Acta Mathematica Sinica, English Series Jan., 2007, Vol. 23, No. 1, pp. 57–64 Published online: May 15, 2006 DOI: 10.1007/s10114-005-0784-1 Http://www.ActaMath.com
Integrated Groups and Smooth Distribution Groups Pedro J. MIANA Departamento de Matem´ aticas, Universidad de Zaragoza, 50009, Spain E-mail:
[email protected] Abstract In this paper, we prove directly that α-times integrated groups define algebra homomorphisms. We also give a theorem of equivalence between smooth distribution groups and α-times integrated groups. Keywords integrated groups, distribution groups, Weyl fractional calculus MR(2000) Subject Classification 47D62, 26A33
Introduction Integrated groups in Banach spaces have been introduced to study abstract “ill-posed” Cauchy problems, for example the Schr¨ odinger problem in Lp (Rn ) with p ≥ 1, see [1]. n-Times integrated semigroups were introduced by Arendt ([2]), and Hieber and Kellerman ([3]) with n ∈ N, and later, Hieber defined α-times integrated semigroups with α ≥ 0 ([4]). Differential operators and differential operators with potentials are examples of integrated groups, see for example [1] and [5]. On the other hand, vector-valued distribution groups appeared also in connection with the Cauchy problem ([6, 7]). For integer order, both concepts are essentially the same with polynomial or exponential growth; see [8, Theorem 3.4] and [9]. In this paper, we present an approach which extends largely these results. We prove directly that α-times integrated groups define algebra homomorphisms, and smooth distribution groups of fractional order are equivalent to α-times integrated groups. These ideas are followed also in the cases of integrated semigroups ([10]). Algebra homomorphisms allow us to prove subordination results on integrated groups; see the last section. Fractional Banach algebras for the convolution product T (α) (τα ) are defined using the Weyl fractional derivation. They are canonical for α-times integrated groups since Bochner–Riesz functions belong to them. Algebras T (α) (τα ) were defined in [11] in the context of quasimultipliers for Banach algebras. Notations and main facts of fractional calculi are presented in the first section; see also [12]. 1
Convolution Fractional Banach Algebras on R
In this section, we review some results proven in [11]. We also give a new result about fractional calculi in Lemma 1. Let D be the class of C ∞ functions with compact support on R and S the Schwartz class on R. For a function f ∈ S , Weyl fractional integrals W+−α f, W−−α f of order α > 0 are defined by ∞ t 1 1 W+−α f (t) := (s − t)α−1 f (s)ds, W−−α f (t) := (t − s)α−1 f (s)ds, Γ(α) t Γ(α) −∞ Received March 16, 2004, Accepted June 10, 2005 Partially supported by the Spanish Project MTM2004-03036, MCYT DGI–FEDER and the DGA Project “An´ alisis Matem´ atico y Aplicaciones” E-12/25
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 133, Number 9, Pages 2569–2575 S 0002-9939(05)07978-5 Article electronically published on March 31, 2005
UNIFORMLY BOUNDED LIMIT OF FRACTIONAL HOMOMORPHISMS PEDRO J. MIANA (Communicated by Jonathan M. Borwein) Abstract. We show that a bounded homomorphism T : L1ω (R+ ) → A is equivalent to a uniformly bounded family of fractional homomorphisms Tα : (α) ACω (R+ ) → A for any α > 0. We add this characterization to the WidderArendt-Kisy´ nski theorem and relate it to α-times integrated semigroups.
1. Introduction Let A be a Banach algebra (with or without identity). For Ω ⊂ C, a family (rλ )λ∈Ω of elements of A is called a pseudo-resolvent if the equation rλ − rµ = (µ − λ)rλ rµ holds for λ, µ ∈ Ω. An example of a pseudo-resolvent is the following. Let R, R+ and C be the sets of real, positive real and complex numbers, respectively. For each λ ∈ R, denote by λ the function t ∈ R+ .
