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SELF-ADJOINT OPERATORS IN KREIN SPACES
Tosheva Nargiza Ahmedovna,
Bukhara State University, Bukhara, Uzbekistan
Qodirov Suhayl Orifjonovich,
Bukhara State University, Bukhara, Uzbekistan
Abstract. 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,
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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
ℂ
2
, consisting of ordered pairs of
complex numbers. In this space, we define an indefinite inner product as follows:
[𝑥, 𝑦] = 𝑥
1
𝑦
1
̅̅̅ − 𝑥
2
𝑦
2
̅̅̅
, where
𝑥 = (𝑥
1
, 𝑥
2
)
and
𝑦 = (𝑦
1
, 𝑦
2
)
.
This space can be decomposed as:
ℂ
2
= 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
1
, 0)
. In this subspace,
the indefinite inner product becomes:
[𝑥, 𝑦] = 𝑥
1
𝑦
1
̅̅̅
, which is the standard
(positive-definite) inner product on complex numbers. Hence,
(𝐾
+
, [⋅,⋅])
is a
Hilbert space.
The subspace
𝐾
−
consists of elements of the form
(0, 𝑥
2
)
. In this subspace,
the inner product is given by:
[𝑥, 𝑦] = −𝑥
2
𝑦
2
̅̅̅
. [1-2].
If we multiply this by a negative sign:
−[𝑥, 𝑦] = 𝑥
2
𝑦
2
̅̅̅
we recover the standard
inner product on
ℂ
, indicating that
(𝐾
−
, −[⋅,⋅])
is also a Hilbert space [3].
Therefore, the space
(ℂ
2
, [⋅,⋅])
admits a decomposition of the form:
ℂ
2
= 𝐾
+
⊕ 𝐾
−
and satisfies the conditions of a Krein space.
Example 2.
Consider the space
𝐿
2
(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
𝐿
2
(ℝ)
into a Krein space, we decompose it into two as follows:
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𝐾
+
: the subspace of functions that vanish for negative values of
𝑡
, i.e.,
𝑓(𝑡) =
0
for
𝑡 < 0
;
𝐾
−
: the subspace of functions that vanish for positive values of
𝑡
, i.e.,
𝑓(𝑡) =
0
for
𝑡 > 0
.
Thus, the space can be written as:
𝐿
2
(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,
(𝐿
2
(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
:
𝐽
2
= 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.
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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
(𝐿
2
(ℝ), [⋅,⋅])
, the indefinite inner product is
defined as:
[𝑓, 𝑔] = ∫
𝑓(𝑡)𝑔(𝑡)sgn(𝑡)𝑑𝑡
∞
−∞
.
The corresponding fundamental symmetry operator
𝐽: 𝐿
2
(ℝ) → 𝐿
2
(ℝ)
is
defined for all
𝑓 ∈ 𝐿
2
(ℝ)
as:
(𝐽𝑓)(𝑡) = sgn(𝑡)𝑓(𝑡)
This operator acts by multiplying each function by the signum function, effectively
flipping the sign of the function on the negative half-line.
The operator
𝐽
has the following properties:
1.
𝐽
2
= 𝐼
(Idempotency)
Proof.
Apply the operator
𝐽
twice:
(𝐽
2
𝑓)(𝑡) = 𝐽(𝐽𝑓)(𝑡) = sgn(𝑡)(𝐽𝑓)(𝑡) = sgn(𝑡)sgn(𝑡)𝑓(𝑡)
Since
sgn(𝑡)
2
= 1
, we have:
𝐽
2
𝑓 = 𝑓
.
Thus,
𝐽
2
= 𝐼
, where
𝐼
is the identity operator.
2.
𝐽
∗
= 𝐽
(Self-adjointness)
Proof.
Let us verify that
𝐽
is self-adjoint with respect to the indefinite inner
product, i.e., for all
𝑓, 𝑔 ∈ 𝐿
2
(ℝ)
,
[𝐽𝑓, 𝑔] = [𝑓, 𝐽𝑔]
.
