Convergence of Vectorial Continued Fractions Related to the Spectral Seminorm

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.

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 .

A formal vectorial continued fraction is an expression of the form

(1.1)

where and are two sequences of elements in .

In order to discuss convergence or divergence of the vectorial continued fraction (1.1), we associate a sequence (called sequence of th approximants) defined by:

(1.2)

By induction, it can be shown that

(1.3)

where the expressions and are determined from recurrence relations

(1.4)

with initial conditions:

(1.5)

and are respectively called th numerator and th denominator of (1.1).

Now, consider the following example.

Let be a nonnull quasinilpotent element in . Consider the vectorial continued fraction defined by

(1.6)

where for each positive integer , we have

(1.7)

So,

(1.8)

Therefore, the series 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), we get

(1.9)

So, the series 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 satisfies

(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 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]).

For all , we have

(2.1)

Remark 2.3.

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

For all , one has

(2.2)

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

Since coefficients of (1.1) are commuting elements in , it is easy to show that for all positive integers and , we have

(2.3)

So,

(2.4)

Now, suppose that for , .

From relations (1.4) and (2.4), we have

(2.5)

Then, consequently

(2.6)

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 .

By an induction argument, it is easy to show that for all , we have

(2.7)

where and for all .

Since for all positive integer , , and all are commuting elements in , from Remark 2.3 above, we have

(2.8)

So,

(2.9)

Then,

(2.10)

From this preceding,

(2.11)

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.

Since the series converges, therefore, there exists a positive integer such that

(2.12)

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

We have

(2.13)

But

(2.14)

Hence,

(2.15)

Theorem 2.9.

Let in the vectorial continued fraction (1.1) for all and be a sequence of commuting elements in . If both series

(2.16)

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.

Now, consider the vectorial continued fraction

(2.17)

where

(2.18)

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.

For all positive integers , consider the quantities

(2.19)

Then,

(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 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.

Then, it follows as in the proof of Theorem 2.5, that the vectorial continued fraction (2.17) diverges and

(2.21)

So,

(2.22)

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.

Since , we have . So the element is invertible in . Its inverse is

(2.23)

So

(2.24)

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.

From recurrence relation (1.5) above, we have

(2.25)

then, and

Now, suppose that for , exists and

Then, from recurrence relation (1.4) above, we have

(2.26)

Put

(2.27)

Appling Lemma 2.11, we have

(2.28)

So is invertible and its inverse satisfies

(2.29)

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.

For positive integers and , we introduce the finite vectorial continued fraction

(2.30)

with initial conditions

(2.31)

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

It is easily shown from (2.30) that

(2.32)

By the repeated use of Lemma 2.11 in each iteration in (2.30) for every and every , we can show that for each and , exists and

(2.33)

We have

(2.34)

Thus, from relations (2.32) and (2.34), we have

(2.35)

where , for .

Then,

(2.36)

Since from (2.33) , then, using Lemma 2.11,

(2.37)

we have and .

Then,

(2.38)

Gradually, we get

(2.39)

Besides, we have and

Thus,

(2.40)

Now, consider , we have

(2.41)

In these inequalities is arbitrary, thus we can choose

Then,

(2.42)

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.

Let be a sequence of commuting elements in such that for each positive integer , where Then, the vectorial continued fraction

(2.43)

converges.

Proof.

By relations (1.4) and (1.5), we have , thus,

(2.44)

And , thus,

(2.45)

By induction, we show that for all

(2.46)

such that

(2.47)

Hence,

(2.48)

So exists for all

Since all are commuting elements, then by Remark 2.3 above

(2.49)

where

(2.50)

We have

(2.51)

Hence,

(2.52)

Therefore, for positive integers and such that , we have

(2.53)

So

(2.54)

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.

Let be a unital complex Banach algebra and a nonnull quasinilpotent element in . Consider the sequence in defined for each positive integer , by

(3.1)

For each positive integer , is then invertible.

Let and be two sequences in defined for each positive integer , by

(3.2)

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 ).

The th approximant and the th approximant of the vectorial continued fraction (1.1) are, respectively, equal to

(3.3)

Obviously, the sequence 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

(3.4)

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