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Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2008, Article ID 768105, 12 pages
doi:10.1155/2008/768105
Research Article
Convergence of Vectorial Continued Fractions
Related to the Spectral Seminorm
M. Hemdaoui and M. Amzil
Laboratoire G.A.F.O, D
´
epartement de Mathmatiques & Informatique, Facult
´
e des Sciences,
Universit
´
e Mohamed Premier, Oujda, Morocco
Correspondence should be addressed to M. Hemdaoui,
Received 07 February 2008; Accepted 16 April 2008
Recommended by Charles Chidume
We show that the spectral seminorm is useful to study convergence or divergence of vectorial
continued fractions in Banach algebras because such convergence or divergence is related to a
spectral property.
Copyright q 2008 M. Hemdaoui and M. Amzil. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
1. Introduction
Let A be a unital complex Banach algebra. We denote by e the unit element of A. · is the
norm of A.Fora ∈A, σa, and ρa denote, respectively, the spectrum and the spectral
seminorm of a.
A formal vectorial continued fraction is an expression of the form
y
0
b
0
a
1
·
b
1
a
2
·
b
2
···
−1
−1
, 1.1
where a
n
n≥1
and b
n
n≥1
are two sequences of elements in A.
In order to discuss convergence or divergence of the vectorial continued fraction 1.1,
we associate a sequence s
n
n≥0
called sequence of nth approximants defined by:
s
0
b
0
,
s
1
b
0
a
1
· b
−1
1
,
s
2
b
0
a
1
·
b
1
a
2
· b
−1
2
−1
,
.
.
.
.
.
.
s
n
b
0
a
1
·
b
1
a
2
·
b
2
··· a
n−1
·
a
n−1
a
n
· b
−1
n
−1
−1
−1
,
.
.
.
.
.
.
1.2
2 Journal of Inequalities and Applications
By induction, it can be shown that
s
n
b
0
p
n
· q
−1
n
, 1.3
where the expressions p
n
and q
n
are determined from recurrence relations
p
n1
p
n
· b
n1
p
n−1
· a
n1
,
q
n1
q
n
· b
n1
q
n−1
· a
n1
,
1.4
with initial conditions:
p
0
0,p
1
a
1
.
q
0
e, q
1
b
1
.
1.5
p
n
and q
n
are respectively called nth numerator and nth denominator of 1.1.
Now, consider the following example.
Let a be a nonnull quasinilpotent element in A. Consider the vectorial continued fraction
defined by
e a
1
4
e a
1
9
e a
···
1
n
2
e a
···
−1
−1
−1
−1
, 1.6
where for each positive integer n>0, we have
b
n
1
n
2
· e a. 1.7
So,
b
n
a
1
n
2
e
≥
a−
1
n
2
. 1.8
Therefore, the series
∞
n1
b
n
diverges.
By Fair 1, Theorem 2.2, we cannot ensure convergence or divergence of the vectorial
continued fraction 1.6. But, if we apply the spectral seminorm to 1.7,weget
ρ
b
n
≤
1
n
2
ρa
1
n
2
. 1.9
So, the series
∞
n1
ρb
n
converges. From Theorem 2.5 in Section 2 below, the vectorial
continued fraction 1.6 diverges according to the spectral seminorm so it diverges also
according to the norm because the spectral seminorm ρ satisfies
ρx ≤x, ∀x ∈A. 1.10
In Section 3, we give another example of a vectorial continued fraction that converges
according to the spectral seminorm and diverges according to the norm algebra.
From the simple and particular example above and the example in Section 3, we see that
to study convergence or divergence of vectorial continued fractions we can use the spectral
seminorm of the algebra to include a large class of vectorial continued fractions.
First, we start by determining necessary conditions upon a
n
and b
n
to ensure the
convergence.
Next, we give sufficient conditions to have the convergence.
M. Hemdaoui and M. Amzil 3
2. Convergence of vectorial continued fractions
In this section, we discuss some conditions upon the elements a
n
and b
n
of the vectorial
continued fraction 1.1with b
0
0 which are necessary to ensure the convergence.
Definition 2.1. The vectorial continued fraction 1.1 converges if q
−1
n
exists starting from a
certain rank N, and the sequence of nth approximants s
n
converges. Otherwise, the vectorial
continued fraction 1.1 diverges.
For future use, we record the following theorem due to P. Wynn.