λ (t) = eλt ,
Take ω ∈ R+ ∪ {0} and let L1ω (R+ ) be the usual Banach algebra with norm given by ∞
|f (t)|eωt dt < +∞, f w := 0 t and the convolution f ∗ g(t) := 0 f (t − s)g(s)ds, with t ≥ 0, as its product. Then (−λ )λ∈(ω,+∞) is a pseudo-resolvent in L1ω (R+ ) and verifies 1 −λ ∗ · · · ∗ −λ ω = , (λ − ω)n
λ ∈ (ω, +∞), n ∈ N.
n times
Moreover, the set (−λ )λ∈(ω,+∞) is linearly dense in L1ω (R+ ). The next result shows the equivalence between a homomorphism T : L1ω (R+ ) → A and a class of pseudo-resolvents; in other words, the family (−λ )λ∈(ω,+∞) is universal for this class of pseudo-resolvents. We present here an early version; see [8] and [3] which is included in [9, Theorem 5.1]. Recently, this result has been called the Widder-Arendt-Kisy´ nski theorem; see for example [5, Theorem 1.1]. Received by the editors February 1, 2003. 2000 Mathematics Subject Classification. Primary 47D62; Secondary 26A33, 46J25. Key words and phrases. Pseudo-resolvents, homomorphisms, integrated semigroups. This work was supported by a grant from Programa Europa, CAI, 2002. This paper was made during a visit to the Charles University in Prague. The author thanks Dr. Eva Fasangova and the Analysis Mathematical Department for the stay in Prague. c 2005 American Mathematical Society Reverts to public domain 28 years from publication
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Semigroup Forum Vol. 73 (2006) 61–74
c 2006 Springer
DOI: 10.1007/s00233-006-0631-4
RESEARCH ARTICLE
Convolution Products in Naturally Ordered Semigroups Pedro J. Miana ∗ Communicated by Jerome A. Goldstein
Abstract In this paper new equalities between two different convolution products in cancellative naturally ordered semigroups (but not in groups) are given. We also give several applications in particular cases N∗ and R+∗ . Keywords: Convolution products, naturally ordered semigroups and groups. AMS Classification: Primary: 22A15; Secondary 22D15, 43A05.
0. Introduction Let L1 (R+ ) be the usual Banach algebra with the convolution product t f ∗ g(t) = f (t − s)g(s) ds, t ≥ 0, f, g ∈ L1 (R+ ). 0
This convolution product ∗ is commutative and associative. However a second convolution product, ◦ , may be considered in L1 (R+ ), ∞ f ◦ g(t) = f (s − t)g(s) ds, t ≥ 0, f, g ∈ L1 (R+ ). t
Both convolution products are independent but a joint approach may be given. One can consider L1 (R+ ) → L1 (R) , f → F , defined by f (t), if t ≥ 0, F (t) := 0, if t < 0, and the usual convolution product in L1 (R) , G ∗ H(t) = with t ∈ R and G, H ∈ L1 (R) . Take f, g ∈ L1 (R+ ) then f ∗ g(t) = F ∗ G(t),
f ◦ g(t) = F˜ ∗ G(t),
∞
−∞
G(t − s)H(s) ds
t ≥ 0,
where F˜ (t) := F (−t) with t ∈ R . ∗ Supported by Project MTM2004-03036, DGI-FEDER, of the MCYT Spain and Project E-64 of D.G.A., Spain.
Semigroup Forum Vol. 71 (2005) 119–133
c 2005 Springer
DOI: 10.1007/s00233-005-0519-8
RESEARCH ARTICLE
Vector-Valued Cosine Transforms Pedro J. Miana ∗ Communicated by Jerome A. Goldstein
Abstract The aim of this paper is to characterize cosine and sine functions in terms of vector-valued cosine transforms on a Banach space X . We also introduce vector-valued sine transforms. Keywords: Homomorphisms, cosine and sine functions, trigonometric convolution products. AMS Classification: Primary 47D09; Secondary 44A35, 42A38.
