J-Self-Adjoint Projections in Krein Spaces

Abstract

Let be a Krein space with fundamental symmetry J. Starting with a canonical block-operator matrix representation of J, we study the regular subspaces of . We also present block-operator matrix representations of the J-self-adjoint projections for the regular subspaces of , as well as for the regular complements of the isotropic part in a pseudo-regular subspace of .

1. Introduction

Throughout this paper, let be a separable complex Hilbert space with the inner product 〈⋅, ⋅〉 and let be the algebra of all bounded linear operators on . A contraction in is an operator Q in such that ∥Q∥ ≤ 1. For an operator , T^∗, σ(T), σ_P(T), , and denote the adjoint, the spectrum, the point spectrum, the range, and the null space of T, respectively. An operator T in is said to be self-adjoint if T = T^∗.

An operator J in is said to be a symmetry (or self-adjoint involution) if J = J^∗ = J⁻¹. If the symmetry J is nonscalar, thenwhich defines an indefinite inner product on , and is called a Krein space ([1–3]).

For , the J-adjoint operator of T is the unique operator T^♯ in satisfyingfor all . It is easy to see that T^♯ = JT^∗J.

Definition 1. If , then (a) T is J-normal if TT^♯ = T^♯T; (b) T is J-self-adjoint if T^♯ = T; (c) T is J-positive if JT ≥ 0; (d) T is J-negative if JT ≤ 0.
An idempotent in is called a projection. A projection is normal if and only if it is self-adjoint. However, there exist J-normal projections which are not J-self-adjoint (see [4]).
For a subspace of the Krein space , denotes the J-orthocomplement of in , that is,It is obvious that equals the usual orthocomplement . Let be the isotropic part of . If , then is said to be J-nondegenerate. Otherwise, is said to be J-degenerate.

Definition 2. If is a subspace of the Krein space , then (a) is positive if , where is the orthogonal projection of onto ; (b) is uniformly positive if for some ϵ > 0; (c) is regular if ; (d) is pseudoregular if and the algebraic sum are closed.
It is well known that a subspace of is regular if and only if it is the range of a (unique) J-self-adjoint projection in (see [2]). Therefore, there is a one-to-one correspondence between regular subspaces of and J-self-adjoint projections in . It is also proved in [5] that a closed subspace is regular if and only if for some ϵ > 0. In consequence, a closed subspace is uniformly positive if and only if it is regular and positive, and due to Proposition 4 in [6], this is the case if and only if it is a regular subspace with a J-positive projection.
Pseudoregular subspaces are important since they enable to generalize some Pontryagin space arguments to general Krein space (see [7]). Pseudoregular subspaces and its properties have been studied extensively by many authors (see [4, 7, 8]). In [4], the authors proved that a closed subspace of is pseudoregular if and only if it is the range of a J-normal projection in . They also showed in the same paper that a pseudoregular subspace admits infinitely many J-normal projections onto it, unless is regular. In [8], Giribet et al. gave a block-operator matrix representation of the fundamental symmetry J depending on a pseudoregular subspace of , and from here on, they characterized the J-self-adjoint projections for the regular complements of in .
In this paper, we give a new block-operator matrix representation of the fundamental symmetry J related to a closed subspace of . This offers an improvement over the result in [8], since we do not need to impose the assumption of the pseudoregularity of . We also study the J-self-adjoint projections for the regular subspaces of , as well as for the regular complements of the isotropic part in a pseudoregular subspace of .
The paper is organized as follows. In Section 2, we give a block-operator matrix representation of the fundamental symmetry J. Therein, we also characterize the regular subspaces of and present a block-operator matrix representation of the J-self-adjoint projections for the regular subspaces of . In Section 3, we study the pseudoregular subspaces of . If is a pseudoregular subspace of and is a regular complement of in , we give a block-operator matrix representation of the J-self-adjoint projection onto .

2. J-Self-Adjoint Projections for the Regular Subspaces

Let be a Krein space with fundamental symmetry J. Then, and are mutually annihilating orthogonal projections. Denote and . We have fundamental decomposition .

Let be a closed subspace of the Krein space , and let be its isotropic part. Denote , , , , , , , and . It is easy to check that , 1 ≤ i ≤ 8, are pairwise orthogonal subspaces of . The operators in this paper are frequently treated as block-operator matrices with respect to the space decomposition:

For a pseudoregular subspace of the Krein space , a block-operator matrix representation of the fundamental symmetry J was obtained with the space decomposition in [8]. We continue the study of the block-operator matrix representation of the fundamental symmetry J, but we do not impose the assumption of the pseudoregularity of the subspace .

