




































©2024 Ada Academica https://adac.eeEur. J. Math. Anal. 4 (2024) 1doi: 10.28924/ada/ma.4.1
Frame Operators for Frames in Krein Spaces

Shah Jahan1 , P. Sam Johnson2,∗
1Department of Mathematics, Central University of Haryana, Haryana 123029, India

shahjahan@cuh.ac.in
2Department of Mathematical and Computational Sciences, National Institute of Technology Karnataka,

Surathkal 575025, India
sam@nitk.edu.in

∗Correspondence: sam@nitk.edu.in

Abstract. In recent years, frames in Krein spaces and several generalizations have been extensivelystudied. In this paper, we propose an alternative way of looking at the notion of frames in Kreinspaces and give a necessary and sufficient condition for a sequence in a Krein space to be a Besselsequence. We observe that a subsequence of a frame in a Krein space need not be a frame. Also,two complementary subsequences are considered in which one of them is a frame for a Krein space.We obtain necessary and sufficient conditions under which the other one is also a frame for the Kreinspace.

1. Introduction
Hilbert space frames were originally introduced by Duffin and Schaeffer [7] to deal with someproblems in non-harmonic Fourier analysis. The linear independence property for a (Hamel) basis,which allows every vector to be uniquely represented as a linear combination is very restrictive forpractical problems. Frames allow each element in the space to be written as a linear combinationof the elements in the frame, but linear independence is not required. Frames can be viewed asredundant bases which are generalization of Riesz bases. This redundancy property sometimes isextremely important in applications such as sampling theory [9], filter banks [3], signal and imageprocessing [6] and so on.

Definition 1.1. [4] Let H be a Hilbert space and I be a countable index set. A collection {fn}n∈I
in a Hilbert space H is said to be a frame for H if there exist a, b > 0 such that

a‖f ‖2 ≤
∑
n∈I
|〈f , fn〉|2 ≤ b‖f ‖2, ∀f ∈ H.

Received: 19 Nov 2023.2020 Mathematics Subject Classification. 42C15, 46C05, 46C20.
Key words and phrases. Krein space; Bessel sequence; frame sequence; frame operator.1

https://adac.ee
https://doi.org/10.28924/ada/ma.4.1
https://orcid.org/0000-0002-5966-9185
https://orcid.org/0000-0003-3461-5380


Eur. J. Math. Anal. 10.28924/ada/ma.4.1 2We now look at the definition of frame which is equivalent to perceive as the map
H 3 f 7→

∑
n∈I
〈f , fn〉fn ∈ H (1)

which is a well-defined bounded positive invertible operator.The bounded linear operator S : H −→ H defined by
Sf =

∑
n∈I
〈f , fn〉fn, f ∈ H,

is known as the frame operator associated to the frame {fn}n∈I . This operator S is bounded invert-ible, positive and self adjoint. It allows to reconstruct each vector in terms of the sequence {fn}n∈Ias follows:
f =

∑
n∈I
〈f , S−1fn〉fn =

∑
n∈I
〈f , fn〉S−1fn. (2)

The formula (2) is known as reconstruction formula associated to {fn}n∈I and if S = I, then thereconstruction formula resembles the Fourier series of f associated with the orthonormal sequence
{fn}n∈I .The concept of indefinite inner product was first found in a paper on quantum field theory byDirac in 1942 [5]. Pontrjagin gave the mathematical interpretation of indefinite inner product.Giribet et al. have introduced and studied frames for Krein spaces [8]. Motivated by the equivalentdefinition of frame as given in (1), in this paper, we propose an alternative way of looking at thenotion of frames in Krein spaces by decomposing the index set I in a natural way and obtain somenew results on frames sequences.The paper is organized as follows. Standard definition of Krein space is given in Section 2along with some notations and examples which will be used in the sequel. In Section 3, we definethe concept of Bessel sequence in Krein spaces and give a necessary and sufficient condition fora sequence to be a Bessel sequence in Krein spaces. In Section 4, we give the definition of framefor Krein space and study operators associated to the frame. In the last section, we study framesequences in Krein spaces. In general, if {fn}n∈I is a frame in a Krein space and {nk} is any infiniteincreasing sequence in I , then {fnk} need not be a frame sequence. We provide some sufficientconditions under which subsequences become frame sequence for the Krein space.