Introduction The Laplace transform is an important tool in the theory of scalar and vector valued differential equations ([2], [3]). Other transforms are also used in the case of scalar differential equations ([10]). Cosine and sine transforms appear, in most cases, related with the Fourier transform. We may define different convolution products in L1 (R+ ) using embeddings of L1 (R+ ) into L1 (R) and taking the usual convolution product in L1 (R) , ∞ f ∗ g(t) = f (t − s)g(s) ds, t ∈ R, (0.1) −∞
with f, g ∈ L1 (R). These convolution products are usually called trigonometric convolution products. Trigonometric convolution products fit perfectly with trigonometric (cosine and sine) transforms, see [10] and here Theorem 1.4. On a Banach space X , cosine and sine functions, (C(t))t≥0 , (S(t))t≥0 ⊂ B(X) play almost the same role as trigonometric functions in the scalar case. For example, cosine and sine functions give the solution of the second order Cauchy problem ([7], [12]). It seems natural to investigate vector-valued cosine and sine transforms. In fact, Marschall was the first to consider a vector-valued cosine transform in [9], where he studied spectral properties and the spectral mapping theorem for cosine functions. Sine functions, or once integrated cosine functions, were introduced by Arendt and Kellermann [4]. They gave examples of sine functions which were ∗ Supported by the Project MTM2004-03036, MCYT, DGI-FEDER and the DGA Project E-12/25.
Journal of Functional Analysis 237 (2006) 1–53 www.elsevier.com/locate/jfa
One-parameter groups of regular quasimultipliers ✩ José E. Galé ∗ , Pedro J. Miana Departamento de Matemáticas, Universidad de Zaragoza, 50009 Zaragoza, Spain Received 25 February 2004; accepted 2 March 2006
Communicated by D. Sarason
Abstract Groups of unbounded operators are approached in the setting of the Esterle quasimultiplier theory. We introduce groups of regular quasimultipliers of growth ω, or ω-groups for short, where ω is a continuous weight on the real line. We study the relationship of ω-groups with families of operators and homomorphisms such as regularized, distribution and integrated groups, holomorphic semigroups, and functional calculi. Some convolution Banach algebras of functions with derivatives to fractional order are needed, which we construct using the Weyl fractional calculus. © 2006 Elsevier Inc. All rights reserved. Keywords: Banach algebras; Multipliers; Quasimultipliers; Groups of unbounded operators; Regularized, distribution and integrated groups; Holomorphic semigroups; Functional calculus; Fractional calculus
Introduction Unbounded operators arise naturally in differential equations, and appear profusely in harmonic analysis. Quite often, they can be suitably treated by functional-theoretical methods, although, even in these cases, such operators are not always easily understood. This paper concerns groups of unbounded operators (e−itH )t∈R , together with their “infinitesimal generators” −iH , which arise as formal solutions of ill-posed Cauchy problems. A typical example is the group (e−it L )t∈R where L is the Laplacian L = − nj=1 ∂ 2 /∂xj2 on Rn , which provides the ✩
This research has been partially supported by Projects BFM2001-1793 and MTM2004-03036 of the M.C.YT.D.G.I./F.E.D.E.R., Spain, and the Project E-12/25 of the D.G. de Aragón, Spain. * Corresponding author. E-mail addresses:
[email protected] (J.E. Galé),
[email protected] (P.J. Miana). 0022-1236/$ – see front matter © 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2006.03.021
Constr. Approx. (2007) 26: 93–113 DOI: 10.1007/s00365-006-0664-1
CONSTRUCTIVE APPROXIMATION © 2007 Springer Science+Business Media, Inc.
