative. The term uniformly definite is defined accordingly.
Subspaces L ± of H are called dual subspaces if L − is nonpositive, L + is nonnegative, and L ± are orthogonal with respect to [·, ·], that is [ f + , f − ] = 0 for all f + ∈ L + and all f − ∈ L − . The subject of the paper is dual definite subspaces. Our attention is mainly focused on dual definite subspaces L ± with additional assumption of the density of their direct At first glance, the density of D in H should imply the maximality of definite subspaces L ± in the Krein space (H, [·, ·]). This assumption is true when L ± are uniformly definite. For the case of dual definite subspaces, Langer [13] constructed a densely defined sum (1.2) for which there exist various extensions to dual maximal definite subspaces L max ± : The decomposition (1.2) is often appeared in the spectral theory of PT -symmetric Hamiltonians [9] as the result of closure of linear spans of positive and negative eigenfunctions and it is closely related to the concept of C-symmetry in PT -symmetric quantum mechanics (PTQM) [7,8]. The description of symmetries C is one of the key points in PTQM and it can be successfully implemented only in the case where the dual subspaces in (1.2) are maximal. This observation give rise to a natural question: how to describe all possible extensions of dual definite subspaces L ± to dual maximal definite subspaces L max ± ? In Sect. 2 this problem is solved with the use of Krein's results on non-densely defined Hermitian contractions [4,11]. The main result (Theorem 2.6) reduces the description of extensions (1.3) to the solution of the operator equation (2.10).
Each direct sum D max of dual maximal definite subspaces L max ± generates an associated Hilbert space (H G , (·, ·) G ) defined in Sect. 3. If L max ± are uniformly definite, then D max coincides with H and H = H G [since the inner product (·, ·) G is equivalent to the original one (·, ·)]. On the other hand, if L max ± are definite subspaces, then H = H G and (·, ·) G is not equivalent to (·, ·). In this case, the direct sum D ⊂ D max may be nondensely defined in the Hilbert space (H G , (·, ·) G ) constructed by D max .
We say that dual definite subspaces L ± are quasi maximal if there exists an extension (1.3) such that the set D remains dense in the Hilbert space (H G , (·, ·) G ) constructed by D max .
In Sect. 4, dual quasi maximal subspaces are characterized in terms of extremal extensions of symmetric operators: Theorems 4.2, 4.5, Corollary 4.6. The theory of extremal extensions [2,3] allows one to classify all possible cases: a Hilbert spaces H G which preserve the density of D exists and is uniquely defined (A); exists and is non-uniquely defined (B); does not exist (C).
Section 5 deals with the operator of C-symmetry. Each pair of dual definite subspaces L ± determines by (5.1) an operator C 0 such that C 2 0 = I and J C 0 is a positive symmetric operator in H. The operator C 0 is called an operator of C-symmetry if J C 0 is a self-adjoint operator in H. In this case, the notation C is used instead of C 0 .
Let C 0 be an operator associated with dual definite subspaces L ± . Its extension to the operator of C-symmetry is equivalent to the construction of dual maximal definite subspaces L max ± in (1.3). This relationship allows one to use the classification (A)−(C) in Sect. 4 for the solution of the following problems: (i) how many operators of Csymmetry can be constructed on the base of dual definite subspaces L ± ? (ii) is it possible to define an operator of C-symmetry as the extension of C 0 by continuity in the new Hilbert space (H G , (·, ·) G )?
The concept of dual quasi maximal subspaces allows one to introduce quasi bases in Sect. 6. The characteristic properties of quasi bases are presented in Theorem 6.3 and Corollaries 6.4-6.6. The relevant examples are given. In particular, complex shifts of eigenfunctions of the harmonic oscillator are considered.
In what follows D(H ), R(H ), and ker H denote, respectively, the domain, the range, and the kernel space of a linear operator H . The symbol H D means the restriction of H onto a set D. Let H be a Hilbert space. Sometimes, it is useful to specify the inner product (·, ·) endowed with H. In this case the notation (H, (·, ·)) will be used.