2. Preliminaries
Let K be a complex vector space with a Hermitian sesquilinear form defined on it. Then wecall (K, [., .]) an inner product space. An element x ∈ K is called neutral, positive, or negativeif [x, x ] = 0, [x, x ] > 0, or [x, x ] < 0 respectively. If K contains positive as well as negativeelements, then it is called an indefinite inner product space, otherwise it is called a semi-definiteinner product space. We refer [1, 2] for basics on indefinite inner product spaces.

https://doi.org/10.28924/ada/ma.4.1


Eur. J. Math. Anal. 10.28924/ada/ma.4.1 3An indefinite inner product space (K, [., .]) is decomposable if it can be written as an orthogonaldirect sum of a neutral subspace K0, a positive definite subspace K+ and a negative definitesubspace K−:
K = K0[+̇]K+[+̇]K−. (3)

Then (3) is known as a fundamental decomposition of K.An indefinite inner product space (K, [., .]) is a Krein space if it can be written as an orthogonaldirect sum of a positive definite subspace K+ and a negative definite subspace K− such that
(K+, [., .]) and (K−,−[., .]) are Hilbert spaces. Let a fundamental decomposition of a Krein space
K be given by

K = K+[+̇]K− (4)
and P± be the orthogonal projections onto K±. The linear map

J = P+ − P−

is called the fundamental symmetry corresponding to (4). Then
(f , g)J = [Jf , g]

is a positive definite inner product on K, called J-inner product corresponding to the fundamentaldecomposition (4). We can write
(f , f )J = [Jf , f ] = [(2P

+ − I)f , f ] = 2[P+f , P+f ]− [f , f ]. (5)
The corresponding norm (called J-norm) is denoted by

‖f ‖J = (f , f )
1
2
J = [Jf , f ]

1
2 .

Example 2.1. Consider K = `2(N), the linear space of square-summable sequences, with

[f , g] =

∞∑
n=1

(−1)nfngn for f = (fn)
∞
n=1, g = (gn)

∞
n=1 ∈ K.

Let K+ =
{
(fn)

∞
n=1 : fn = 0 if n is odd

}
and K− =

{
(fn)

∞
n=1 : fn = 0 if n is even

}
. Then K =

K+[+̇]K−, where K+ and K− are complete with respect to the induced norm and hence K is a
Krein space.

Theorem 2.1. [2] Let K be a Krein space. Then the following are equivalent:(1) There exists a fundamental decomposition of K.(2) There exists a maximal uniformly positive ortho-complemented subspace.(3) There exists a maximal uniformly negative ortho-complemented subspace.(4) There exists a mapping J in K such that J = J∗ = J−1.

https://doi.org/10.28924/ada/ma.4.1


Eur. J. Math. Anal. 10.28924/ada/ma.4.1 4Different fundamental decompositions induce different J-norms. Hence various norms can bedefined on a Krein space by choosing different underlying fundamental decompositions.
Example 2.2. Let K be a two-dimensional vector space with basis {e1, e2} and an indefinite inner
product defined by [e1, e1] = 1, [e2, e2] = −1 and [e1, e2] = 0. If we take Y = span{e1}, then it is a
maximal uniformly positive definite subspace and hence there exists a fundamental decomposition of
K with K+ = Y and K− = span{e2}. Choosing K+n = span{(n, 1)} and K−n = span{(1, n)} where
n > 1, we get several fundamental decompositions. The corresponding fundamental symmetries Jn
are given by

Jn =

(
n2+1
n2−1

−2n
n2−1

2n
n2−1

−(n2+1)
n2−1

)
.

Here we can see that the fundamental symmetries Jn satisfy J2n = In, [Jnf , g] = [f , Jng] and
[Jnf , Jng] = [f , g] for all f , g ∈ K.

3. Frame Operator for Frames in Krein Spaces
Let K be a Krein space and let {fn}n∈I be a sequence in K. In relation to the sequence {fn}n∈I ,the index set I is decomposed as I+ = {n ∈ I : [fn, fn] ≥ 0} and I− = {n ∈ I : [fn, fn] < 0}. It iseasy to observe that `2(I) is the orthogonal direct sum of `2(I+) and `2(I−).

