SELF-ADJOINT OPERATORS IN KREIN SPACES

Аннотация

In this paper, positive, negative, and neutral elements and subspaces are defined using the inner product. The Krein space is defined, and the spectral properties of 𝐽 − self-adjoint operators in this space are studied. The relationships between self-adjoint operators in Hilbert space and 𝐽 − self-adjoint operators in Krein space are explored. Key words: Inner product, Hilbert space, Krein space, Fundamental symmetry, J-orthonormal basis, self-adjoint operators. Introduction. Krein spaces represent a rich and intriguing generalization of Hilbert spaces in functional analysis. Unlike Hilbert spaces, which are characterized by positive-definite inner products, Krein spaces allow for indefinite inner products, meaning that the “length” (or norm) of a vector can be positive, negative, or zero. This feature makes Krein spaces a powerful framework for addressing mathematical and physical problems that cannot be adequately handled within the confines of traditional Hilbert space theory. A Krein space is essentially a vector space equipped with an indefinite inner product (𝐾,[⋅,⋅]), which can be decomposed orthogonally into the direct sum of two Hilbert subspaces: 𝐾 = 𝐾 ⊕ 𝐾 + − Here, both (𝐾 ,[⋅,⋅]) and (𝐾 ,−[⋅,⋅]) are Hilbert spaces with positive- + − definite inner products, though the decomposition itself is not unique. Importantly, https://scientific-jl.com/luch/ 5 1 Часть-42_ Том-3_ Апрель-2025 while the inner products may vary across decompositions, the resulting topologies (in terms of convergence, continuity, and other properties) remain equivalent. Example 1. Let us consider the space ℂ2, consisting of ordered pairs of complex numbers. In this space, we define an indefinite inner product as follows: [𝑥,𝑦] = 𝑥 𝑦̅̅̅ − 𝑥 𝑦̅̅̅, where 𝑥 = (𝑥 ,𝑥 ) and 𝑦 = (𝑦 ,𝑦 ). 1 1 2 2 1 2 1 2 This space can be decomposed as: ℂ2 = ℂ ⊕ ℂ, where the first copy of ℂ represents the subspace 𝐾 , and the second copy + represents the subspace 𝐾 . The subspace 𝐾 consists of elements of the form (𝑥 ,0). In this subspace, + 1 the indefinite inner product becomes: [𝑥,𝑦] = 𝑥 𝑦̅̅̅, which is the standard 1 1 (positive-definite) inner product on complex numbers. Hence, (𝐾 ,[⋅,⋅]) is a + Hilbert space. The subspace 𝐾 consists of elements of the form (0,𝑥 ). In this subspace, − 2 the inner product is given by: [𝑥,𝑦] = −𝑥 𝑦̅̅̅. [1-2]. 2 2 If we multiply this by a negative sign: −[𝑥,𝑦] = 𝑥 𝑦̅̅̅ we recover the standard 2 2 inner product on ℂ, indicating that (𝐾 , −[⋅,⋅]) is also a Hilbert space [3]. Therefore, the space (ℂ2,[⋅,⋅]) admits a decomposition of the form: ℂ2 = 𝐾 ⊕ 𝐾 + − and satisfies the conditions of a Krein space. Example 2. Consider the space 𝐿2(ℝ), consisting of square-integrable functions. With the standard inner product [𝑓,𝑔] = ∫ 𝑓(𝑡)𝑔(𝑡)𝑑𝑡 this space forms a Hilbert space [4-10]. Now, let us define an indefinite inner product on this space using the signum function sgn(𝑡): [𝑓,𝑔] = ∫ 𝑓(𝑡)𝑔(𝑡)sgn(𝑡)𝑑𝑡 This inner product is indefinite, since the signum function sgn(𝑡) can take both positive and negative values. To make 𝐿2(ℝ) into a Krein space, we decompose it into two as follows: https://scientific-jl.com/luch/ 5 2 Часть-42_ Том-3_ Апрель-2025 𝐾 : the subspace of functions that vanish for negative values of 𝑡, i.e., 𝑓(𝑡) = + 0 for 𝑡 0. Thus, the space can be written as: 𝐿2(ℝ) = 𝐾 ⊕ 𝐾 . + − (𝐾 ,[⋅,⋅]) is a Hilbert space: since sgn(𝑡) is positive on 𝐾 , the inner product + + reduces to the standard positive-definite form [11-18]. (𝐾 ,−[⋅,⋅]) is also a Hilbert space: on 𝐾 , where sgn(𝑡) is negative, applying a minus sign to the inner product restores positive-definiteness. Therefore, with this decomposition, (𝐿2(ℝ),[⋅,⋅]) becomes a Krein space. Fundamental Symmetry. Let 𝐾 be a Krein space with the orthogonal decomposition: 𝑲 = 𝑲 ⊕ 𝑲 . + − Define the operator 𝐽: 𝐾 → 𝐾 by: 𝐽(𝑥 + 𝑥 ) = 𝑥 − 𝑥 + − + − where 𝑥 ∈ 𝐾 and 𝑥 ∈ 𝐾 . This operator is called the fundamental symmetry + + − − operator. The operator 𝐽 has the following properties [15-25]: 1. Idempotency: 𝐽2 = 𝐼 where 𝐼 is the identity operator. 2. Self-adjointness: 𝐽∗ = 𝐽 meaning that 𝐽 is self-adjoint with respect to the indefinite inner product. Introducing a Hilbert Space Structure in a Krein Space: he fundamental symmetry operator 𝑱 allows us to define a positive-definite inner product in the Krein space 𝑲. This inner product is given by: (𝑥,𝑦) = [𝐽𝑥,𝑦], for all 𝑥,𝑦 ∈ 𝐾 This definition yields a Hilbert space structure, as it satisfies the following properties: Positive definiteness: (𝑥,𝑥) = [𝐽𝑥,𝑥] ≥ 0,and (𝑥,𝑥) = 0 if and only if 𝑥 = 0. https://scientific-jl.com/luch/ 5 3 Часть-42_ Том-3_ Апрель-2025 Linearity and symmetry: The inner product is linear in the first argument and conjugate symmetric: (𝑥,𝑦) = (𝑦,𝑥). Example 3. In the Krein space (𝐿2(ℝ),[⋅,⋅]), the indefinite inner product is defined as: [𝑓,𝑔] = ∫ 𝑓(𝑡)𝑔(𝑡)sgn(𝑡)𝑑𝑡 . The corresponding fundamental symmetry operator 𝐽:𝐿2(ℝ) →