Reproducing Kernel in Krein Spaces

OSMIN FERRER1, DIEGO CARRILLO2, ARNALDO DE LA BARRERA3

1Department of Mathematics, Sucre University, Sincelejo, COLOMBIA

2Basic Sciences Department, Caribbean University Corporation, Sincelejo, COLOMBIA

3Department of Mathematics, University of Pamplona, Pamplona, COLOMBIA

Abstract: - This article describes a new form to introduce a reproducing kernel for a Krein space based on

orthogonal projectors enabling to describe the kernel of a Krein space as the difference between the kernel of

definite positive subspace and the kernel of definite negative subspace corresponding to kernel of the associated

Hilbert space. As application, the authors obtain some basic properties of both kernels for Krein spaces and

exhibit that each kernel is uniquely determined by the Krein space given. The methods and results employed

generalize the notion of reproducing kernel given in Hilbert spaces to the context of spaces endowed with

indefinite metric.

Key-Words: - Hilbert space, Krein space, reproducing kernel, indefinite metric.

Received: March 21, 2021. Revised: November 17, 2021. Accepted: December 22, 2021. Published: January 11, 2022.

1 Introduction

A Hilbert space endowed with a function such that

reproduces every element of space through the

internal product is called a Hilbert space with

reproducing kernel or a reproducing kernel Hilbert

space. The theory of these spaces plays an important

role in many branches of mathematics such that

functional analysis, harmonic analysis, operator

theory and quantum mechanics [1], [2] and [3]. The

concept of reproducing kernel was used first at the

early twentieth century by Zaremba [4]. The same

author studied other functions that satisfy a property

of reproducing in the theory of integral equations,

called definite positive kernels [5]. Later, Bergman

[6] introduced reproducing kernel in one and several

variables for the class of harmonic functions and the

class of analytic functions, called kernel functions.

After them many authors have studied the same idea

as Szego [7], even in recent years there has been

considerable interest in studying the reproducing

properties in functional analysis, harmonic analysis,

operator theory, quantum mechanics, among other

fields; which can be seen in [8], [9], [10] and [11].

In [12] a reproducing kernel structure was

introduced for Hilbert spaces with functions where

the differences of point evaluations are limited and

in [13] they used the reproducing kernel to study

model spaces generated by certain types of singular

inner functions. Motivated by these researches, in

this paper we extend the notion of reproducing

kernel to more general spaces such that Krein spaces

[14], [15], [16] and [17]. The principal research

method used in this work is the hypothetical-

deductive method. In this work we characterize the

existence of a reproducing kernel for a Krein space

through two important Hilbert subspaces, called

definite positive subspace and definite negative

subspace. By using these subspaces, we describe

the kernel of a Krein space as the difference

between the kernel of definite positive subspace and

the kernel of definite negative subspace, while the

sum of these subspaces corresponds to kernel of the

associated Hilbert space. Also, we prove some basic

properties of both kernels for spaces with indefinite

metric and we exhibit that each kernel is uniquely

determined by the Krein space given. From this, we

generalize the notion of reproducing kernel given in

Hilbert spaces to the context of spaces endowed

with indefinite metric. The results of our research

can be potentially important for mathematicians

who working in areas such that functional analysis,

harmonic analysis, operator theory and quantum

mechanics and others affine areas.

2 Problem Formulation

Starting from the reproducing kernel defined in

Hilbert spaces, the generalization of this concept

was pursued in spaces with undefined metrics, more

specifically in Krein spaces. It was even sought to

verify some basic properties for both kernel and in

turn show that each kernels is uniquely determined

by the given Krein space.

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Definition 2.1. Let be a Hilbert space of

complex-valued functions on a set . Given

, we denoted by the internal product of

and . The complex-valued function

denoted by is

called a reproducing kernel for , if this

function satisfies:

i) For each the function defined by

is an element of .

ii) For each and , the reproducing

property is fulfilled.

Proposition 2.2. Let be a Hilbert space with

reproducing kernel , then

A Hilbert space of complex-valued functions on

a set is called a Hilbert space with a reproducing

kernel (often abbreviated EHNR) if there exists a

reproducing kernel associated to .