Definition 3.1. A sequence {fn}n∈I in a Krein space K is called a Bessel sequence if there exists
a constant B > 0 such that ∑

n∈I
|[fn, f ]|2 ≤ B‖f ‖2, for all f ∈ K. (6)

The constant B in the inequality (6) is called a Bessel bound for {fn}n∈I .

Theorem 3.1. Let {fn}n∈I be a sequence in a Krein space K. Then {fn}n∈I is a Bessel sequence
with a Bessel bound B if and only if the operators T+ : `2(I+) −→ K+ defined by T+{cn}n∈I+ =∑
n∈I+ cnfn and T− : `2(I−) −→ K− defined by T−{cn}n∈I− =

∑
n∈I− cnfn are well defined

bounded operators and ‖T‖ ≤
√
B, where T = T+ + T−.

Proof. Suppose first that {fn}n∈I is a Bessel sequence with a Bessel bound B. Let {cn}n∈I+ ∈ `2(I+)and {cn}n∈I− ∈ `2(I−). Let `,m ∈ I− such that ` > m. Then∥∥∥∥∥∑̀
n=1

cnfn −
m∑
n=1

cnfn

∥∥∥∥∥ =

∥∥∥∥∥ ∑̀
n=m+1

cnfn

∥∥∥∥∥
= sup

‖g‖=1
|[
∑̀
n=m+1

cnfn, g]|

≤ sup
‖g‖=1

∑̀
n=m+1

|[cnfn, g]|

https://doi.org/10.28924/ada/ma.4.1


Eur. J. Math. Anal. 10.28924/ada/ma.4.1 5

≤

( ∑̀
n=m+1

|cn|2
) 1
2

sup
‖g‖=1

( ∑̀
n=m+1

|[fn, g]|2
) 1
2

≤
√
B

( ∑̀
n=m+1

|cn|2
) 1
2

.

Since {cn}n∈I− ∈ `2(I−), {∑`n=1 |cn|2} is a Cauchy sequence in C. The above calculation showsthat {∑`n=1 cnfn}`∈I− is a Cauchy sequence in K−, and so it is convergent. Hence T− is welldefined. With similar arguments and by considering `,m ∈ I+ with ` > m one may prove that
{
∑`
n=1 cnfn}`∈I+ is a Cauchy sequence in K+ and so it is convergent. Thus T− and T+ arewell defined and boundedness follows from the above calculation. Clearly T− and T+ are linear.Conversely, suppose that T− and T+ are well defined bounded linear operators with their adjoints

T−∗ : K− −→ `2(I−) and T+∗ : K+ −→ `2(I+) defined by T−∗f = {−[f , fn]}n∈I− and T+∗f =
{[f , fn]}n∈I+ respectively. Since the adjoint of a bounded operator is bounded, ‖T−∗‖ = ‖T−‖ and
‖T+∗‖ = ‖T+‖. Also we have

‖T+∗f ‖2 ≤ ‖T+‖2‖f ‖2, for all f ∈ K+
and

‖T−∗f ‖2 ≤ ‖T−‖2‖f ‖2, for all f ∈ K−.So ‖T ∗f ‖2 ≤ ‖T‖2‖f ‖2, for all f ∈ K. Hence {fn}n∈I is a Bessel sequence. �

Corollary 3.2. Let {fn}n∈I be a sequence in a Krein space K such that both
∑
n∈I+ cnfn and∑

n∈I− cnfn are convergent for all {cn}n∈I− ∈ `2(I−) and {cn}n∈I+ ∈ `2(I+). Then {fn}n∈I is a
Bessel sequence.

The condition (6) remains unchanged regardless how the elements of {fn}n∈I are numbered.
Corollary 3.3. Let {fn}n∈I be a Bessel sequence in a Krein space K. Then

∑
n∈I+ cnfn and∑

n∈I− cnfn converge unconditionally for all {cn}n∈I+ ∈ `2(I+) and {cn}n∈I− ∈ `2(I−) respectively.

