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, mhemdaoui@yahoo.fr 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, σa, and ρa 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 n1  p n · b n1  p n−1 · a n1 , q n1  q n · b n1  q n−1 · a n1 , 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  ∞ n1 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  ρa 1 n 2 . 1.9 So, the series  ∞ n1 ρb 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 ρx ≤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 1.1with 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 n1 − s n  p n1 · q −1 n1 − 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 n1 q −1 n1 . 2.1 Remark 2.3. In the commutative case, Theorem 2.2 above becomes as follows. For all n ∈ N, one has s n1 − s n −1 n  in1  i1 a i  · q −1 n1 · 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 2n1 are all quasinilpotent elements in A. Proof. In fact, from relation 1.5 above, we have q 1  b 1 .Soρq 1 ρb 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, ρq 2n−1 0. From relations 1.4 and 2.4,wehave ρ  q 2n1  ≤ ρ  q 2n  · ρ  b 2n1   ρ  q 2n−1  · ρ  a 2n1  . 2.5 Then, ρq 2n1 0, consequently ∀n ≥ 0; ρ  q 2n1   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  ∞ n1 ρb n  diverges. Proof. Suppose  ∞ n1 ρb 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 2n1  ≤ exp  K 2n1  , 2.7 where K 0  0andK n   n k1 ρb 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 2n1 − s 2n  q −1 2n1 · q −1 2n , ∀n ≥ E  N 2   1. 2.8 So, ρ  d n  ≥  ρ  d −1 n  −1   ρ  q 2n1 · q 2n  −1 , ∀n ≥ E  N 2   1. 2.9 Then, ρ  d n  ≥  ρ  q 2n1  −1 ·  ρ  q 2n  −1 , ∀n ≥ E  N 2   1. 2.10 From this preceding, ρ  d n  ≥ 1 exp  K 2n1  · 1 exp  K 2n  − 1 ≥ 1 exp  2K 2n1  ≥ 1 exp2K > 0, 2.11 where K   ∞ n1 ρb 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 Kerρ{x ∈A|ρx0}. The quotient vectorial space A/ Kerρ becomes a normed vectorial space with norm defined by ˙ρ ˙xρx,x∈ ˙x. “˙x denotes the class of x modulo Kerρ.” Generally, the normed vectorial space A/ Kerρ is not complete. Its complete normed vectorial space is  A/ Kerρ witch is a Banach space. So, ρ-Cauchy sequences in A,ρ converge in the Banach space  A/ Kerρ . 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 |ψb n | converges. Lemma 2.8. Let u n  n be a sequence of commuting elements in A. If the series  n≥1 ρu n  converges, then, there exists a positive integer N ≥ 1 such that for every positive integer k ≥ 1, the finite product  k p1 e  u Np  is invertible and ρ-bounded and its inverse is also ρ-bounded. Proof. Since the series  n≥1 ρu 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 p1 eu Np  is invertible as finite product of invertible elements. We have ρ  k  p1  e  u Np   ≤ k  p1  1  ρ  u Np  ≤ ∞  p1  1  ρ  u Np  . 2.13 But  k  p1  e  u Np   −1  k  p1  e  u Np  −1  k  p1 ∞  n0 −1 n u n Np . 2.14 Hence, ρ  k  p1  e  u Np   −1  ≤ k  p1 ∞  n0 ρ n  u Np   k  p1 1 1 − ρ  u Np   1  k p1  1 − ρ  u Np  ≤ 1  ∞ p1  1 − ρ  u Np  . 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 2p1  ,  n≥0 ρ  b 2p1  · 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 ρb 2p1  and  n≥0 ρb 2p1  · ρ 2 b 2p  converge, it follows that the series  n≥0 ρb 2p1  · ρb 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 p1 1  b 2N2p1 · b 2N2p  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 2N2k1 · θ −1 k−1 · θ −1 k ,c 2k−1  −b 2N2k1 · b 2 2N2k · 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 2N2k1 · θ −1 k ,V 2k  q 2N2k1 · θ −1 k , U 2k1  p 2N2k · θ k ,V 2k1  q 2N2k · θ k , c 2k  b 2N2k1 · θ −1 k−1 · θ −1 k ,c 2k−1  −b 2N2k1 · b 2 2N2k · 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 ρb 2p1 ,  n≥0 ρb 2p1  · ρb 2p  2 converge and from Lemma 2.8 above θ k and θ −1 k are bounded, we conclude that the series  k≥1 ρc k  converges. Then, it follows as in the proof of Theorem 2.5, that the vectorial continued fraction 2.17 diverges and ρ  U 2k1 · V −1 2k1 − U 2k · V −1 2k   ρ  p 2N2k · q −1 2N2k − p 2N2k1 · q −1 2N2k1  ≥ exp  2 ∞  k1 ρ  c k   > 0. 2.21 So, ρ  s 2N2k1 − s 2N2k  ≥ exp  2 ∞  k1 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, σb −1 · c ⊂ B0, 1. Then, the element b  c is invertible and its inverse satisfies ρb  c −1  ≤ ρb −1 /1 − ρb −1 · c. Proof. Since σb −1 · c ⊂ B0, 1,wehaveρb −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 · ∞  n0 −1 n  b −1 · c  n . 