Dual Maximal Subspaces
Let (H, [·, ·]) be a Krein space with fundamental symmetry J . Denote The subspaces H ± of H are orthogonal with respect to the initial inner product (·, ·) as well as with respect to the indefinite inner product where K + : H + → H − and K − : H − → H + are strong contractions 2 with domains D(K + ) = M + ⊆ H + and D(K − ) = M − ⊆ H − , respectively. Therefore, the pair of subspaces L ± is determined by the formula and P + = 1 2 (I + J ) and P − = 1 2 (I − J ) are orthogonal projection operators on H + and H − , respectively.
By the construction, T 0 is a strong contraction in H such that Proof It sufficient to establish that the orthogonality of L ± with respect to [·, ·] is equivalent to the symmetricity of T 0 . For every x ± ∈ M ± , This relation is equivalent to for every f = x + + x − and g = y + + y − from the domain of T 0 . Therefore, T 0 is a symmetric operator.
The operator T 0 characterizes the 'deviation' of definite subspaces L ± with respect to H ± and it allows to characterize the additional properties of L ± . Lemma 2.2 [10] Let L ± be dual definite subspaces (i.e., the operator T 0 satisfies the condition of Lemma 2.1). Then By virtue of Lemmas 2.1, 2.2, the extension of dual definite subspaces L ± to dual maximal definite subspaces L max ± is equivalent to the extension of T 0 to a self-adjoint strong contraction T anticommuting with J . In this case cf. (2.2), In what follows we assume that the direct sum D = L + [+]L − of dual definite subspaces L ± is a dense set in H.
The next result is well known and it can be established by various methods (see, e.g., [6,15] Proof For the construction of L max ± we should prove the existence of a self-adjoint strong contractive extension T ⊃ T 0 which anticommutes with J . The existence of a self-adjoint contractive extension T ⊃ T 0 is well known [5,11]. However, we cannot state that T anticommutes with J . To overcome this inconvenience we modify T as follows: The operator T is a self-adjoint contraction which anticommutes with J . Moreover, T is an extension of T 0 (since J T 0 = −T 0 J ). Therefore, the nonnegative/nonpositive subspaces L max ± defined by (2.5) are dual and L max ± ⊃ L ± . The existence of neutral elements (i.e.,

Remark 2.4
It follows from the proof of Theorem 2.3 that each self-adjoint contractive extension T ⊃ T 0 is a strong contraction.
The set of all self-adjoint contractive extensions of T 0 forms an operator interval [T μ , T M ] [5, §108], [11]. The end points of this interval: T μ and T M are called the hard and the soft extensions of T 0 , respectively. Corollary 2.5 Let L ± be dual definite subspaces. Then their extension to the dual maximal subspaces L max ± can be defined by (2.5) with Proof Let us prove that By virtue of [5, p. 380], the operators T μ and T M have the form: where T is a self-adjoint contractive extension of T 0 anticommuting with J (its existence was proved in Theorem 2. Due to the proof of Theorem 2.3, for the construction of T in (2.6) we use an arbitrary self-adjoint contraction T ⊃ T 0 . In particular, choosing T = T μ and using (2.7), we complete the proof.
In general, the extension of dual definite subspaces L ± to dual maximal definite subspaces L max ± is not determined uniquely. To describe all possible cases we use the formula [2,11] which gives a one-to-one correspondence between all self-adjoint contractive extensions T of T 0 and all nonnegative self-adjoint contractions X in the subspace  Let X 0 = 1 2 I be a solution of (2.10). Then the nonnegative self-adjoint contraction X 1 = I − X 0 is also a solution of (2.10). Moreover, each self-adjoint nonnegative contraction X α = (1 − α)X 0 + α X 1 , α ∈ [0, 1] is the solution of (2.10) Therefore, either dual maximal definite subspaces L max ± ⊃ L ± are determined uniquely or there are infinitely many such extensions.

