- Research Article
- Open Access

# Convergence of Vectorial Continued Fractions Related to the Spectral Seminorm

- M. Hemdaoui
^{1}Email author and - M. Amzil
^{1}

**2008**:768105

https://doi.org/10.1155/2008/768105

© M. Hemdaoui and M. Amzil 2008

**Received:**7 February 2008**Accepted:**16 April 2008**Published:**30 April 2008

## Abstract

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.

## Keywords

- Positive Integer
- Large Class
- Spectral Property
- Recurrence Relation
- Banach Algebra

## 1. Introduction

Let be a unital complex Banach algebra. We denote by the unit element of . is the norm of . For , and denote, respectively, the spectrum and the spectral seminorm of .

where and are two sequences of elements in .

Now, consider the following example.

Therefore, the series diverges.

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 and to ensure the convergence.

Next, we give sufficient conditions to have the convergence.

## 2. Convergence of Vectorial Continued Fractions

In this section, we discuss some conditions upon the elements and of the vectorial continued fraction (1.1) (with ) which are necessary to ensure the convergence.

Definition 2.1.

The vectorial continued fraction (1.1) converges if exists starting from a certain rank , and the sequence of th approximants 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]).

Remark 2.3.

In the commutative case, Theorem 2.2 above becomes as follows.

Since convergence or divergence of the vectorial continued fraction (1.1) is not affected by the value of the additive term , we omit it from subsequent discussion (i.e., ).

Now, we give a proposition that extends a result due to Wall [3] in the case of scalar continued fractions.

Proposition 2.4.

The vectorial continued fraction (1.1) where its terms are commuting elements in diverges, if its odd partial denominators are all quasinilpotent elements in .

Proof.

In fact, from relation (1.5) above, we have . So

Now, suppose that for , .

So infinitely many denominators are not invertible.

The vectorial continued fraction (1.1) diverges.

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 , for all , and be a sequence of commuting elements in . If the vectorial continued fraction (1.1) converges according to spectral seminorm, then, the series diverges.

Proof.

Suppose is a converging series, and there exists a positive integer such that exists, for all .

where and for all .

where

So, the sequence is not a -Cauchy sequence in

Remark 2.6.

In a Banach algebra if denotes the spectral seminorm in it is not a multiplicative seminorm in general.

Consider the vectorial subspace of defined by The quotient vectorial space becomes a normed vectorial space with norm defined by " denotes the class of modulo ."

Generally, the normed vectorial space is not complete. Its complete normed vectorial space is witch is a Banach space. So, -Cauchy sequences in converge in the Banach space .

Remark 2.7.

Whenever is commutative, the vectorial continued fraction (1.1) diverges, if for one character , the series converges.

Lemma 2.8.

Let be a sequence of commuting elements in .

If the series converges, then, there exists a positive integer such that for every positive integer , the finite product is invertible and -bounded and its inverse is also -bounded.

Proof.

Hence, for the product is invertible as finite product of invertible elements.

Theorem 2.9.

converge, then, the vectorial continued fraction (1.1) diverges.

Proof of Theorem 2.9.

Since both series and converge, it follows that the series converges too.

Therefore, from Lemma 2.8 above, there exists a positive integer such that for , the quantity is invertible.

We will suppose that exists for all (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.

This lemma is proved by the same argument given by Wall [3, Lemma 6.1] for scalar continued fractions.

Lemma 2.10 shows that and are respectively the th numerator and th denominator of the vectorial continued fraction (2.17).

Since both series converge and from Lemma 2.8 above and are bounded, we conclude that the series converges.

This shows that the sequence of th approximants is not a -Cauchy sequence in .

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 , .

Before stating the proof of Theorem 2.13, we give the following lemmas.

Lemma 2.11.

Let and be two commuting elements in such that the spectrum of is satisfied, . Then, the element is invertible and its inverse satisfies

Proof.

Lemma 2.12.

Let , and be two sequences of elements in such that for each positive integer , the spectra of and lie in the open ball . Then, for each positive integer , exists and

Where is the th denominator of the vectorial continued fraction (1.1).

Proof.

then, and

Now, suppose that for , exists and

Therefore, exists. So, for all , is invertible and

Theorem 2.13.

Let , and be commuting terms of the vectorial continued fraction (1.1) such that for each positive integer , the spectra of and lie in the open ball . Then, the vectorial continued fraction (1.1) converges.

Proof of Theorem 2.13.

where is the th approximant of the continued fraction (1.1).

We have

where , for .

we have and .

Besides, we have and

In these inequalities is arbitrary, thus we can choose

Hence, the sequence of th approximants of the vectorial continued fraction (1.1) is a -Cauchy sequence in .

Consequently, converges and from Lemma 2.12, exists thus the vectorial continued fraction (1.1) converges.

Theorem 2.14.

converges.

Proof.

So exists for all

It follows that is a -Cauchy sequence in .

## 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.

For each positive integer , is then invertible.

Consider the vectorial continued fraction (1.1) formed with the sequences and Using recurrence relations (1.4) and (1.5), we can easily show that for each positive integer , (thus is invertible, for all ).

Obviously, the sequence is not a Cauchy sequence according to the norm, so the vectorial continued fraction (1.1) does not converge in norm.

The sequence of the th approximants converges according to the spectral seminorm.

Consequently, the vectorial continued fraction (1.1) converges according to the spectral seminorm to the value

## Authors’ Affiliations

## References

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*Analytic Theory of Continued Fractions*. D. Van Nostrand, New York, NY, USA; 1948:xiii+433.MATHGoogle Scholar - von Koch H:
**Sur un théorèrme de Stieltjes et sur les fonctions définies par des fractions continues.***Bulletin de la Société Mathématique de France*1895,**23:**33–40.MATHMathSciNetGoogle Scholar - Peng ST, Hessel A:
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## Copyright

This article is published under license to BioMed Central Ltd. 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.