Given that
𝐽𝑓(𝑡) = sgn(𝑡)𝑓(𝑡)
we compute:
[𝐽𝑓, 𝑔] = ∫ (𝐽𝑓)(𝑡)𝑔(𝑡)
̅̅̅̅̅̅sgn(𝑡)𝑑𝑡
∞
−∞
= ∫ sgn(𝑡)𝑓(𝑡)𝑔(𝑡)
̅̅̅̅̅̅sgn(𝑡)𝑑𝑡
∞
−∞
= ∫ 𝑓(𝑡)
∞
−∞
sgn(𝑡)𝑔(𝑡)
̅̅̅̅̅̅̅̅̅̅̅̅̅̅sgn(𝑡)𝑑𝑡 = ∫ 𝑓(𝑡)(𝐽𝑔)(𝑡)
̅̅̅̅̅̅̅̅̅𝑠𝑔𝑛(𝑡)𝑑𝑡
∞
−∞
= [𝑓, 𝐽𝑔].
Therefore,
𝐽
is self-adjoint:
𝐽 = 𝐽
∗
.
The fundamental symmetry operator also allows us to define a Hilbert space
inner product as:
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(𝑓, 𝑔) = [𝐽𝑓, 𝑔] = ∫ sgn(𝑡)𝑓(𝑡)𝑔(𝑡)
̅̅̅̅̅̅sgn(𝑡)𝑑𝑡
∞
−∞
Since
sgn(𝑡)
2
= 1
, this simplifies to the standard inner product on
𝐿
2
(ℝ)
:
(𝑓, 𝑔) =
∫
𝑓(𝑡)𝑔(𝑡)
̅̅̅̅̅̅𝑑𝑡
∞
−∞
Thus, this inner product induces the strong topology of a Hilbert space on
the Krein space
𝐿
2
(ℝ)
.
The operator
𝐽
is called the fundamental symmetry operator, as it
transforms the indefinite inner product in a Krein space into a positive-
definite one. Using this operator, one can define a Hilbert space structure on
the Krein space. The ability to reconstruct a Hilbert space topology from an
indefinite inner product is of significant practical importance in physics and
quantum mechanics [8-22].
One of the key concepts in Krein spaces is the notion of a J-orthonormal
basis.
𝑱
–
Orthonormal bases
.
Let
(𝐾, [⋅,⋅])
be a Krein space with a fundamental
symmetry operator
𝑱
. A sequence
{𝒆
𝒏
} ⊂ 𝑲
is called a
𝑱
-orthonormal basis if it
satisfies the following conditions:
1.
𝐽
–
orthonormality:
[𝑒
𝑚
, 𝑒
𝑛
] = 𝑓(𝑥) = {
0, 𝑚 ≠ 𝑛
±1, 𝑚 = 𝑛
2.
Completeness:
Every vector
𝑥 ∈ 𝐾
can be represented in the form:
𝑥 =
∑
𝐽𝑒
𝑛
∞
𝑛=1
,
meaning that any vector in the space can be expressed using the J-orthonormal
basis.
A
𝐽
-orthonormal basis in a Krein space functions similarly to an orthonormal
basis in a Hilbert space. Such bases:
•
Facilitate convenient representations of vectors and operators;
•
Serve as a fundamental tool for spectral analysis and operator
representation;
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•
Exist in every Krein space—every Krein space possesses at least one J-
orthonormal basis.
This type of basis plays a crucial role in analyzing elements within the Krein space
and studying the spectral properties of operators.
Classification of Vectors and Subspaces: The presence of an indefinite inner
product in Krein spaces leads to a classification of vectors and subspaces into the
following categories:
Type of Vector or
Subspace
Condition
Property
Positive vector
[𝑥, 𝑥] > 0
Has a positive “length”
Negative vector
[𝑥, 𝑥] < 0
Has a negative “length”
Neutral vector
[𝑥, 𝑥] = 0
“Self-orthogonal”
Positive subspace
∀𝑥 ≠ 0, [𝑥, 𝑥] > 0
All nonzero vectors have
positive length
Negative subspace
∀𝑥 ≠ 0, [𝑥, 𝑥] < 0
All nonzero vectors have
negative length
Neutral subspace
∀𝑥, [𝑥, 𝑥] = 0
All vectors are self-
orthogonal
This classification is one of the fundamental distinctions between Krein
spaces and Hilbert spaces, introducing the concepts of positive and negative
vectors. Neutral vectors and subspaces do not appear in standard Hilbert spaces but
play an important role in the theory of Krein spaces.