II.
The above results as well as the results in the sequel can be rewritten with the use of the Cayley transform of T 0 : (2.11) The operator G 0 is a closed densely defined positive symmetric operator in H with and such that It follows from Remark 2.4 that every nonnegative self-adjoint extension Self-adjoint positive extensions G of G 0 are in one-to-one correspondence with the set of contractive self-adjoint extensions of T 0 : (2.13) In particular, the Friedrichs extension G μ of G 0 corresponds to the operator T μ , while the Krein-von Neumann extension G M is the Cayley transform of T M . The relation (2.7) between T μ and T M is rewritten as follows [12]: 14) It follows from (2.11) and Lemmas 2.1, 2.2 (see also [12,Proposition 4.2]) that dual maximal definite subspaces L max ± ⊃ L ± is in one-to-one correspondence with positive self-adjoint extensions G of G 0 satisfying the additional condition:

The Case of Maximal Uniformly Definite Subspaces
Let L max ± be dual maximal uniformly definite subspaces. Then: Relation (3.1) illustrates the variety of possible decompositions of the Krein space (H, [·, ·]) onto its maximal uniformly positive/negative subspaces. This property is characteristic for a Krein space and, sometimes, it is used for its definition [6].
With decomposition (3.1) one can associate a new inner product in H: . By virtue of (2.5), the relations f ± = (I + T )x ± , g ± = (I + T )y ± , x ± , y ± ∈ H ± hold. Taking (2.13) into account we rewrite (3.2) as follows: Here G is a bounded 3 positive self-adjoint operator with 0 ∈ ρ(G). Therefore, the dual subspaces L max ± determine the new inner product which is equivalent to the initial one (·, ·). The subspaces L max ± are mutually orthogonal with respect to (·, ·) G in the Hilbert space (H, (·, ·) G ).
Summing up: the choice of various dual maximal uniformly definite subspaces L max ± generates infinitely many equivalent inner products (·, ·) G of the Hilbert space H but it does not change the initial Krein space (H, [·, ·]).

The Case of Maximal Definite Subspaces
Assume that L max ± are dual maximal definite subspaces. Then the direct sum is a dense set in the Hilbert space (H, (·, ·)). The corresponding positive self-adjoint operator G is unbounded.
Similarly to the previous case, with direct sum (3.4) one can associate a new inner product (·, ·) G = (G·, ·) defined on D(G) = D max by the formula (3.2). The inner product (·, ·) G is not equivalent to the initial one and the linear space D max endowed with (·, ·) G is a pre-Hilbert space.
Let H G be the completion of D max with respect to (·, ·) G . The Hilbert space H G does not coincide with H. The dual subspaces L max ± are orthogonal with respect to (·, ·) G and, by construction, the new Hilbert space (H G , (·, ·) G ) can be decomposed as follows: whereL max ± are the completion of L max ± with respect to (·, ·) G . The decomposition (3.5) can be considered as a fundamental decomposition of the new Krein space (H G , [·, ·] G ) with the indefinite inner product where Let D[G] be the energetic linear manifold constructed by the positive self-adjoint operator G. In other words, D[G] denotes the completion of D(G) = D max with respect to the energetic norm The set of elements D[G] coincides with D( √ G) and the energetic linear manifold is a Hilbert space (D[G], (·, ·) en ) with respect to the energetic inner product  Proof Indeed, taking (3.2) and (3.6) into account, The obtained relation can be extended onto Summing up: the choice of various dual maximal definite subspaces L max ± generates infinitely many Krein