Proposition 2.3. Let be a Hilbert space with

reproducing kernel , then satisfies the

following properties:

i)

ii)

iii)

Also, iv) for all the following conditions are

equivalent:

a)

b)

c)

Theorem 2.4. Let be a Hilbert space with a

reproducing kernel , the is uniquely

determined by .

The proofs of above results can be seen in [4].

2.2 Krein Spaces

Definition 2.5. A topological space with an internal

product and a fundamental decomposition

is called a Krein space, if the

topology on the orthogonal sum coincides with the

original topology on the space and the subspaces

are Hilbert spaces.

Note that if a Krein space admits a decomposition

, we can define an internal product

. In addition, there

exist projections and associated to the spaces

and respectively, such that and

for all . The operator

defined by is called

fundamental symmetry. Observe that , also

,h) and (1)

The product is called J-internal product and

is used to study the linear operators acting

on the Krein space . For further

information on Krein spaces, we refer to [15] and

[16].

Definition 2.6. If and are

Krein spaces with fundamental symmetries and

respectively, then

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2.1 Kernels on Hilbert Spaces

Observe that the adjoint operator of an operator

is given in natural form and the equality

is valid. More precisely if we

consider the internal J-product on the Hilbert space

we denoted the adjoint by where is

the fundamental symmetry of the Krein space.

Definition 2.7. Let be a linear operator with dense

domain on and codomain on .

The adjoint operator is defined by

,

where is given by

,

for all and .

The adjoint operator can be expressed through

the adjoint which is called J-adjoint and satisfy

for all and . Moreover

[16], we have the equality

(2)

A linear operator is called self-adjoined if ,

and J-self adjoint if . In addition, a linear

operator is invertible if its range and domain are

the entire space. The following result can be seen in

[14].

Lemma 2.8. Let be a Krein space with

associated fundamental symmetry and an

orthogonal projection commuting with , then both

and are Krein spaces with

fundamental symmetries and

respectively.

3 Problem Solution

The reproducing kernel for a Kerin spaces is defined

from the kernels of the defined subspaces that form

the associated Hilbert space.

3.1 Existence of Kernel for Krein Spaces

Proposition 3.1. Let be a Krein space of

complex-valued functions on a set . If admits a

decomposition with reproducing

kernels and for the

Hilbert spaces and

respectively, then is a reproducing kernel

for the Hilbert space associated to the

fundamental symmetry .

Proof. For each function in , there exist

functions , in the spaces and

respectively, such and

Suppose that is a reproducing kernel for and

that is a reproducing kernel for , then for

each , and are elements in and

respectively. From which we concluded that

given by

belong to .

Also, the reproducing property is fulfilled for

and in and respectively.

Therefore, for any and for each

we have that and

Hence, for each with

and any

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Consequently and then

is a reproducing kernel for

Observe that, in the above reasoning,

define a reproducing kernel in

the following form.

Definition 3.2. Let be a Krein space of

complex-valued functions on a set . The function

is said to be a reproducing kernel for

if the following conditions are true:

i) For each the function defined by

belong to .

ii) For any and , the reproducing

property is fulfilled for

We can observe that the above definition

generalizes the notion of kernel given in the second

section for a Hilbert space , taking the

fundamental symmetry as the identity operator.

Example 3.3. Consider with the standard

basis where is the sequence

whose term is 1 and the other terms are zero.

Now, let be the class of functions

such that and

. If we take the internal product

given by ( )

Then is a Krein space with

fundamental decomposition

, where

and

Thus, each can be written in the form

Also, for the fundamental symmetry

we have that

. That is,

Finally, if and

is defined as:

Then, both and functions are reproducing

kernels for the Hilbert spaces and

, respectively. For each ,

the function defined by

belong to and satisfy (for odd and even)

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and zero in the other cases. In addition, the

reproducing property is verified immediately for

even and , because

and , in case odd. In a

similar form we can prove this property for the

kernel .