The following example illustrates that how the norm of a single element actually depends uponthe choice of fundamental decomposition. If a frame is defined relative to fundamental decom-position, then frame bounds will vary arbitrarily when difference fundamental decompositions areconsidered.
Example 3.2. Consider the two dimensional Minkowski space K = R2 with the inner product
[f , g] = f1g1−f2g2 where f = (f1, f2), g = (g1, g2) ∈ R2. Consider the fundamental decompositions
with K+n = span{(n+1n ,

n−1
n )} and K−n = span{(n−1n ,

n+1
n )} where n > 1. Then we get

‖f ‖2Jn =
1

4
[(2n + 2/n)(f 21 + f

2
2 ) + 4f1f2(1/n − n)].

Let f = (1, 1) and g = (1, 0). Then ‖f ‖2Jn =
2
n and ‖g‖2Jn =

1
2(n +

1
n ).

https://doi.org/10.28924/ada/ma.4.1


Eur. J. Math. Anal. 10.28924/ada/ma.4.1 6Let {fn}n∈I be a Bessel sequence in K. Then both {fn}n∈I+ ⊂ K+ and {fn}n∈I− ⊂ K− areBessel sequences. Define T+ : `2(I+) −→ K+ and T− : `2(I−) −→ K− by
T+{cn} =

∑
n∈I+

cnfn

and
T−{cn} =

∑
n∈I−

cnfn

respectively. Then T+ and T− are both bounded linear operators and are called synthesis opera-tors. Define T+∗ : K+ −→ `2(I+) and T−∗ : K− −→ `2(I−) by
T+∗f = {[f , fn]}n∈I+

and
T−∗f = {[f , fn]}n∈I−are called analysis operators. Thus S+ = T+T+∗ : K+ −→ K+ given by S+f = T+T+∗f =∑

n∈I+ [f , fn]fn and S− = T−T−∗ : K− −→ K− given by S−f = T−T−∗f =
∑
n∈I− [f , fn]fn.Therefore the frame operator S = S+ + S− : K −→ K is defined by

Sf = (S+ + S−)f =
∑
n∈I+

[f , fn]fn +
∑
n∈I−

[f , fn]fn.

Esmeral et al. [5] have given a defintion of frame which involves fundamental symmetry of theKrein space K. As shown in the Example 3.2, for sufficiently large values on n, J-norms of elementsof K can be too small or too large. Thus we propose the following definition for frame in Kreinspaces.
Definition 3.3. Let {fn}n∈I be a Bessel sequence in K. The sequence {fn}n∈I} is said to be a
frame if the frame operator S = S+ + S− : K −→ K defined by

Sf = (S+ + S−)f =
∑
n∈I+

[f , fn]fn +
∑
n∈I−

[f , fn]fn

is a bounded positive invertible operator.

The following is the frame decomposition theorem for the Krein space K, which states that if
{fn}n∈I is a frame for a Krein space K, then every element can be written as a linear combinationof frame elements.
Theorem 3.4. Let {fn}n∈I be a frame for a Krein space K with the frame operator S. Then

f =
∑
n∈I+

[f , S+
−1
fn]fn +

∑
n∈I−

[f , S−
−1
fn]fn, for all f ∈ K, (7)

and both the series converges unconditionally for all f ∈ K.

https://doi.org/10.28924/ada/ma.4.1


Eur. J. Math. Anal. 10.28924/ada/ma.4.1 7

Proof. Since the operator S is self adjoint and invertible, we have
f = SS−1f

=
∑
n∈I
[S−1f , fn]fn

=
∑
n∈I+

[S+
−1
f , fn]fn +

∑
n∈I−

[S−
−1
f , fn]fn

=
∑
n∈I+

[f , S+
−1
fn]fn +

∑
n∈I−

[f , S−
−1
fn]fn

Since {fn}n∈I is a frame with [f , S+−1fn] ∈ `2(I+) and [f , S−−1fn]fn ∈ `2(I−), the unconditionalconvergence follows from Corollary 3.3. �

4. Frame Sequences in Krein Spaces
We begin this section with the following definitions for frame sequence

Definition 4.1. Let K be a Krein space. A sequence {fn}n∈I ∈ K is called a(a) frame sequence if it is a frame for [fn] = span{fn : n ∈ I}.(b) exact if removal of an arbitrary fn render the collection {fn} no longer a frame for the Krein
space K.(c) near exact if it can be made exact by removing finitely many elements from it.