Theorem 2.2 2. For all n ∈
N,wehave
s
n1
− s
n
p
n1
· q
−1
n1
− p
n
· q
−1
n
−1
n
a
1
b
−1
1
q
0
a
2
q
−1
2
q
1
a
3
q
−1
3
q
2
a
4
q
−1
4
···q
n−2
a
n
q
−1
n
q
n−1
a
n1
q
−1
n1
.
2.1
Remark 2.3. In the commutative case, Theorem 2.2 above becomes as follows.
For all n ∈
N, one has
s
n1
− s
n
−1
n
in1
i1
a
i
· q
−1
n1
· q
−1
n
. 2.2
Since convergence or divergence of the vectorial continued fraction 1.1 is not affected
by the value of the additive term b
0
, we omit it from subsequent discussion i.e., b
0
0.
Now, we give a proposition that extends a result due to Wall 3 inthecaseofscalar
continued fractions.
Proposition 2.4. The vectorial continued fraction 1.1 where its terms are commuting elements in A
diverges, if its odd partial denominators b
2n1
are all quasinilpotent elements in A.
Proof. In fact, from relation 1.5 above, we have q
1
b
1
.Soρq
1
ρb
1
0.
Since coefficients of 1.1 are commuting elements in A,itiseasytoshowthatforall
positive integers n and m,wehave
a
m
· q
n
q
n
· a
m
; b
m
· q
n
q
n
· b
m
. 2.3
So,
ρ
a
m
· q
n
≤ ρ
a
m
· ρ
q
n
; ρ
b
m
· q
n
≤ ρ
b
m
· σ
q
n
. 2.4
Now, suppose that for n ≥ 1, ρq
2n−1
0.
From relations 1.4 and 2.4,wehave
ρ
q
2n1
≤ ρ
q
2n
· ρ
b
2n1
ρ
q
2n−1
· ρ
a
2n1
. 2.5
Then, ρq
2n1
0, consequently
∀n ≥ 0; ρ
q
2n1
0. 2.6
So infinitely many denominators q
n
are not invertible.
The vectorial continued fraction 1.1 diverges.
4 Journal of Inequalities and Applications
Theorem 2.5 below gives a necessary condition for convergence according to the spectral
seminorm. This result is an extension of von Koch Theorem 4, concerning the scalar case. A
similar theorem was given by Fair 1 for vectorial continued fractions according to the norm
convergence.
Theorem 2.5. Let a
n
e, for all n ≥ 1,andb
n
be a sequence of commuting elements in A.Ifthe
vectorial continued fraction 1.1 converges according to spectral seminorm, then, the series
∞
n1
ρb
n
diverges.
Proof. Suppose
∞
n1
ρb
n
is a converging series, and there exists a positive integer N such that
q
−1
n
exists, for all n ≥ N.
By an induction argument, it is easy to show that for all n ∈
N,wehave
ρ
q
2n
− e
≤ exp
K
2n
− 1,ρ
q
2n1
≤ exp
K
2n1
, 2.7
where K
0
0andK
n
n
k1
ρb
k
; for all n ≥ 1.
Since for all positive integer n, a
n
e,andallb
n
are commuting elements in A,from
Remark 2.3 above, we have
d
n
s
2n1
− s
2n
q
−1
2n1
· q
−1
2n
, ∀n ≥ E
N
2
1. 2.8
So,
ρ
d
n
≥
ρ
d
−1
n
−1
ρ
q
2n1
· q
2n
−1
, ∀n ≥ E
N
2
1. 2.9
Then,
ρ
d
n
≥
ρ
q
2n1
−1
·
ρ
q
2n
−1
, ∀n ≥ E
N
2
1. 2.10
From this preceding,
ρ
d
n
≥
1
exp
K
2n1
·
1
exp
K
2n
− 1
≥
1
exp
2K
2n1
≥
1
exp2K
> 0, 2.11
where K
∞
n1
ρb
n
.
So, the sequence s
n
n≥0
is not a ρ-Cauchy sequence in A.
Remark 2.6. In a Banach algebra A if ρ denotes the spectral seminorm in A it is not a
multiplicative seminorm in general.
Consider the vectorial subspace of A defined by Kerρ{x ∈A|ρx0}. The
quotient vectorial space A/
Kerρ
becomes a normed vectorial space with norm defined by
˙ρ ˙xρx,x∈ ˙x. “˙x denotes the class of x modulo Kerρ.”