Definition and Principal Results
Let L ± be dual definite subspaces and let G 0 be the corresponding symmetric operator. Each positive self-adjoint extension G of G 0 with condition (2.15) determines the Hilbert space (H G , (·, ·) G ). Obviously, each dual maximal definite subspaces are quasi maximal. For dual uniformly definite subspaces, the concept of quasi-maximality is equivalent to maximality, i.e., each quasi maximal uniformly definite subspaces have to be maximal uniformly definite.
In general case of definite subspaces, the closure of dual quasi maximal subspaces L ± with respect to (·, ·) G coincides with subspacesL max ± in the fundamental decomposition (3.5), i.e., the closure of . It is natural to suppose that the quasi maximality can be characterized in terms of the corresponding positive self-adjoint extensions G of G 0 . For this reason, we recall

Remark 4.3
In general, an extremal extension G of G 0 is not determined uniquely. Let G i , i = 1, 2 be extremal extensions of G 0 that satisfy (2.15). By virtue of (3.2), the operator defined as W f = f for f ∈ D(G 0 ) and extended by continuity onto (H G 1 , (·, ·) G 1 ) is a unitary mapping between H G 1 and H G 2 . Moreover, W J G 1 = J G 2 W , where J G i are the fundamental symmetry operators corresponding to the fundamental decompositions . Therefore, the indefinite inner products of these spaces satisfy the relation Proof Let G 0 be a unique nonnegative self-adjoint extension G. Then, G = G μ = G M and, by virtue of (2.14), J G = G −1 J . Therefore, the operator G determines dual maximal subspaces L max ± ⊃ L ± . Furthermore, G is an extremal extension (since the Friedrichs extension and the Krein-von Neumann extension are extremal). In view of Theorem 4.2, L ± are quasi maximal. Thus, the condition (i) ensures the quasi maximality of L ± .
The condition (ii) is equivalent to (i) due to (2.14) and (2.15). The equivalence (i) and (iii) follows form [11,Theorem 9]. The condition (i) reformulated for the Cayley transform T 0 of G 0 [see (2.11)] means that T 0 has a unique self-adjoint contractive extension T = T μ = T M . The latter is equivalent to (iv) due to [11,Theorem 6].
Assume that dual subspaces L ± do not satisfy conditions of Proposition 4.4. Then  Conversely, let a hypermaximal neutral subspace M 1 be given. Then M = M 1 ⊕ J M 1 and the orthogonal projection X on M 1 is the solution of (2.10). The formula (2.9) with given X determines the self-adjoint strong contraction T anticommuting with J and its Cayley transform G defines the Hilbert space (H G , (·, ·) G ) in which D(G 0 ) is a dense set. Proof It follows from the proof of Theorem 4.5 that a hypermaximal neutral subspace M 1 determines the dual maximal subspaces L max ± ⊃ L ± such that D is a dense set in the Hilbert space (H G , (·, ·) G ) associated with D max . Precisely, the orthogonal projection X on M 1 defines the required subspaces L max ± by the formulas (2.5) and (2.9). Therefore, one can construct infinitely many such extensions L max ± because there are infinitely many hypermaximal neutral subspaces in the Krein space (M, [·, ·]).
If X is the orthogonal projection on M 1 , then I − X is the orthogonal projection on the hypermaximal neutral subspace J M 1 . These operators are solutions of (2.10). In this case, the nonnegative self-adjoint contractions X α = (1 − α)X + α(I − X ), α ∈ (0, 1) are solutions of (2.10) and they also determine dual maximal subspaces L max ± (α) ⊃ L ± via (2.5) and (2.9). The linear manifold D cannot be dense in the Hilbert space (H G , (·, ·) G ) associated with L max + (α)[+]L max − (α) because X α loses the property of being projection operator (since X 2 α = X α ). subspaces ⇐⇒ the subspaces L ± are not quasi maximal, the total amount of possible extensions L ± → L max ± is not specified (a unique extension is possible as well as infinitely many ones), the linear manifold D is not dense in the Hilbert space (H G , (·, ·) G ) associated with D max .