Example 4.2. Let {fn}n∈I be a sequence of unit orthonormal vectors in Krein space K and {nk}
be any infinite increasing subset of I. Then {fnk} is a frame sequence.

Let {fn}n∈I be a frame for a Krein space K and {nk} be any infinite increasing sequence in I .Then {fnk} need not be a frame sequence.
Example 4.3. Let {fn}n∈I be a sequence of unit orthonormal vectors in a Krein space K. Define
a sequence {hn}n∈I ∈ K by hn = 1√

n
fn, n ∈ I. Let nk = nk−1 + (k − 1), k ∈ I and n0 = ±1.

Then {nk} is an infinite increasing sequence in I . Define another sequence {gn}n∈I ∈ K by
g1 = h1, gnk = gnk+1 = gnk+2 = · · · = gnk+1 − 1 = hk , k ≥ 2. Then {gn}n∈I is a tight frame for K.
But note that {gnk} = {hk} is not a frame sequence.

The following theorem gives a necessary and sufficient condition for a existence of a subsequenceto be a frame for a Krein space K.
Theorem 4.1. Let {fn}n∈I be a frame for a Krein space K and let {mk} and {nk} be two infinite in-
creasing sequences in I with {m+k }∪{n

+
k } = I+ and {m−k }∪{n

−
k } = I−. If {fmk}mk∈I is a frame, then

{fnk}nk∈I is a frame if and only if there exists a bounded linear operator T : `2(I) −→ `2(I) such
that T = T++T−, where T+ : `2(I+) −→ `2(I+) defined by T+{[f +nk , f

+]} = {[f +mk , f
+]}, f + ∈ K+

and T− : `2(I−) −→ `2(I−) is defined by T−{[f −nk , f
−]} = {[f −mk , f

−]}, f − ∈ K−.

https://doi.org/10.28924/ada/ma.4.1


Eur. J. Math. Anal. 10.28924/ada/ma.4.1 8

Proof. Suppose that {fmk}mk∈I is a frame with lower frame bounds A and A′ . Then∑
mk∈I+

|[fmk , f
+]|2 =

∑
nk∈I+

‖[T+{[f +nk , f
+]}‖

≤ ‖T+‖
∑
nk∈I+

|[fnk , f
+]|2.

So, we have ∑
nk∈I+

|[fnk , f
+]|2 ≥

∑
mk∈I+ |[fmk , f

+]|2

‖T+‖

≥
A

‖T‖‖f
+‖2.

Similarly, we have ∑
nk∈I−

|[fnk , f
−]|2 ≥

A
′

‖T‖‖f
−‖2.

Hence {fnk}nk∈I is a frame for the Krein space K.Conversely, suppose that {fnk}nk∈I is a frame for the Krein space K. Then there exist operators
T1
+ : `2(I+) −→ K1 given by T1+{[f +nk , f +]} −→ f + and T ∗1 + : K1 −→ `2(I+) given by T ∗1 +f + =

{[f +nk , f
+]} and similarly there exist operators T1− : `2(I−) −→ K2 given by T1−{[f −nk , f −]} = f −and T−1
∗
: K2 −→ `2(I−) given by T−1

∗
f − = {[f −nk , f

−]}. Also, since {fmk}mk∈I is a frame forthe Krein space K, there exist operators T+2 : `2(I+) −→ K1 given by T2+{[f +mk , f +]} = f + and
T+2
∗ : K1 −→ `2(I+) given by T+2 ∗f + = {[f +mk , f +]}.Similarly, there exist operators T2− : `2(I−) −→ K2 given by T2−{[f −mk , f −]} = f − and T−2 ∗ :

K2 −→ `2(I−) given by T−2
∗f − = {[f −mk , f

−]}. Then T+ = T+2
∗
T1
+ : `2(I+) −→ `2(I+) is abounded linear operator such that T+{[f +nk , f +]} = {[f +mk , f +]}, f + ∈ K+ and T− = T−2