Generally, the normed vectorial space A/
Kerρ
is not complete. Its complete normed
vectorial space is
A/
Kerρ
witch is a Banach space. So, ρ-Cauchy sequences in A,ρ converge
in the Banach space
A/
Kerρ
.
M. Hemdaoui and M. Amzil 5
Remark 2.7. Whenever A is commutative, the vectorial continued fraction 1.1 diverges, if for
one character ψ, the series
n≥1
|ψb
n
| converges.
Lemma 2.8. Let u
n
n
be a sequence of commuting elements in A.
If the series
n≥1
ρu
n
converges, then, there exists a positive integer N ≥ 1 such that for every
positive integer k ≥ 1, the finite product
k
p1
e u
Np
is invertible and ρ-bounded and its inverse is
also ρ-bounded.
Proof. Since the series
n≥1
ρu
n
converges, therefore, there exists a positive integer N ≥ 1
such that
ρ
u
n
< 1; ∀n ≥ N. 2.12
Hence, for k ≥ 1 the product
k
p1
eu
Np
is invertible as finite product of invertible elements.
We have
ρ
k
p1
e u
Np
≤
k
p1
1 ρ
u
Np
≤
∞
p1
1 ρ
u
Np
. 2.13
But
k
p1
e u
Np
−1
k
p1
e u
Np
−1
k
p1
∞
n0
−1
n
u
n
Np
. 2.14
Hence,
ρ
k
p1
e u
Np
−1
≤
k
p1
∞
n0
ρ
n
u
Np
k
p1
1
1 − ρ
u
Np
1
k
p1
1 − ρ
u
Np
≤
1
∞
p1
1 − ρ
u
Np
.
2.15
Theorem 2.9. Let in the vectorial continued fraction 1.1 a
n
e for all n ≥ 1 and b
n
n∈N
be a
sequence of commuting elements in A. If both series
n≥0
ρ
b
2p1
,
n≥0
ρ
b
2p1
· ρ
2
b
2p
2.16
converge, then, the vectorial continued fraction 1.1 diverges.
Proof of Theorem 2.9. Since both series
n≥0
ρb
2p1
and
n≥0
ρb
2p1
· ρ
2
b
2p
converge, it
follows that the series
n≥0
ρb
2p1
· ρb
2p
converges too.
Therefore, from Lemma 2.8 above, there exists a positive integer N ≥ 1 such that for
k ≥ 1, the quantity θ
k
k
p1
1 b
2N2p1
· b
2N2p
is invertible.
6 Journal of Inequalities and Applications
Now, consider the vectorial c ontinued fraction
c
1
c
2
c
3
···
−1
−1
−1
, 2.17
where
c
2k
b
2N2k1
· θ
−1
k−1
· θ
−1
k
,c
2k−1
−b
2N2k1
· b
2
2N2k
· θ
k−1
· θ
k
,k 1, 2, 2.18
We will suppose that q
−1
n
exists for all n ≥ N otherwise, from Definition 2.1, the vectorial
continued fraction 1.1 diverges.
Before continuing the proof, we give the following lemma that will be used later.
Lemma 2.10. For all positive integers k ≥ 1, consider the quantities
U
2k
p
2N2k1
· θ
−1
k
,V
2k
q
2N2k1
· θ
−1
k
,
U
2k1
p
2N2k
· θ
k
,V
2k1
q
2N2k
· θ
k
,
c
2k
b
2N2k1
· θ
−1
k−1
· θ
−1
k
,c
2k−1
−b
2N2k1
· b
2
2N2k
· θ
k−1
· θ
k
,
k 1, 2, ,
θ
0
e
.
2.19
Then,
U
k
U
k−1
· c
k
U
k−2
,
V
k
V
k−1
· c
k
V
k−2
, ∀k ≥ 2.
2.20
This lemma is proved by the same argument given by Wall 3, Lemma 6.1 for scalar
continued fractions.
Lemma 2.10 shows that U
n
and V
n
are respectively the nth numerator and nth
denominator of the vectorial continued fraction 2.17.
Since both series
n≥0
ρb
2p1
,
n≥0
ρb
2p1
· ρb
2p
2
converge and from Lemma 2.8
above θ
k
and θ
−1
k
are bounded, we conclude that the series
k≥1
ρc
k
converges.