Auxiliary Statement
Let L ± be dual definite subspaces and let L max ± ⊃ L ± be dual maximal definite subspaces. The subspaces L max Therefore, { f n } ( f n ∈ D(G)) is a Cauchy sequence in (H G , (·, ·) G ) if and only if { x n } is a Cauchy sequence in (H, (·, ·)). This means that a one-to-one correspondence between H G and H can be established as follows: Let as assume that F ∈ H G is orthogonal to where x 0 runs D(T 0 ). By virtue of (2.3) and (4.4), γ = 0. Therefore, F γ = 0.

How to Construct Dual Quasi Maximal Subspaces?
We consider below an example (inspired by [1,13]) which illustrates a general method of the construction of dual quasi maximal subspaces.
Let {γ + n } and {γ − n } be orthonormal bases of subspaces H ± in the fundamental decomposition (2.1). Every φ ∈ H has the representation where {c ± n } ∈ l 2 (N). The operator is a self-adjoint strong contraction anticommuting with the fundamental symmetry J of the Krein space (H, [·, ·]). The subspaces L max ± defined by (2.5) with the operator T above are dual maximal definite. But they cannot be uniformly definite since T = 1, see Lemma 2.2.
Let us fix elements χ ± ∈H, and define the following subspaces of H ± : are quasi maximal for 1 2 < δ ≤ 3 2 . In particular, 1 2 < δ ≤ 1 corresponds to the case (A); the case (B) holds when 1 < δ ≤ 3 2 . Proof First of all we note that D = L + [+]L − is a dense set in H for 1 2 The subspaces L ± in (4.8) are the restriction of dual maximal subspaces L max ± = (I + T )H. Let us show that, for 1 2 < δ ≤ 1, the set D is dense in the Hilbert space (H G , (·, ·) G ) associated with L max ± . Due to Lemma 4.7, one should check (4.4). In view of (4.5), (4.6): On that is impossible for 1 2 < δ ≤ 1. Therefore, relation (4.4) holds and L ± are quasi maximal subspaces. By [1, Proposition 6.3.9], the dual subspaces L ± have a unique extension to dual maximal subspaces when 1 2 < δ ≤ 1 that corresponds to the case (A). If 1 < δ ≤ 3 2 , then the set D cannot be dense in the Hilbert space (H G , (·, ·) G ) considered above. However, for such δ, the dual subspaces L ± can be extended to different pairs of dual maximal subspaces [1, Proposition 6.

The Uniqueness of Dual Maximal Extension L max
± ⊃ L ± Does Not Mean that L ± are Quasi Maximal Let us assume that χ + = 0 and χ − is defined as in (4.7). Then M + = H + and the subspace L + in (2.14) coincides with L max + . This means that the dual maxi-mal definite subspaces L max ± ⊃ L ± are determined uniquely. Precisely, L max where L [⊥] + denotes the maximal negative subspace orthogonal to L + with respect to the indefinite inner product [·, ·].
Reasoning by analogy with the proof of Proposition 4.8 we obtain that the dual definite subspaces L max + and L − are quasi maximal (the case (A) of the classification above) for 1 2 < δ ≤ 1. If 1 < δ ≤ 3 2 , then the direct sum L max + [+]L − cannot be dense in the Hilbert space (H G , (·, ·) G ) constructed by L max ± . Since the extension L max ± ⊃ L ± is determined uniquely, the subspaces L max + and L − cannot be quasi maximal (the case (C) of the classification above).

Operators of C-Symmetry Associated with Dual Maximal Definite Subspaces
An operator C 0 associated with dual definite subspaces L ± is defined as follows: its domain D(C 0 ) coincides with L + [+]L − and If C 0 is given, then the corresponding dual subspaces L ± are recovered by the formula L ± = 1 2 (I ± C 0 )D(C 0 ).

