- Email: [email protected]
Tauberian-type theorem for (e)-convergent sequences
C. R. Acad. Sci. Paris, Ser. I 351 (2013) 177–179
Contents lists available at SciVerse ScienceDirect
C. R. Acad. Sci. Paris, Ser. I www.sciencedirect.com
Complex Analysis/Functional Analysis
Tauberian-type theorem for (e )-convergent sequences ✩ Mubariz T. Karaev Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 23 January 2013 Accepted after revision 28 February 2013 Available online 8 April 2013
We prove a Tauberian-type theorem for (e )-convergent sequences, which were introduced by the author in Karaev (2010) [4]. Our proof is based on the Berezin symbols technique of operator theory in the reproducing kernel Hilbert space. © 2013 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.
Presented by the Editorial Board
1. Introduction The notion of (e )-convergence for sequences was introduced by the author in [4]. Recall that (see [4]) if H = H(Ω) is a Hilbert space of complex-valued functions on some set Ω with reproducing kernel:
kH,λ ( z) :=
en (λ)en ( z),
n0
where e := (en ( z))n0 is an orthonormal basis of H, and (an )n0 is any sequence of complex numbers, then we say that ∞ 2 the sequence (an )n0 is (e )-convergent to a if n=0 an |en (λ)| is convergent for all λ ∈ Ω and
lim ∞
λ→ξ
∞
1
n=0 |en
(λ)|2
2
an en (λ) = a
n =0
for every ξ ∈ ∂Ω . It is easy to verify that for H = H 2 (D) (Hardy space on the unit disc D = { z ∈ C: | z| < 1} of the complex plane C) and H = L a2 (C) (Fock space) our definition of (e )-summability coincides with Abel and Borel summability, respectively. We also recall that the so-called Berezin symbol of a bounded linear operator A on H is the function:
A (λ) := A kH,λ ( z), kH,λ ( z) (λ ∈ Ω), where kH,λ ( z) := kH,λ is the normalized reproducing kernel of H (see, for instance, Berezin [1,2]). Clearly, kH,λ H = H,λ H ∞ ( n=0 |en (λ)|2 )1/2 . For any bounded sequence (an )n0 of complex numbers, let D (a ) denote the diagonal operator on H defined by k
( z)
k
D (ak ) en ( z) = an en ( z),
n = 0, 1 , 2 , . . . ,
with respect to the orthonormal basis e = (en ( z))n0 of H. ✩
Supported by King Saud University, Deanship of Scientific Research, College of Science Research Center. E-mail address: [email protected].
1631-073X/$ – see front matter © 2013 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved. http://dx.doi.org/10.1016/j.crma.2013.02.016
178
M.T. Karaev / C. R. Acad. Sci. Paris, Ser. I 351 (2013) 177–179
Following Nordgren and Rosenthal [5], we say that a functional Hilbert space H(Ω) is standard if the underlying set Ω is a subset of a topological space and if the boundary ∂Ω is non-empty and has the property that ( kH,λ ) converges weakly to 0 whenever λ tends to a point on the boundary. For H(Ω) a standard functional Hilbert space and A a compact operator A (λ) tends to 0 whenever λ tends to an a point on the boundary ∂Ω (since compact operators send weakly on H(Ω), convergent sequences into strongly convergent ones). In this sense, the Berezin symbol of a compact operator on a standard functional Hilbert space vanishes on the boundary. If {ak }k0 is a bounded sequence, then an easy calculus shows that (see [4]): ∞ 2 1 D (ak ) (λ) = ∞ an en (λ) 2 n=0 |en (λ)|
(1.1)
n =0
for all λ ∈ Ω , which gives the following criteria for the (e )-summability method: the sequence (ak )k0 is (e )-convergent D (ak ) (λ) = a for every ξ ∈ ∂Ω . to a if and only if limλ→ξ On the other hand, if H is a standard functional Hilbert space and (ak )k0 converges to a, then by considering that D (ak ) (λ) = D (ak −a) + a and D (ak −a) is a compact operator on the standard functional Hilbert space H, we obtain that (see formula (1.1)):
a = lim D (ak−a ) (λ) + a = lim D (ak ) (λ) λ→ξ
λ→ξ
= lim ∞ λ→ξ
∞
1
n=0 |en
(λ)|2
2
an en (λ) ,
n =0
which shows that (ak )k0 (e )-converges to a. This shows that the functional Hilbert space is regular in the sense that it transforms (see [4]). The purpose of the present article is to solve a third important a Tauberian-type theorem for (e )-convergent sequences. (The good classical summability methods are the books by Hardy [3], by Powell
(e )-summability method associated with a standard convergent sequences into (e )-convergent sequences problem for (e )-convergence, namely, we will prove references for Tauberian-type theorems for valuable and Shah [7], and also by Postnikov [6].)