Definition 1.
Let,
𝐻 = 𝐻
1
⊕ 𝐻
2
be a Hilbert space, and let the operator
𝐽 = (
𝐼
0
0
−𝐼
)
(1)
be defined on it.
If
𝐴
is a densely defined linear operator on
𝐻
such that the operator
𝐽𝐴
is
self-adjoint in
𝐻
, then
𝐴
is called
𝐽
-self-adjoint.
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If
𝐽𝐴
is symmetric in the Hilbert space
𝐻
, then
𝐴
is called
𝐽
-symmetric.
It is known that every J-symmetric operator is also J-self-adjoint.
If we define a new indefinite inner product on
𝐻
by:
[•,•] = (𝐽 •,•)
then a
bounded linear operator
𝐴
is J-self-adjoint if and only if:
[𝐴𝑥, 𝑦] = [𝑥, 𝐴𝑦]
, for all
𝑥, 𝑦 ∈ 𝐻
.
The Hilbert space
𝐻
, equipped with the indefinite inner product
[•.•]
, becomes a
Krein space, and every
𝐽
-self-adjoint operator is self-adjoint with respect to this
indefinite inner product.
Let
𝐴
be a block operator matrix of the form:
𝐴 = (
𝐴
11
𝐴
12
𝐴
21
𝐴
22
)
.
Then, the operator
𝐴
is self-adjoint if and only if the following conditions hold:
𝐴
11
∗
= 𝐴
11
, 𝐴
22
∗
= 𝐴
22
, 𝐴
21
= 𝐴
12
∗
The operator
𝐴
is
𝐽
-self-adjoint if and only if:
𝐴
11
∗
= 𝐴
11
, 𝐴
22
∗
= 𝐴
22
, 𝐴
21
= −𝐴
12
∗
.
Theorem 1.
Assume that either
dim𝐻
1
≥ 2
or
dim𝐻
2
≥ 2
. If the numerical
range
𝑊
2
(𝐴) ⊂ ℝ
then the following hold:
𝐴
11
∗
= 𝐴
11
, 𝐴
22
∗
= 𝐴
22
, and either
𝐴
has a block triangular form (i.e.,
𝐴
12
= 0 ⋅
or
𝐴
21
= 0
), or there exists a real number
𝛾 ∈ 𝑅, 𝛾 ≠ 0
, such that:
𝐴 = (
𝐴
11
𝐴
12
𝛾𝐴
21
𝐴
22
)
.
In the latter case, the operator
𝐴
is similar to the block operator matrix:
𝐴̃ = (
𝐴
11
𝐴
12
sgn(𝛾)𝐴̃
12
𝐴
22
),
𝐴̃ = √|𝛾|𝐴
12
If
sgn(𝛾) = 1
, then
𝐴̃
is a
𝐽
-self-adjoint operator.
Lemma 1.
Let
𝐴
𝑖𝑗
: 𝐻
𝑗
→ 𝐻
𝑖
,
for
𝑖, 𝑗 = 1,2, 𝑖 ≠ 𝑗
, be densely defined closed
operators such that
(𝐴
12
𝑓
2
, 𝑓
1
)(𝐴
21
𝑓
1
, 𝑓
2
) ∈ ℝ,
∀𝑓
𝑗
∈ 𝐷(𝐴
𝑖𝑗
)
.
(2)
Then one of the following holds:
𝐴
12
= 0, 𝐴
21
= 0
or
𝐴
21
⊂ 𝛾𝐴
12
∗
, for some
𝛾 ∈
ℝ
(and
𝐴
21
= 𝛾𝐴
12
∗
if both
𝐴
12
and
𝐴
21
are bounded)
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58
Proof.
Suppose that
(𝐴
12
𝑓
2
, 𝑓
1
) ≠ 0
for some
𝑓
𝑗
∈ 𝐷(𝐴
𝑖𝑗
), 𝑖. 𝑗 = 1.2, 𝑖 ≠ 𝑗
.