Hermite Matrix-Valued Functions Associated to Matrix Differential Equations Jos´e E. Gal´e, Pedro J. Miana, and Ana Pe˜na Abstract. Some sequences of matrix polynomials have been introduced recently as solutions of certain second-order differential equations, which can be seen as appropriate generalizations, to the matrix setting, of classical orthogonal polynomials. In this paper, we consider families (in a complex parameter) of matrix-valued special functions of Hermite type, which arise as natural extensions of the aforementioned matrix polynomials of the same type. We show that such families are solutions of corresponding differential equations and enjoy several structural properties. In particular, they satisfy a Rodrigues formula expressed in terms of the Weyl fractional calculus. We also show that, unlike the scalar case, a second-order differential operator having such a family as a set of joint eigenfunctions need not be unique.
0. Introduction The interest in orthogonal matrix polynomials seems to go back to the early work by M. G. Krein. Over the years, quite a number of papers have been devoted to different aspects of this subject, in such a way that there is at present a cohesive, well-established theory which makes up a field of active research, see [10], [4], [8] and references therein. However, it is only very recently that the theory has been enriched with a systematic study about the existence of families of new, matrix-valued, orthogonal polynomials, to be obtained as solutions (or “eigenvectors”) of suitable second-order differential equations with matrix coefficients. Here, by “new” families we understand families which are not those of the scalar case, and such that they are orthogonal with respect to a positivedefinite weight matrix of measures on the real line [4]. After some convincing arguments, the equations that are proposed we deal with in [4] have the form Y (t)A2 (t) + Y (t)A1 (t) + Y (t)A0 (t) = Y (t), where Ai is a matrix polynomial with degree less than or equal to i (i = 1, 2, 3), and is a Hermitian matrix independent of t. Moreover, the coefficients Ai are related to a given Date received: July 28, 2005. Date revised: August 16, 2006. Date accepted: September 19, 2006. Communicated by Edward B. Saff. Online publication: March 15, 2007. AMS classification: Primary 39B42; Secondary 26A33. Key words and phrases: Matrix differential equation, Hermite function, Matrix eigenfunction, Rodrigues’ formula, Fractional calculus. 93
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J. E. Gal´e, P. J. Miana, and A. Pe˜na
Proof. Let L be a differential operator of degree 2 such that there exists ± (ν, a) ∈ M2 (C) satisfying ± ± L Hν,a,1 = ± (ν, a)Hν,a,1
(ν ∈ C\N).
± For ν ∈ C and t ∈ R, let ± (ν, t) denote the determinant of the matrix Hν,a,1 (t), i.e., ± 2 2 ± 2 2 ± 2 ± (ν, t) = ((a 2 ν + 2)/2)(h ± ν ) (t) + a ν(ν − 1)h ν (t)h ν−2 (t) − a ν (h ν−1 ) (t) ± 2 2 2 ± 2 2 ± ± = (h ± ν ) (t) − a ν (h ν−1 ) (t) + (a ν/2)h ν (t)[h ν (t) + 2(ν − 1)h ν−2 (t)] ± 2 2 2 ± 2 2 ± = (h ± ν ) (t) − a ν (h ν−1 ) (t) ± (a ν/2)h ν (t)2th ν−1 (t) ± ± 2 2 = (h ± ν ) (t) + (a ν/2)h ν−1 (t)h ν+1 (t).