∗
T1
− :

`2(I−) −→ `2(I−) is a bounded linear operator such that T−{[f −nk , f −]} = {[f −mk , f −]}, f − ∈
K−. �

Next, we give a sufficient condition for two subsequences of a frame for K to be a frame sequence.
Theorem 4.2. Let {fn}n∈I be a frame for a Krein space K. Let {mk} and {nk} be two infinite
increasing sequences in I with {m+k } ∪ {n

+
k } = I+ and {m−k } ∪ {n

−
k } = I−. Let K1 = [f +mk ] ∩ [f

+
nk
].

If K1 is a finite dimensional space, then {f +mk} and {f +nk } are frame sequences for K+. Further, if
K2 = [f −mk ] ∩ [f

−
nk
] is finite dimensional, then {f −mk} and {f −nk } are frame sequences for K−.

Proof. Let {`+k } be a finite subsequence of {n+k } such that K1 = [f`k ]`k∈I+ . Since K1 is finitedimensional, {f +`k } is a frame for K1. Let A′ and B
′ be the frame bounds for {f +`k }. Consider

{fnk}nk∈I+ , let f + ∈ {fnk}nk∈I+ be any element. Now, if f +[⊥]K1, then∑
n∈I+

[f +, fn]
2 =

∑
nk∈I+

[f +, fnk ]
2

≥ A‖f +‖2.

https://doi.org/10.28924/ada/ma.4.1


Eur. J. Math. Anal. 10.28924/ada/ma.4.1 9Also, if f + ∈ K1, then ∑
nk∈I+

[f +, fnk ]
2 ≥

∑
`k∈I+

[f +, flk ]
2

≥ A
′‖f +‖2.

Otherwise, we have
f + =

∑
αk fnk

=
∑

αi fn +
∑

αj fj , i ∈ {nk}\{`k}, j ∈ {`k}

= (f +)
′
+ (f +)

′′
,where (f +)′ [⊥]K1 and (f +)′′ ∈ K1.

Thus ∑
nk∈I+

[f +, fnk ]
2 =

∑
[f +, fn]

2 +
∑
[f +, fj ]

2, i ∈ {nk}\{`k}, j ∈ {`k}

=
∑
[(f +)

′
+ (f +)

′′
, fn]

2 +
∑
[(f +)

′
+ (f +)

′′
, fj ]
2

=
∑
[(f +)

′
, fn]

2 +
∑
[(f +)

′′
, fj ]
2

≥ A‖(f +)′‖2 + A′‖(f +)′′‖

≥ min
{A
2
,
A
′

2

}
‖f +‖2.

Hence {fnk}nk∈I+ is a frame sequence for K+. Similarly we can show that {fmk}mk∈I+ is a framesequence for K+. Also, in a similar way, one can prove that {fnk}nk∈I− and {fmk}mk∈I− are framesequences for K−. �

Corollary 4.3. Let {fn}n∈I be a frame for a Krein space K. Let {mk} and {nk} be two infinite
increasing sequences in I with {m+k } ∪ {n

+
k } = I+ and {m−k } ∪ {n

−
k } = I−. Let {fmk}mk∈I+ and

{fnk}nk∈I+ be frames for [fmk ]mk∈I+ and [fnk ]nk∈I+ respectively and let {fmk}mk∈I− and {fnk}nk∈I−
be frames for [fmk ]mk∈I− and [fnk ]nk∈I− respectively. If {g+i } = {fmk}mk∈I+∪{fnk}nk∈I+ and {g−i } =
{fmk}mk∈I− ∪ {fnk}nk∈I− , then {g+i } and {g−i } are frame sequences.