Proposition 5.1
The following are equivalent: (i) C 0 is determined by dual subspaces L ± with the use of (5.1); (ii) C 0 satisfies the relation C 2 0 f = f for all f ∈ D(C 0 ) and J C 0 is a closed densely defined positive symmetric operator in H.
Proof In view of (5.1), and Taking into account the definition (2.11) of G 0 and formulas (3.2), (3.3), we obtain that G 0 = J C 0 , where G 0 is a closed densely defined positive symmetric operator in The Cayley transform T 0 of G 0 = J C 0 is a symmetric strong contraction in H (since G 0 is positive symmetric). Moreover, the condition C 2 0 = I on D(C 0 ) means that T 0 satisfies (2.4). Substituting T 0 into (2.2) we obtain the required dual definite subspaces L ± which generate C 0 with the help of (5.1).
We say that C 0 is an operator of C-symmetry if C 0 is associated with dual maximal definite subspaces L max ± . In this case, the notation C will be used instead of C 0 . An operator of C-symmetry admits the presentation C = J e Q , where Q is a self-adjoint operator in H such that J Q = −Q J [12]. Let C 0 be an operator associated with dual definite subspaces L ± . Its extension to the operator of C-symmetry C is equivalent to the construction of dual maximal definite subspaces L max ± ⊃ L ± . By Theorem 2.3, each operator C 0 can be extended to an operator of C-symmetry which, in general, is not determined uniquely. Its choice C ⊃ C 0 determines the new Hilbert space (H G , (·, ·) If L ± are quasi maximal, then there exists a dual maximal extension L max ± ⊃ L ± such that C 0 is extended to C by continuity in the new Hilbert space (H G , (·, ·)

Quasi Bases
A Therefore, { f n } is an orthonormal basis in the Hilbert space (H G , (·, ·) G ). The inverse implication (ii) → (i) is obvious.
(ii) → (iii). In view of (5.2), C = J e Q and G = e Q , where Q is a selfadjoint operator anticommuting with J . In this case, the relation (6.2) takes the form (e Q/2 f n , e Q/2 f m ) = δ nm . Hence, {g n = e Q/2 f n } is an orthonormal sequence in H. Its completeness will be established with the use of Lemma 4.7. Before doing this we note that the dual maximal definite subspaces L max ± = 1 2 (I ± C)D(C) corresponding to C = J e Q are also given by (2.5) with T = − tanh Q 2 [12]. Therefore, the bounded operator in (4.3) coincides with cosh −1 Q/2, where cosh Q/2 = 1 2 (e Q/2 + e −Q/2 ).
Since g n = e Q/2 f n and f n ∈ D(G) = D(e Q ), every g n belongs to the domain of definition of cosh Q/2 and (I − tanh Q/2) cosh Q/2g n = (cosh Q/2 − sinh Q/2)g n = e −Q/2 g n = f n . (6.3) Comparing the obtained relation with (2.2) and taking into account that the subspaces L ± ⊂ L max ± coincide with the closure of span{ f ± n }, we conclude that D(T 0 ) = M − ⊕ M + coincides with the closure of span{cosh Q/2g n }. Therefore, Let u ∈ H be orthogonal to {g n }. Then 0 = (u, g n ) = (cosh −1 Q/2u, cosh Q/2g n ) and hence, cosh −1 Q/2u ∈ R( ) ∩ (H (M − ⊕ M + )). By virtue of Lemma 4.7, cosh −1 Q/2u = 0. This means that u = 0 and {g n } is a complete orthonormal sequence in H, i.e., {g n } is a basis in H.
It follows from (2.2) and (6.3), that cosh Q/2g n belongs to one of the subspaces H ± (depending on either f n ∈ L + or f n ∈ L − ). The same property is true for g n because g n = cosh Q/2g n and : H ± → H ± .
(iii) → (ii). Since g n = e Q/2 f n belongs to H ± , we get J g n = ±g n = J e Q/2 f n = e −Q/2 J f n . Therefore, g n ∈ D(e Q/2 ) ∩ D(e −Q/2 ). This means that the sequence {cosh Q/2g n } is well defined and f n ∈ D(e Q ).
For given Q we define the operator of C-symmetry C = J e Q and G = e Q . By analogy with (6.2), Therefore, { f n } is an orthonormal sequence in (H G , (·, ·) G ). It follows form (6.4) that the sequence {G f n } is biorthogonal to { f n } (in the Hilbert space H). Hence, G f n = sign([ f n , f n ])J f n and C f n = sign([ f n , f n ]) f n . The latter means that C is an extension of the operator C 0 defined by (5.1) and the dual maximal definite subspaces L max ± determined by C are extensions of the dual definite subspaces L ± generated as the closures of span{ f ± n }. This fact and (6.3) lead to the conclusion that L ± are determined by (2.2), where D(T 0 ) = M − ⊕ M + coincides with the closure of span{cosh Q/2g n }.
Assume that { f n } is not complete in (H G , (·, ·) G ). Then the direct sum of L ± cannot be dense in H G and, by Lemma 4.7 (since = cosh −1 Q/2) there exists p = cosh −1 Q/2u = 0 such that, for all g n , 0 = ( p, cosh Q/2g n ) = (cosh −1 Q/2u, cosh Q/2g n ) = (u, g n ) that is impossible (since {g n } is a basis of H). The obtained contradiction implies that { f n } is an orthonormal basis of H G . [g, C f n ] f n , e Q/2 g = ∞ n=1 [g, C f n ]e Q/2 f n (6.5) where the series converge in the Hilbert spaces (H G , (·, ·) G ) and (H, (·, ·)), respectively. Proof