2. A Tauberian-type theorem for (e )-convergent sequences In this section, we use the Berezin symbols technique to prove a Tauberian-type theorem for (e )-convergent sequences. Namely, we will apply a result of Nordgren and Rosenthal [5, Corollary 2.8], which states that an operator on a standard functional Hilbert space is compact if and only if all the Berezin symbols of all unitarily equivalent operators vanish on the boundary. Let H = H(Ω), e = (en ( z))n0 and kH,λ be the same as in the definition of (e )-convergent sequences. Also, let 21 denote the unit sphere of the sequences space l2 :
21 := (xm )m0 ∈ 2 : (xm ) 2 = 1 . Our result is the following. Theorem. Let (an )n0 be a bounded sequence of complex numbers such that (an )n0 (e )-converges to a. Denoting by kH,λ the reproducing kernel of the standard functional Hilbert space H at λ, we assume that +∞
+∞ 2
(am − a) xmn en (λ) = o kH,λ 2
m =0
(2.1)
n =0
+∞ for every double sequence (xmn )m ,n=0 with (xmn )m = 1 (∀n 0) and (xmn )n = 1 (∀m 0), whenever λ tends to a point in the boundary of Ω . Then (an )n converges to a in the usual sense.
Proof. First note that if U is a unitary operator on H and bmn := U en ( z), em ( z) (m, n = 0, 1, 2, . . .) are its matrix elements, then it is classical (and trivial) that both sequences (bmn )m and (bmn )n are in the unit sphere 21 of 2 . Then we have
U −1 D (an −a) U (λ) = U −1 D (an −a) U kH,λ , kH,λ
= =
1
kH,λ 2 1
kH,λ 2
U
−1
D (an −a) U
en (λ)en ( z),
n0
D (an −a)
n0
en (λ)U en ( z),
n0
n0
en (λ)en ( z)
en (λ)U en ( z)
M.T. Karaev / C. R. Acad. Sci. Paris, Ser. I 351 (2013) 177–179
= = =
=
1
kH,λ 2 1
kH,λ 2 1
kH,λ 2 1
kH,λ 2
179
D (an −a) en (λ) bmn em ( z), en (λ) bmn em ( z) n0
m0
en (λ)(am − a)bmn em ( z),
n0 m0
m0
∞
m0
en (λ)bmn em ( z)
n0 m0
(am − a) en (λ)bmn em ( z), en (λ)bmn em ( z) n0
∞ 2 (am − a) bmn en (λ)
m =0
n0
m0
n0
n =0
for every unitary operator U and λ ∈ Ω . Hence, by using condition (2.1) of the theorem, we assert that
U −1 D (an −a) U (λ) → 0 as λ tends to a point on the boundary ∂Ω , as desired. Then, by the above-mentioned result of Nordgren and Rosenthal [5, Corollary 2.8], we deduce that D (an −a) is a compact operator on H, and consequently an − a → 0 (n → ∞), that is an converges to a, which proves the theorem. 2 Acknowledgement I am grateful to the referee for his useful remarks. References [1] [2] [3] [4] [5] [6] [7]
F.A. Berezin, Covariant and contravariant symbols for operators, Math. USSR-Izv. 6 (1972) 1117–1151. F.A. Berezin, Quantization, Math. USSR-Izv. 8 (1974) 1109–1163. G.H. Hardy, Divergent Series, Clarendon Press, Oxford, 1956. M.T. Karaev, (e )-Convergence and related problem, C. R. Acad. Sci. Paris, Ser. I 348 (2010) 1059–1062. E. Nordgren, P. Rosenthal, Boundary values of Berezin symbols, Oper. Theory Adv. Appl. 73 (1994) 362–368. A.G. Postnikov, Tauberian Theory and Its Applications, Proc. Steklov Inst. Math., vol. 144, Amer. Math. Soc., 1980. R.E. Powell, S.M. Shah, Summability Theory and Applications, Prentice-Hall, 1988.