Then, from condition (2), we obtain [5-20]:
(𝐴
21
𝑓
1
, 𝑓
2
)
(𝑓
1
, 𝐴
12
𝑓
2
)
=
(𝐴
12
𝑓
2
, 𝑓
1
)(𝐴
21
𝑓
1
, 𝑓
2
)
(𝐴
12
𝑓
2
, 𝑓
1
)(𝑓
1
, 𝐴
12
𝑓
2
)
∈ 𝑅 (3)
Assume
𝐴
12
≠ 0
. Since
𝐴
21
is densely defined, there exist elements
𝑥
0
∈
𝐷(𝐴
21
)
and
𝑦
0
∈ 𝐷(𝐴
12
)
such that
(𝑥
0
, 𝐴
12
𝑦
0
) ≠ 0
. For arbitrary
𝑢 ∈ 𝐷(𝐴
21
)
,
𝛾 ∈
𝐷(𝐴
12
)
define the function:
𝑓
𝑢,𝛾
(𝑧) =
(𝐴
21
(𝑥
0
+ 𝑧𝑢), 𝑦 + 𝑧̅𝛾)
(𝑥
0
+ 𝑧𝑢, 𝐴
12
(𝑦
0
+ 𝑧̅𝛾))
=
=
(𝐴
21
𝑥
0
, 𝑦
0
) + 𝑧((𝐴
21
𝑥
0
, 𝛾) + (𝐴
21
𝑢, 𝑦
0
)) + 𝑧
2
(𝐴
21
𝑢, 𝛾)
(𝑥
0
, 𝐴
12
𝑦
0
) + 𝑧((𝑥
0
, 𝐴
12
𝛾) + (𝑢, 𝐴
12
𝑦
0
)) + 𝑧
2
(𝑢, 𝐴
12
𝛾)
; 𝑧 ∈ 𝐶
Since
(𝑥
0
, 𝐴
12
𝑦
0
) ≠ 0
, the denominator is not identically zero. Thus,
𝑓
𝑢,𝛾
(⋅)
is a rational function on
ℂ
with at most two poles.
𝑓
𝑢,𝛾
(𝑧) = 𝑓
𝑢,𝛾
(0) = (𝐴
2
, 𝑥
0
, 𝑦
0
)/(𝑥
0
, 𝐴
12
𝑦
0
) = 𝛾 ∈ 𝑅
or
(𝐴
21
𝑥
0
, 𝑦
0
) + 𝑧((𝐴
21
𝑥
0
, 𝑣) + (𝐴
21
𝑢, 𝑦
0
)) + 𝑧
2
(𝐴
21
𝑢, 𝑣) =
= 𝛾 ((𝑥
0
, 𝐴
12
𝑦
0
) + 𝑧((𝑥
0
, 𝐴
12
𝑦
0
) + (𝑢, 𝐴
12
𝑣)) + 𝑧
2
(𝑢, 𝐴
12
𝑣)) , 𝑧 ∈ 𝐶\{𝜁
1
, 𝜁
2
}
By comparing the coefficients on both sides, for all
𝑢 ∈ 𝐷(𝐴
21
), 𝑣 ∈ 𝐷(𝐴
12
)
, we
obtain:
(𝐴
2
, 𝑢, 𝑣) = 𝛾(𝑢, 𝐴
12
𝑣)
.
This implies
𝛾𝐴
12
⊂ 𝐴
21
∗
or, equivalently,
𝐴
21
⊂ 𝛾𝐴
12
∗
. This completes the
proof of Lemma 1.
Conclusion: Krein spaces provide a powerful theoretical framework for
analyzing problems involving indefinite inner products. They serve as a
generalization of Hilbert spaces and enable the study of phenomena that do not
arise in conventional Hilbert space theory. The indefinite inner product introduces
new classes of vectors, subspaces, and operators, giving rise to a distinct and
intricate spectral theory [10-25].
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The significance of Krein spaces extends beyond theoretical mathematics.
They play an important role in various applied fields such as quantum mechanics,
signal processing, and control theory. The ability to work with indefinite metrics
makes Krein spaces a vital tool for analyzing complex systems and dynamic
processes.
Although substantial research has already been conducted on Krein spaces,
many open problems and research directions remain. These include the study of
invariant subspaces, the development of numerical methods for Krein space
operators, and the exploration of new applications across scientific disciplines.
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