Theorem 1. Let be a Krein space with fundamental symmetry J, and let be a closed subspace of . Then, J has the operator matrix representation:with respect to space decomposition (4), where I_i is the identity operator on the corresponding space , i = 1, 2, 4, 7, 8, U is an isometric isomorphism from onto , is an isometric isomorphism from onto , and Q is a self-adjoint contraction on with 0, ±1 ∉ σ_P(Q).

Proof. It is clear that and are subspaces of and and are subspaces of . So implies and and implies and . Since , we have for i ≠ 6. Moreover, since , we get for i ≠ 3. Noting that J is self-adjoint, then J has the operator matrix representation:with respect to space decomposition (4), where I_i is the identity operator on the corresponding space , i = 1, 2, 7, 8, , , and J₄₄ and J₅₅ are self-adjoint contractions on and , respectively: Let J₃₆ = U. Since J² = I, it is easy to see that UU^∗ = I₃ and U^∗U = I₆. Thus, U is an isometric isomorphism from onto . Let J₄₄ = Q. If and x = (0, 0, 0, ξ, 0, 0, 0, 0)^T, thenSince ∥J∥ = 1, and Jx = x. It follows that , and hence, . This implies ξ = 0. Thus, 1 ∉ σ_P(Q). Analogously, −1 ∉ σ_P(Q) and ±1 ∉ σ_P(J₅₅).
Moreover, if and x = (0, 0, 0, ξ, 0, 0, 0, 0)^T, thenfor all . It follows that , and hence, . So ξ = 0, and hence, 0 ∉ σ_P(Q).
Let , that is,Then, J′ is a symmetry. So we haveand by Proposition 5 in [9], there exists a contraction from into such thatThen, by a direct calculation, equation impliesNoting that −1 ∉ σ_P(Q) and −1 ∉ σ_P(J₅₅), it follows thatBy (11) and (ii) of (13), we obtainBy (i) and (ii) of (13), (I₄ − Q) (VV^∗ − I₄) = 0, and since I₄ − Q is injective, VV^∗ = I₄. By (ii) and (iii) of (13), (V^∗V − I₅) (I₅ − J₅₅) = 0, and since I₅ − J₅₅ is a self-adjoint operator with dense range in , V^∗V = I₅. Thus, is an isometric isomorphism from onto , and by (ii) of (13) again,Now we see thatand J has the asserted operator matrix.

Lemma 1 (see [5]). A closed subspace is regular if and only if for some ϵ > 0. In this case, the J-self-adjoint projection onto is determined as , where stands for the Moore–Penrose inverse of .

Theorem 2. Let be a Krein space with fundamental symmetry J, and let be a closed subspace of . Write J in (5) with respect to space decomposition (4). Then, is regular if and only if and 0 ∉ σ(Q). In this case, and with respect to the space decomposition , J and the J-self-adjoint projection E onto have operator matrix representations:andrespectively.

Proof. It is clear that has the operator matrix representation:with respect to space decomposition (4), and by a direct calculation,Then, by Lemma 1, is regular if and only if and Q² ≥ ϵI₄ for some ϵ > 0, and since Q is self-adjoint, this is the case if and only if and 0 ∉ σ(Q).
If is regular, then , and hence, . By Theorem 1, J has the operator matrix representation given in (17), and since with respect to , we haveThen, by Lemma 1, we getwith respect to .
Recall that a closed subspace of is uniformly positive if and only if it is a regular subspace with a J-positive projection in . We give a characterization of the uniformly positive subspaces of the Krein space .

Corollary 1. Let be a Krein space with fundamental symmetry J, and let be a closed subspace of . Write J in (5) with respect to space decomposition (4). Then, is uniformly positive if and only if , , and Q is a positive operator in with 0 ∉ σ(Q). In this case, the block-operator matrix representation of the J-positive projection E onto is given bywith respect to the space decomposition .

Proof (Necessity). Suppose that the closed subspace is uniformly positive. Then, is regular, and suppose that E is the J-self-adjoint projection onto . By Theorem 2, , 0 ∉ σ(Q), and with respect to , J and E have operator matrix representations (17) and (18), respectively. By a direct calculation, we haveand since JE ≥ 0, we see that and Q ≥ 0.

Sufficiency. Suppose that , , and Q is a positive operator in with 0 ∉ σ(Q). Then, by Theorem 1, J has the operator matrix representation:with respect to , and by Theorem 2, is regular and the J-self-adjoint projection E onto has the operator matrix representation:with respect to . It follows thatand hence, is uniformly positive.