Contents lists available at SciVerse ScienceDirect
C. R. Acad. Sci. Paris, Ser. I www.sciencedirect.com
Complex Analysis/Functional Analysis
Tauberian-type theorem for (e )-convergent sequences ✩ Mubariz T. Karaev Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 23 January 2013 Accepted after revision 28 February 2013 Available online 8 April 2013
We prove a Tauberian-type theorem for (e )-convergent sequences, which were introduced by the author in Karaev (2010) [4]. Our proof is based on the Berezin symbols technique of operator theory in the reproducing kernel Hilbert space. © 2013 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.
Presented by the Editorial Board
1. Introduction The notion of (e )-convergence for sequences was introduced by the author in [4]. Recall that (see [4]) if H = H(Ω) is a Hilbert space of complex-valued functions on some set Ω with reproducing kernel:
kH,λ ( z) :=
en (λ)en ( z),
n0
where e := (en ( z))n0 is an orthonormal basis of H, and (an )n0 is any sequence of complex numbers, then we say that ∞ 2 the sequence (an )n0 is (e )-convergent to a if n=0 an |en (λ)| is convergent for all λ ∈ Ω and
lim ∞
λ→ξ
∞
1
n=0 |en
(λ)|2
2
an en (λ) = a
n =0
for every ξ ∈ ∂Ω . It is easy to verify that for H = H 2 (D) (Hardy space on the unit disc D = { z ∈ C: | z| < 1} of the complex plane C) and H = L a2 (C) (Fock space) our definition of (e )-summability coincides with Abel and Borel summability, respectively. We also recall that the so-called Berezin symbol of a bounded linear operator A on H is the function:
A (λ) := A kH,λ ( z), kH,λ ( z) (λ ∈ Ω), where kH,λ ( z) := kH,λ is the normalized reproducing kernel of H (see, for instance, Berezin [1,2]). Clearly, kH,λ H = H,λ H ∞ ( n=0 |en (λ)|2 )1/2 . For any bounded sequence (an )n0 of complex numbers, let D (a ) denote the diagonal operator on H defined by k
( z)
k
D (ak ) en ( z) = an en ( z),
n = 0, 1 , 2 , . . . ,
with respect to the orthonormal basis e = (en ( z))n0 of H. ✩
Supported by King Saud University, Deanship of Scientific Research, College of Science Research Center. E-mail address: [email protected].
1631-073X/$ – see front matter © 2013 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved. http://dx.doi.org/10.1016/j.crma.2013.02.016
178
M.T. Karaev / C. R. Acad. Sci. Paris, Ser. I 351 (2013) 177–179
Following Nordgren and Rosenthal [5], we say that a functional Hilbert space H(Ω) is standard if the underlying set Ω is a subset of a topological space and if the boundary ∂Ω is non-empty and has the property that ( kH,λ ) converges weakly to 0 whenever λ tends to a point on the boundary. For H(Ω) a standard functional Hilbert space and A a compact operator A (λ) tends to 0 whenever λ tends to an a point on the boundary ∂Ω (since compact operators send weakly on H(Ω), convergent sequences into strongly convergent ones). In this sense, the Berezin symbol of a compact operator on a standard functional Hilbert space vanishes on the boundary. If {ak }k0 is a bounded sequence, then an easy calculus shows that (see [4]): ∞ 2 1 D (ak ) (λ) = ∞ an en (λ) 2 n=0 |en (λ)|
(1.1)
n =0
for all λ ∈ Ω , which gives the following criteria for the (e )-summability method: the sequence (ak )k0 is (e )-convergent D (ak ) (λ) = a for every ξ ∈ ∂Ω . to a if and only if limλ→ξ On the other hand, if H is a standard functional Hilbert space and (ak )k0 converges to a, then by considering that D (ak ) (λ) = D (ak −a) + a and D (ak −a) is a compact operator on the standard functional Hilbert space H, we obtain that (see formula (1.1)):
a = lim D (ak−a ) (λ) + a = lim D (ak ) (λ) λ→ξ
λ→ξ
= lim ∞ λ→ξ
∞
1
n=0 |en
(λ)|2
2
an en (λ) ,
n =0
which shows that (ak )k0 (e )-converges to a. This shows that the functional Hilbert space is regular in the sense that it transforms (see [4]). The purpose of the present article is to solve a third important a Tauberian-type theorem for (e )-convergent sequences. (The good classical summability methods are the books by Hardy [3], by Powell
(e )-summability method associated with a standard convergent sequences into (e )-convergent sequences problem for (e )-convergence, namely, we will prove references for Tauberian-type theorems for valuable and Shah [7], and also by Postnikov [6].)