Finally, the function is a

reproducing kernel for the Hilbert space

according the Proposition 3.1, where

for each , we can deduce that

Indeed,

In consequence is a reproducing kernel for the

Krein space

3.2 Kernel for Krein Spaces

In this section we prove some useful results to

establish relations between reproducing kernel for

the Krein spaces and the orthogonal projectors [14].

Lemma 3.4. Let be a Krein space of

complex-valued functions on a non-empty set with

fundamental symmetry and an orthogonal

projection commuting with . If is a

reproducing kernel for , then is a

reproducing kernel for .

Proof. If is a reproducing kernel for then, for

each belong to and hence belong

to for all . On the other hand, if and

then Also, if

then belong to . Thus, is a projection in and

. Hence,

satisfies the reproducing property and then is a

reproducing kernel for .

Proposition 3.5. If is a Krein space of

complex-valued functions on with a fundamental

symmetry , then the following are equivalent:

i) is a reproducing kernel for the Krein space

ii) is a reproducing kernel for the Hilbert space

Proof. Let be an element of , then if and

only if . Since is an operator acting from

into itself and satisfy the equality , this

property implies that . Therefore,

the equivalence of kernel is proved because by

hypotheses, for each

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The following result generalizes the Proposition 2.3

given for Hilbert spaces.

Proposition 3.6. Let be a Krein space of

complex-valued functions on a set and let

be a reproducing kernel. Then, for each

x and y in , satisfies the following properties:

i)

ii)

iii)

Also,

iv) For each the following conditions are

equivalent:

a)

b)

c)

Proof. Let be a reproducing kernel for a

Krein space and . The

properties ii) and iv) are clear. Also, according

to Proposition 3.5,

Finally, by Cauchy Schwarz’s inequality, it follows

iv),

The Example 3.3 exhibits relationship among the

reproducing kernels of , and . The

relationship among these kernels is proved in the

following theorem.

Theorem 3.7. Let be a Krein space of

complex-valued functions on a set with a

decomposition . If is a

reproducing kernel for , then there exist

reproducing kernels , for Hilbert spaces

and respectively, such

Proof. Let be a reproducing kernel for the Krein

space and an element of , then

the function defined by belong

to and there exist , in the Hilbert spaces

and respectively, such . Now

it must be proved that the functions

and given by

are reproducing kernels for the spaces and

respectively. Indeed, for each the

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functions and defined by

and meet the

following

whereby for each the functions and

are elements of the Hilbert spaces and

respectively. Moreover, from this it is easily seen

that As well if then

for each by using the

reproducing property of kernel N. Therefore,

Similarly, if then and

Thus, the reproducing properties of reproducing

kernels and are proved for the Hilbert

spaces and , respectively.

In other words, Theorem 3.7 guarantees us a natural

way to define the nucleus of reproduction for a

Krein space. Even with theorem 3.8 we have the

implication that said nucleus of reproduction is

independent of the fundamental symmetry since any

other nucleus is equivalent to that of the previous

theorem.

Theorem 3.8. Let be a Krein space of

complex-valued functions on a set . The

reproducing kernel is uniquely determined by the

Krein space .

Proof. Suppose that a Krein space with

fundamental symmetry has a reproducing kernel

. If is another reproducing kernel for , by

Proposition 3.6 it follows that for all

Additionally, by the reproducing property of and

, we obtain the equalities

Then, for each . Thus, for any

,

4 Conclusion

We describe the kernel of a Krein space as the

difference between the kernel of definite positive

subspace and the kernel of definite negative

subspace. Also, we prove some basic properties of

both kernels for Krein spaces and we exhibit that

each kernel is uniquely determined by the Krein

space given. In consequence, we generalize the

notion of reproducing kernel given in Hilbert spaces

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to the context of spaces endowed with indefinite

metric.

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Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

Osmin Ferrer carried out the conceptualization,

methodology and supervision.

Diego Carrillo was responsible for the writing of the

original draft.

Arnaldo de la Barrera wrote the corresponding

review and editing.

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(Attribution 4.0 International, CC BY 4.0)

This article is published under the terms of the

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