Then, it follows as in the proof of Theorem 2.5, that the vectorial continued fraction 2.17
diverges and
ρ
U
2k1
· V
−1
2k1
− U
2k
· V
−1
2k
ρ
p
2N2k
· q
−1
2N2k
− p
2N2k1
· q
−1
2N2k1
≥ exp
2
∞
k1
ρ
c
k
> 0.
2.21
So,
ρ
s
2N2k1
− s
2N2k
≥ exp
2
∞
k1
ρ
c
k
> 0, ∀k ≥ 0. 2.22
This shows that the sequence of nth approximants s
n
n≥1
is not a ρ-Cauchy sequence in A.
M. Hemdaoui and M. Amzil 7
Now, we state Theorem 2.13 to give a sufficient condition to have convergence of the
vectorial continued fraction 1.1.
A similar theorem was given by Peng and Hessel 5, to study convergence of the
vectorial continued fraction 1.1 in norm where for each positive integer n, a
n
e.
Before stating the proof of Theorem 2.13, we give the following lemmas.
Lemma 2.11. Let b and c be two commuting elements in A such that the spectrum of b
−1
· c is satisfied,
σb
−1
· c ⊂ B0, 1. Then, the element b c is invertible and its inverse satisfies ρb c
−1
≤
ρb
−1
/1 − ρb
−1
· c.
Proof. Since σb
−1
· c ⊂ B0, 1,wehaveρb
−1
· c < 1. So the element b c is invertible in A.Its
inverse is
b c
−1
b
−1
e b
−1
· c
−1
b
−1
·
∞
n0
−1
n
b
−1
· c
n
. 2.23
So,
ρ
b c
−1
≤ ρ
b
−1
·
∞
n0
ρ
n
b
−1
· c
ρ
b
−1
1 − ρ
b
−1
· c
. 2.24
Lemma 2.12. Let ∈0, 1, a
n
n∈N
and b
n
n∈N
be two sequences of elements in A such that for each
positive integer n ≥ 1, the spectra of a
n
· b
−1
n
and b
−1
n
lie in the open ball B0, 1/2. Then, for each
positive integer n ≥ 1, q
−1
n
exists and ρq
−1
n
· q
n−1
<.
Where q
n
is the nth denominator of the vectorial continued fraction 1.1.
Proof. From recurrence relation 1.5 above, we have
q
0
e, q
1
b
1
, 2.25
then, q
−1
1
b
−1
1
and ρq
−1
1
· q
0
ρb
−1
1
≤ 1/2<.
Now, suppose that for n ≥ 2, q
−1
n−1
exists and ρq
−1
n−1
· q
n−2
<.
Then, from recurrence relation 1.4 above, we have
q
n
q
n−1
· b
n
q
n−2
· a
n
q
n−1
·
b
n
q
−1
n−1
· q
n−2
· a
n
. 2.26
Put
c q
−1
n−1
· q
n−2
· a
n
,b b
n
. 2.27
Appling Lemma 2.11,wehave
ρ
b
−1
· c
≤ ρ
q
−1
n−1
· q
n−2
· ρ
b
−1
n
· a
n
<
1
2
. 2.28
So b
n
q
−1
n−1
· q
n−2
· a
n
is invertible and its inverse satisfies
ρ
b
n
q
−1
n−1
· q
n−2
· a
n
−1
<
1/2
1 − 1/2
<
1/2
1 − 1/2
<. 2.29
Therefore, q
−1
n
exists. So, for all n ≥ 0, q
n
is invertible and ρq
−1
n
· q
n−1
<.
8 Journal of Inequalities and Applications
Theorem 2.13. Let ∈0, 1, a
n
and b
n
be commuting terms of the vectorial continued fraction 1.1
such that for each positive integer n ≥ 1, the spectra of a
n
· b
−1
n
and b
−1
n
lie in the open ball B0, 1/2.
Then, the vectorial continued fraction 1.1 converges.
Proof of Theorem 2.13. For positive integers n ≥ 1andm ≥ 1, we introduce the finite vectorial
continued fraction
s
n
m
a
n1
·
b
n1
a
n2
·
b
n2
··· a
nm−1
·
b
nm−1
a
nm
· b
−1
nm
−1
−1
−1
2.30
with initial conditions
s
n
0
0,s
0
m
s
m
, 2.31
where s
m
is the mth approximant of the continued fraction 1.1.