2.23 So, ρ  b  c −1  ≤ ρ  b −1  · ∞  n0 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 B0, 1/2. Then, for each positive integer n ≥ 1, q −1 n exists and ρq −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 ρq −1 1 · q 0 ρb −1 1  ≤ 1/2<. Now, suppose that for n ≥ 2, q −1 n−1 exists and ρq −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 ρq −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 B0, 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 n1 ·  b n1  a n2 ·  b n2  ··· a nm−1 ·  b nm−1  a nm · b −1 nm  −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 n1 ·  b n1  s n1 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 n1  s n1 m−1  −1 exists and ρ  s n m  <. 2.33 We have  b n1  s n1 m  −1 −  b n1  s n1 m−1  −1   b n1  s n1 m  −1 ·  s n1 m−1 − s n1 m  ·  b n1  s n1 m−1  −1 . 2.34 Thus, from relations 2.32 and 2.34,wehave s n m1 − s n m  a n1 ·  b n1  s n1 m  −1 −  b n1  s n1 m−1  −1   a n1 ·  b n1  s n1 m  −1 ·  s n1 m−1 − s n1 m  ·  b n1  s n1 m−1  −1  a n1 · b −2 n1 · K m ·  s n1 m − s n1 m−1  · K m−1 , 2.35 where K m e  b −1 n1 · s n1 m  −1 ,form ∈ N ∗ . Then, ρ  s n m1 − s n m  ≤ ρ  a n1 · b −1 n1  · ρ  b −1 n1  · ρ  K m  · ρ  s n1 m − s n1 m−1  · ρ  K m−1  . 2.36 Since from 2.33 ρb −1 n1 · s n1 m  ≤ ρb −1 n1  · ρs n1 m  ≤ 1/2 2 < 1/2, then, using Lemma 2.11, ρ  K m  ≤ 1 1 − ρ  b −1 n1 · s n1 m  < 2, for m ∈ N ∗ , 2.37 we have ρa n1 · b −1 n1  ≤ 1/2 and ρb −1 n1  ≤ 1/2. M. Hemdaoui and M. Amzil 9 Then, ρ  s n m1 − s n m  ≤  2 · ρ  s n1 m − s n1 m−1  . 2.38 Gradually, we get ρ  s n m1 − s n m  ≤  2m · ρ  s nm 1 − s nm 0  . 2.39 Besides, we have s nm 0  0ands nm 1  a nm1 · b −1 nm1 . Thus, ρ  s n m1 − s n m  ≤  2m · ρ  s nm 1    2m · ρ  a nm1 · b −1 nm1  < 1 2  2m1 . 2.40 Now, consider m>1,p≥ 1, we have ρ  s n mp − s n m  ≤ ip−1  i0 ρ  s n mi1 − s n mi  ≤ 1 2 ·  ip−1  i0  2m2i1   1 2 ·  2m1  1 −  2p  1 −  2 ≤ 1 2 ·  2m1 1 −  2 . 2.41 In these inequalities n is arbitrary, thus we can choose n  0. Then, ρ  s mp − s m  ≤ 1 2 ·  2m1 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, σa 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  k2  s k − s k−1   s 1  n  k2 d k q −1 k q −1 k−1 , 2.49 where d k −1 k−1 ik  i1  − a i   ik  i1 a i . 2.50 We have 0 ≤ ρ  d k  ≤ ik  i1 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 2kk  1 . 2.52 Therefore, for positive integers n and m such that n>m,wehave ρ  s n − s m  ≤ n  m1 ρ  d k · q −1 k · q −1 k−1  ≤ n  km1 1 2kk  1 < 1 m  1 . 2.53 So, ρ  s n − s m  ≤ ∞  km1 1 2kk  1 < 1 m  1 . 2.54 It follows that s n  n≥1 is a ρ-Cauchy sequence in A. [...]... to the spectral seminorm Consequently, the vectorial continued fraction 1.1 converges according to the spectral seminorm to the value ∞ n 1 e/k 2 k 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,”... Obviously, the sequence sn 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 s2n ρ s2n − 1 − s2n ∞ k 1 ·e k2 1 1 ρ u2n 2 ∞ ρ nT k n 1 1 ·e k2 1 ≤ 1 2 ρ T ≤ nρ T 1 2n2 ∞ k n 1 k2 1 1 2n2 1 2 ∞ k n −→ 0, 1 −→ 0 when n −→ ∞ k2 1 3.4 The sequence sn n≥0 of the nth approximants converges according to the. .. bn ⎩b e − an , ∀n ≥ 2 n Consider the vectorial continued fraction 1.1 formed with the sequences an n∈N and bn n∈N Using recurrence relations 1.4 and 1.5 , we can easily show that for each positive integer n ≥ 1, qn e thus qn is invertible, for all n ≥ 1 The 2n th approximant and the 2n 1 th approximant of the vectorial continued fraction 1.1 are, respectively, equal to n s2n n p2n uk T k 1 n s2n 1... 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 un T 1 · e n2 3.1 For each positive integer n > 0, un is then invertible Let an n∈N and... 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´ or` rme de Stieltjes et sur les fonctions d´ finies par des fractions continues,” e e e Bulletin de la Soci´ t´ Math´ matique de France, vol 23, pp 33–40, 1895 ee e 5 S T Peng and A Hessel, Convergence of noncommutative continued fractions, ” SIAM Journal on Mathematical Analysis, vol . 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. 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. 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