Convergence of Vectorial Continued Fractions Related to the Spectral Seminorm
© M. Hemdaoui and M. Amzil 2008
Received: 7 February 2008
Accepted: 16 April 2008
Published: 30 April 2008
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.
Now, consider the following example.
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.
Next, we give sufficient conditions to have the convergence.
2. Convergence of Vectorial Continued Fractions
For future use, we record the following theorem due to P. Wynn.
Theorem 2.2 ().
In the commutative case, Theorem 2.2 above becomes as follows.
Now, we give a proposition that extends a result due to Wall  in the case of scalar continued fractions.
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 , concerning the scalar case. A similar theorem was given by Fair  for vectorial continued fractions according to the norm convergence.
converge, then, the vectorial continued fraction (1.1) diverges.
Proof of Theorem 2.9.
Before continuing the proof, we give the following lemma that will be used later.
This lemma is proved by the same argument given by Wall [3, Lemma 6.1] for scalar continued fractions.
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 , 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.
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.
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.
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 ).
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