It is easily shown from 2.30 that
s
n
m
a
n1
·
b
n1
s
n1
m−1
−1
. 2.32
By the repeated use of Lemma 2.11 in each iteration in 2.30 for every n ≥ 1 and every m ≥ 1,
we can show that for each n and m, b
n1
s
n1
m−1
−1
exists and
ρ
s
n
m
<. 2.33
We have
b
n1
s
n1
m
−1
−
b
n1
s
n1
m−1
−1
b
n1
s
n1
m
−1
·
s
n1
m−1
− s
n1
m
·
b
n1
s
n1
m−1
−1
. 2.34
Thus, from relations 2.32 and 2.34,wehave
s
n
m1
− s
n
m
a
n1
·
b
n1
s
n1
m
−1
−
b
n1
s
n1
m−1
−1
a
n1
·
b
n1
s
n1
m
−1
·
s
n1
m−1
− s
n1
m
·
b
n1
s
n1
m−1
−1
a
n1
· b
−2
n1
· K
m
·
s
n1
m
− s
n1
m−1
· K
m−1
,
2.35
where K
m
e b
−1
n1
· s
n1
m
−1
,form ∈ N
∗
.
Then,
ρ
s
n
m1
− s
n
m
≤ ρ
a
n1
· b
−1
n1
· ρ
b
−1
n1
· ρ
K
m
· ρ
s
n1
m
− s
n1
m−1
· ρ
K
m−1
. 2.36
Since from 2.33 ρb
−1
n1
· s
n1
m
≤ ρb
−1
n1
· ρs
n1
m
≤ 1/2
2
< 1/2, then, using Lemma 2.11,
ρ
K
m
≤
1
1 − ρ
b
−1
n1
· s
n1
m
< 2, for m ∈
N
∗
, 2.37
we have ρa
n1
· b
−1
n1
≤ 1/2 and ρb
−1
n1
≤ 1/2.
M. Hemdaoui and M. Amzil 9
Then,
ρ
s
n
m1
− s
n
m
≤
2
· ρ
s
n1
m
− s
n1
m−1
. 2.38
Gradually, we get
ρ
s
n
m1
− s
n
m
≤
2m
· ρ
s
nm
1
− s
nm
0
. 2.39
Besides, we have s
nm
0
0ands
nm
1
a
nm1
· b
−1
nm1
.
Thus,
ρ
s
n
m1
− s
n
m
≤
2m
· ρ
s
nm
1
2m
· ρ
a
nm1
· b
−1
nm1
<
1
2
2m1
. 2.40
Now, consider m>1,p≥ 1, we have
ρ
s
n
mp
− s
n
m
≤
ip−1
i0
ρ
s
n
mi1
− s
n
mi
≤
1
2
·
ip−1
i0
2m2i1
1
2
·
2m1
1 −
2p
1 −
2
≤
1
2
·
2m1
1 −
2
.
2.41
In these inequalities n is arbitrary, thus we can choose n 0.
Then,
ρ
s
mp
− s
m
≤
1
2
·
2m1
1 −
2
. 2.42
Hence, the sequence s
m
m∈N
of mth approximants of the vectorial continued fraction 1.1 is a
ρ-Cauchy sequence in A.
Consequently, s
m
converges and from Lemma 2.12, q
−1
n
exists thus the vectorial
continued fraction 1.1 converges.
Theorem 2.14. Let a
n
be a sequence of commuting elements in A such that for each positive integer
n ≥ 1, σa
n
{α
n
}, where 0 ≤ α
n
≤ 1/4. Then, the vectorial continued fraction
a
1
e − a
2
e − a
3
e − a
4
e −···
−1
−1
−1
−1
2.43
converges.
Proof. By relations 1.4 and 1.5,wehaveq
1
e, thus,
σ
q
1
β
1
with β
1
1 1
2
1. 2.44
And q
2
q
1
− q
0
a
2
e − a
2
, thus,
σ
q
2
β
2
with β
2
1 α
2
≥ 1 −
1
4
3
4
· β
1
. 2.45
10 Journal of Inequalities and Applications
By induction, we show that for all n ≥ 2
σ
q
n
β
n
, 2.46
such that
β
n
≥
n 1
2n
β
n−1
,
β
n
β
n−1
− α
n
β
n−2
.
2.47
Hence,
β
n
≥
n 1
2
n
β
0
≥
n 1
2
n
> 0; ∀n ≥ 1. 2.48
So q
−1
n
exists for all n ≥ 1.