2. A Tauberian-type theorem for (e )-convergent sequences In this section, we use the Berezin symbols technique to prove a Tauberian-type theorem for (e )-convergent sequences. Namely, we will apply a result of Nordgren and Rosenthal [5, Corollary 2.8], which states that an operator on a standard functional Hilbert space is compact if and only if all the Berezin symbols of all unitarily equivalent operators vanish on the boundary. Let H = H(Ω), e = (en ( z))n0 and kH,λ be the same as in the definition of (e )-convergent sequences. Also, let 21 denote the unit sphere of the sequences space l2 :
21 := (xm )m0 ∈ 2 : (xm ) 2 = 1 . Our result is the following. Theorem. Let (an )n0 be a bounded sequence of complex numbers such that (an )n0 (e )-converges to a. Denoting by kH,λ the reproducing kernel of the standard functional Hilbert space H at λ, we assume that +∞
+∞ 2
(am − a) xmn en (λ) = o kH,λ 2
m =0
(2.1)
n =0
+∞ for every double sequence (xmn )m ,n=0 with (xmn )m = 1 (∀n 0) and (xmn )n = 1 (∀m 0), whenever λ tends to a point in the boundary of Ω . Then (an )n converges to a in the usual sense.
Proof. First note that if U is a unitary operator on H and bmn := U en ( z), em ( z) (m, n = 0, 1, 2, . . .) are its matrix elements, then it is classical (and trivial) that both sequences (bmn )m and (bmn )n are in the unit sphere 21 of 2 . Then we have
U −1 D (an −a) U (λ) = U −1 D (an −a) U kH,λ , kH,λ
= =
1
kH,λ 2 1
kH,λ 2
U
−1
D (an −a) U
en (λ)en ( z),
n0
D (an −a)
n0
en (λ)U en ( z),
n0
n0
en (λ)en ( z)
en (λ)U en ( z)
M.T. Karaev / C. R. Acad. Sci. Paris, Ser. I 351 (2013) 177–179
= = =
=
1
kH,λ 2 1
kH,λ 2 1
kH,λ 2 1
kH,λ 2
179
D (an −a) en (λ) bmn em ( z), en (λ) bmn em ( z) n0
m0
en (λ)(am − a)bmn em ( z),
n0 m0
m0
∞
m0
en (λ)bmn em ( z)
n0 m0
(am − a) en (λ)bmn em ( z), en (λ)bmn em ( z) n0
∞ 2 (am − a) bmn en (λ)
m =0
n0
m0
n0
n =0
for every unitary operator U and λ ∈ Ω . Hence, by using condition (2.1) of the theorem, we assert that
U −1 D (an −a) U (λ) → 0 as λ tends to a point on the boundary ∂Ω , as desired. Then, by the above-mentioned result of Nordgren and Rosenthal [5, Corollary 2.8], we deduce that D (an −a) is a compact operator on H, and consequently an − a → 0 (n → ∞), that is an converges to a, which proves the theorem. 2 Acknowledgement I am grateful to the referee for his useful remarks. References [1] [2] [3] [4] [5] [6] [7]
F.A. Berezin, Covariant and contravariant symbols for operators, Math. USSR-Izv. 6 (1972) 1117–1151. F.A. Berezin, Quantization, Math. USSR-Izv. 8 (1974) 1109–1163. G.H. Hardy, Divergent Series, Clarendon Press, Oxford, 1956. M.T. Karaev, (e )-Convergence and related problem, C. R. Acad. Sci. Paris, Ser. I 348 (2010) 1059–1062. E. Nordgren, P. Rosenthal, Boundary values of Berezin symbols, Oper. Theory Adv. Appl. 73 (1994) 362–368. A.G. Postnikov, Tauberian Theory and Its Applications, Proc. Steklov Inst. Math., vol. 144, Amer. Math. Soc., 1980. R.E. Powell, S.M. Shah, Summability Theory and Applications, Prentice-Hall, 1988.