Remark 1. Let be a subspace of the Krein space . Then, is said to be negative (resp., uniformly negative) if (resp., for some ϵ > 0). If more is closed, then is uniformly negative if and only if it is regular and negative, or equivalently, if and only if it is a regular subspace with a J-negative projection. Arguing as in Corollary 1, we can also give a characterization of the uniformly negative subspaces of the Krein space . With the notation as in Corollary 1, a closed subspace is uniformly negative if and only if , , and −Q is a positive operator in with 0 ∉ σ(Q). In this case, the block matrix representation of the J-negative projection F onto is given bywith respect to the space decomposition .
In the end of this section, we give an alternative proof of Theorem 2.3 in [5].

Corollary 2. Let be a regular subspace of with the J-self-adjoint projection E. Then, E is uniquely written as E = E₁ + E₂ with E₁, a J-positive projection, and E₂, a J-negative projection satisfying .

Proof. Write J and E in (17) and (18), respectively. Since Q is a self-adjoint operator in , there are unique positive operators Q₊ and Q₋ in such that Q = Q⁺ − Q⁻ and Q⁺Q⁻ = Q⁻Q⁺ = 0 (see [10]). Note that 0 ∉ σ(Q), Q⁺, and Q⁻ have closed ranges in , and . Letwith respect to . Then, E₁ and E₂ are projections, and Q⁺Q⁻ = Q⁻Q⁺ = 0 implies E = E₁ + E₂ and .
Moreover, sincewhere , we see that E₁ is J-positive. Similarly, we have JE₂ ≤ 0, and hence, E₂ is J-negative.
Now, we prove the uniqueness of E₁ and E₂. Since JE₁ and JE₂ are self-adjoint operators in , we haveNoting that JE = JE₁ − (−JE₂), where JE₁ and −JE₂ are the positive operators in , we see that JE₁ and −JE₂ are the positive part and the negative part of the self-adjoint operator JE, respectively. So JE₁ and −JE₂, and hence, E₁ and E₂ are unique.

3. J-Self-Adjoint Projections for the Pseudoregular Subspaces

In this section, we study the pseudoregular subspaces of . If is a pseudoregular subspace of and is a complement of in , we present a block-operator matrix representation of the J-self-adjoint projection onto .

Lemma 2. Let be a Krein space with fundamental symmetry J, and let be a closed subspace of . Then, the following statements are equivalent:(a) is pseudoregular(b)There exists a regular subspace such that , where denotes the direct [⋅, ⋅]-orthogonal sum(c)If , then is regular(d) is regular(e)The operator Q in (5) is invertible

Proof. Due to Proposition 4.1 in [4], (a) ⟺ (b) ⟺ (c). (c) ⟹ (d): since , this is immediate. (d) ⟹ (b): since and , for all and . Thus, , and hence, (d) ⟹ (b) is clear. (d) ⟺ (e): arguing as Theorem 2, is regular if and only if the operator Q in (5) is invertible.If be a pseudoregular subspace of and be a complement of in , then by Lemma 2, is a regular subspace. The following theorem gives a block-operator matrix representation of the J-self-adjoint projection onto .

Theorem 3. Let be a Krein space with fundamental symmetry J, and let be a pseudoregular subspace of . Write J in (5) with respect to space decomposition (4). If is a complement of in , then the J-self-adjoint projection E onto has the operator matrix representation:with respect to the space decomposition , where , i = 1, 2, 4.

Proof. If is a complement of in , then is regular. Suppose that E is the J-self-adjoint projection onto . Since , we have , and hence,Moreover, since , we get . Noting that and , then E has the operator matrix representation:with respect to the space decomposition , where .
Let . By a direct calculation, , and since E² = E, it follows that P² = P. Thus, P is a projection on . Furthermore, since and , we haveSo P is the identity on , and hence,Then,where , and since JE is self-adjoint, we haveSince U is an isometric isomorphism from onto and 0 ∉ σ(Q), it also follows thatNow, we see thatMoreover, since E² = E, we haveThus, E has the asserted operator matrix.

Corollary 3. Let be a Krein space with fundamental symmetry J, and let be a pseudoregular subspace of . Write J in (5) with respect to space decomposition (4). Then, the J-self-adjoint projection E onto has the operator matrix representation:with respect to space decomposition (4).

Proof. By Theorem 3, the J-self-adjoint projection E onto has operator matrix representation (33) with respect to space decomposition (4). Moreover, since , the equation impliesIt follows that , and . Noting that U is an isometric isomorphism from onto and Q is a self-adjoint contraction with dense range in , we get that E₁₆ = 0, E₂₆ = 0, and E₄₆ = 0. Thus, E has the asserted operator matrix.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the NSF of China (No. 11601339) and Natural Science Foundation of Jiangsu Province of China (No. BK20171421).