Corollary 6.6
If eigenfunctions { f n } of a J -symmetric operator H form a quasi basis in H, then there exists an operator of C-symmetry C such that the operator H restricted on span{ f n } turns out to be essentially self-adjoint in the Hilbert space (H G , (·, ·) G ) generated by C.
Proof Due to Theorem 6.3 there exists an operator C such that { f n } is a basis of (H G , (·, ·) G ). The restriction of C on span{ f n } coincides with the operator C 0 defined by (5.1) (here L ± are the closures of span{ f ± n }). It is easy to see that Taking for all f , g ∈ span{ f n }. Hence H is symmetric in (H G , (·, ·) G ) and eigenvalues of H corresponding to eigenfunctions f n must be real. Therefore, R(H ± i I ) ⊃ span{ f n } and the operator H is essentially self-adjoint in H G .

Examples
Quasi-bases can be easy constructed with the use of Theorem 6.3. Indeed let us consider an orthonormal basis {g n } of H such that each g n belongs to one of the subspaces H ± in the fundamental decomposition (2.1). Let Q be a self-adjoint operator in H, which anticommutes with J . If all g n belong to the domain of definition of e −Q/2 then f n = e −Q/2 g n is an J -orthonormal system of the Krein space (H, [·, ·]). Assuming additionally that { f n } is complete in H, we get an example of quasi basis. I. Let H = L 2 (R) and let J = P be the space parity operator P f (x) = f (−x). The subspaces H ± of the fundamental decomposition (2.1) coincide with the subspaces of even and odd functions of L 2 (R).
The Hermite functions g n (x) = 1 2 n n! √ π H n (x)e −x 2 /2 , H n (x) = e x 2 /2 (x − d dx ) n e −x 2 /2 are eigenfunctions of the harmonic oscillator and they form an orthonormal basis of L 2 (R). The functions g n are either odd or even functions. Therefore, g n ∈ H + or g n ∈ H − . Since Hermitian functions are entire functions, the complex shift of g n can be defined: f n (x) = g n (x + ia), a ∈ R\{0}, n = 0, 1, 2, . . . The operator Q is self-adjoint in L 2 (R) and in anticommutes with P. Therefore, { f n } is a quasi basis of L 2 (R). The functions { f n } are simple eigenfunctions of the P-symmetric operator Therefore, H restricted on span{ f n } is essentially self-adjoint in the new Hilbert space (H G , (·, ·) G ), where H G is the completion of span{ f n } with respect to the norm: The eigenfunctions g n are either even or odd functions.