Proof. The proof of the corollary follows from the Theorem 4.2 and the fact that {fmk}mk∈I+ and
{fnk}nk∈I+ are frames for the [fmk ]mk∈I+ and [fnk ]nk∈I+ respectively. �

Finally, we give a sufficient condition for the exactness of frames in a Krein space K.
Theorem 4.4. Let {fn}n∈I be a frame for a Krein space (K, [., .]) with bounds A , A′ and B, B′

such that fn 6= 0, for all n ∈ I. If for every infinite increasing sequence {nk} ∈ I+ and {mk} ∈ I−,
{fnk}nk∈I+ and {fmk}mk∈I− are frame sequences with bounds A, B and A′ , B′ respectively, then
{fn}n∈I is an exact frame.

https://doi.org/10.28924/ada/ma.4.1


Eur. J. Math. Anal. 10.28924/ada/ma.4.1 10

Proof. Suppose on the contrary that {fn}n∈I is not an exact frame. Then, there existsm ∈ I such that
fm ∈ [fn], i 6= m. Let {nk} be an increasing sequence in I , given by nk = k, k = 1, 2, 3, . . . , m−1and nk = k + 1, k = m,m + 1, . . . . Since {fnk}nk∈I+ is a frame for K+ and {fnk}nk∈I− is a framefor K− with bounds A, B and A′ , B′ respectively, we have

A‖f ‖2 ≤
∑
n 6=m
n∈I+

[f , fn]
2 ≤ B‖f ‖2, for all f ∈ K+ (8)

and
A
′‖f ‖2 ≤

∑
n 6=m
n∈I−

|[f , fn]|2 ≤ B
′‖f ‖2, for all f ∈ K−. (9)

Since {fn}n∈I is a frame for the Krein space (K, [., .]), by (8), we have [f , fm] = 0 for all f ∈ K+. Inparticular, [fm, fm] = 0. This gives fm = 0. Also, by (8), |[f −, fm]| = 0, for all f ∈ K−. In particular
|[fm, fm]| = 0. This implies that fm = 0 which is a contradiction. Hence {fnk}nk∈I is an exactframe. �

AcknowledgementsThe present work of the first author is partially supported by University Grants Commission(UGC), Government of India. The present work of the second author is partially supported byScience and Engineering Research Board (SERB), Government of India (Reference Number :TAR/2022/000219).
References

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10.1007/978-3-642-65567-8.[3] P. G. Casazza and G. Kutyniok, Finite frames: Theory and applications, Springer Science & Business Media, 2012.
https://doi.org/10.1007/978-0-8176-8373-3.[4] I. Daubechies, A. Grossmann and Y. Meyer, Painless nonorthogonal expansions, J. Math. Phys. 27 (1986) 1271-1283.
https://doi.org/10.1063/1.527388.[5] P. A. M. Dirac, The physical interpretation of the quantum dynamics, Proc. R. Soc. Lond. Ser. A, 113 (1927) 621-641.
https://doi.org/10.1098/rspa.1927.0012.[6] D. L. Donoho and M. Elad, Optimally sparse representation in general (nonorthogonal) dictionaries via `1 mini-mization, Proc. Nat. Acad. Sci. 100 (2003) 2197-2202. https://doi.org/10.1073/pnas.0437847100.[7] R. J. Duffin and A. C. Schaeffer, A class of nonharmonic fourier series, Trans. Amer. Math. Soc. 72 (1952) 341-366.
https://doi.org/10.1090/S0002-9947-1952-0047179-6.[8] J. Giribet, A. Maestripieri, F. M. Pería and P. G. Massey, On frames for krein spaces, J. Math. Anal. Appl. 393 (2012)122-137. https://doi.org/10.1016/j.jmaa.2012.03.040.[9] D. Han, K. Kornelson, D. Larson and E. Weber, Frames for undergraduates, vol. 40. American Mathematical Soc.,2007.

https://doi.org/10.28924/ada/ma.4.1
https://doi.org/10.1007/978-3-642-65567-8
https://doi.org/10.1007/978-3-642-65567-8
https://doi.org/10.1007/978-0-8176-8373-3
https://doi.org/10.1063/1.527388
https://doi.org/10.1098/rspa.1927.0012
https://doi.org/10.1073/pnas.0437847100
https://doi.org/10.1090/S0002-9947-1952-0047179-6
https://doi.org/10.1016/j.jmaa.2012.03.040

	1. Introduction
	2. Preliminaries
	3. Frame Operator for Frames in Krein Spaces
	4. Frame Sequences in Krein Spaces
	References