Since all a
n
are commuting elements, then by Remark 2.3 above
s
n
s
1
n
k2
s
k
− s
k−1
s
1
n
k2
d
k
q
−1
k
q
−1
k−1
, 2.49
where
d
k
−1
k−1
ik
i1
− a
i
ik
i1
a
i
. 2.50
We have
0 ≤ ρ
d
k
≤
ik
i1
ρ
a
i
≤
1
4
k
. 2.51
Hence,
ρ
d
k
q
−1
k
q
−1
k−1
≤ ρ
d
k
ρ
q
−1
k
ρ
q
−1
k−1
1
β
k
β
k−1
ρ
d
k
≤
1
4
k
2
k
k 1
2
k−1
k
1
2kk 1
.
2.52
Therefore, for positive integers n and m such that n>m,wehave
ρ
s
n
− s
m
≤
n
m1
ρ
d
k
· q
−1
k
· q
−1
k−1
≤
n
km1
1
2kk 1
<
1
m 1
. 2.53
So,
ρ
s
n
− s
m
≤
∞
km1
1
2kk 1
<
1
m 1
. 2.54
It follows that s
n
n≥1
is a ρ-Cauchy sequence in A.
M. Hemdaoui and M. Amzil 11
3. Example
Here, we give an example of a vectorial continued fraction that converges according to the
spectral seminorm and does not converge according to the norm.
Let A be a unital complex Banach algebra and T a nonnull quasinilpotent element in A.
Consider the sequence in A defined for each positive integer n>0, by
u
n
T
1
n
2
· e. 3.1
For each positive integer n>0, u
n
is then invertible.
Let a
n
n∈N
and b
n
n∈N
be two sequences in A defined for each positive integer n>0,
by
a
n
⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
a
1
1
2
· u
1
a
2n
−e, ∀n ≥ 1,
a
2n1
−u
n1
· u
−1
n
, ∀n ≥ 1,
b
n
⎧
⎨
⎩
b
1
e
b
n
e − a
n
, ∀n ≥ 2.
3.2
Consider the vectorial continued fraction 1.1 formed with the sequences a
n
n∈N
and
b
n
n∈N
. Using recurrence relations 1.4 and 1.5, we can easily show that for each positive
integer n ≥ 1, q
n
e thus q
n
is invertible, for all n ≥ 1.
The 2nth approximant and the 2n 1th approximant of the vectorial continued
fraction 1.1 are, respectively, equal to
s
2n
p
2n
n
k1
u
k
n
k1
T
1
k
2
· e nT
n
k1
1
k
2
· e,
s
2n1
p
2n1
n
k1
u
k
u
n1
2
nT
n
k1
1
k
2
· e
1
2
T
1
n 1
2
· e
.
3.3
Obviously, the sequence s
n
n≥0
is not a Cauchy sequence according to the norm, so the
vectorial continued fraction 1.1 does not converge in norm.
Now, we use the spectral seminorm, we have
ρ
s
2n1
− s
2n
1
2
ρ
u
2n1
≤
1
2
ρT
1
2n
2
1
2
1
2n
2
−→ 0,
ρ
s
2n
−
∞
k1
1
k
2
· e
ρ
nT
∞
kn1
1
k
2
· e
≤ nρT
∞
kn1
1
k
2
∞
kn1
1
k
2
−→ 0whenn −→ ∞.
3.4
The sequence s
n
n≥0
of the nth approximants converges according to the spectral seminorm.
Consequently, the vectorial continued fraction 1.1 converges according to the spectral
seminorm to the value
∞
kn1
e/k
2
.
12 Journal of Inequalities and Applications
References
1 W. Fair, “Noncommutative continued fractions,” SIAM Journal on Mathematical Analysis,vol.2,no.2,
pp. 226–232, 1971.
2 P. Wynn, “Continued fractions whose coefficients obey a noncommutative law of multiplication,”
Archive for Rational Mechanics and Analysis, vol. 12, no. 1, pp. 273–312, 1963.
3 H. S. Wall, Analytic Theory of Continued Fractions, D. Van Nostrand, New York, NY, USA, 1948.
4 H. von Koch, “Sur un th
´
eor
`
erme de Stieltjes et sur les fonctions d
´
efinies par des fractions continues,”
Bulletin de la Soci
´
et
´
eMath
´
ematique de France, vol. 23, pp. 33–40, 1895.
5 S. T. Peng and A. Hessel, “Convergence of noncommutative continued fractions,” SIAM Journal on
Mathematical Analysis, vol. 6, no. 4, pp. 724–727, 1975.