Fix n ∈ N. Clearly, the function ± (n, ·) is a nonzero polynomial (recall that h + n = ± is the Hermite polynomial) and therefore the set Z (n) of (real) zeros of (n, ·) (−1)n h − n is finite or empty. Take t ∈ / Z (n). By continuity, there exists a disk t (n), centered at n, in the complex plane such that ± (ν, t) = 0 for all ν ∈ t (n). This implies that ± (t) is invertible, so Hν,a,1 ± ± ± (ν, a) = (L Hν,a,1 )(t)[Hν,a,1 (t)]−1
if ν ∈ t (n)\{n}. Hence the function ν → ± (ν, a) is continuous (analytic, indeed) on t (n)\{n} and there exists the limit ± (n, a) := lim ± (ν, a) ν→n
± ± (which does not depend on t). Thus we have L Hn,a,1 = ± (n, a)Hn,a,1 on R\Z (n). + In fact, again by continuity, the equality holds on all of R. Since Pn,a,1 = Hn,a,1 = n − + − (−1) Hn,a,1 (this is readily seen), we obtain that (n, a) = (n, a) and then the last equation is
L Pn,a,1 = ± (n, a)Pn,a,1 on R. It follows that D2± (a, {ν}) ⊂ D2 (a, {n}) and therefore the dimension of D2± (a, {ν}) is less than or equal to 4, see [2]. Moreover, the operators D j,a ( j = 1, 2, 3, 4) are linearly independent ([2]) and they belong to D2± (a, {ν}) by Proposition 4.1. Hence, D2+ (a, {ν}) = D2− (a, {ν}) = D2 (a, {n}) as we wanted to show. Acknowledgements. The authors wish to thank Dr. Tom Koornwinder for valuable comments and references on special functions. They also thank Dr. Mirta Castro for enlighting remarks on reference [2], and the referee for interesting observations and questions which motivated Section 4. This paper has been partly supported by Projects MTM2004-03036 and MTM2006-13000, DGI-FEDER, of the MCYT, Spain, and Project E-64, D.G. Arag´on, Spain.
Integr. equ. oper. theory 57 (2007), 327–337 c 2006 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/030327-11, published online December 26, 2006 DOI 10.1007/s00020-006-1457-x
Integral Equations and Operator Theory
Extensions of Well-Boundedness and C m-Scalarity Jos´e E. Gal´e, Pedro J. Miana and Detlef M¨ uller Dedicated to the memory of Tadeusz Pytlik
Abstract. We consider some extensions of well-boundedness and C m -scalarity by using fractional calculus, and prove some theorems accordingly. These results are applied to the usual Laplacian on Rn and sub-Laplacians on nilpotent Lie groups. Mathematics Subject Classification (2000). Primary 47B40, 26A33; Secondary 47D03, 34L40. Keywords. Well-bounded operators, C m -scalar operators, functional calculus, fractional derivation, Laplacian.
1. Well-boundedness and C m scalarity Let A be a possibly unbounded, closed, densely defined operator on a Banach space X, with spectrum σ(A) contained in (a, b) ⊂ R, where −∞ ≤ a < b ≤ ∞. Let denote B(X) the usual Banach algebra of bounded operators on X. When X is a Hilbert space and A is a self-adjoint operator on X then there exists a projectionvalued measure Ω → E(Ω) from the Borel subsets Ω of (a, b) into B(X), such that (1) A= λ dE(λ). σ(A)
The research of first and second authors has been partly supported by the Project MTM200403036 of the M.C.YT.-DGI/F.E.D.E.R., Spain, and the Project E-12/25, D. G. Arag´ on, Spain. Part of the research of second author was developed in the Christian-Albrechts Universit¨ at in Kiel, while he was enjoying a HARP-postdoctoral position in the European Harmonic Analysis Network, HARP, IHP 2002-06.
c Proceedings of the Edinburgh Mathematical Society (2007) 50, 725–735 DOI:10.1017/S0013091505000520 Printed in the United Kingdom
ALGEBRA HOMOMORPHISMS FROM REAL WEIGHTED L1 ALGEBRAS PEDRO J. MIANA Department of Mathematics, Universidad de Zaragoza, C/Pedro Cerbuna, 12-50009 Zaragoza, Spain (
[email protected]) (Received 13 April 2005)
Abstract We characterize algebra homomorphisms from the Lebesgue algebra L1ω (R) into a Banach algebra A. As a consequence of this result, every bounded algebra homomorphism Φ : L1ω (R) → A is approached through a uniformly bounded family of fractional homomorphisms, and the Hille–Yosida theorem for C0 -groups is proved. Keywords: pseudo-resolvents; algebra homomorphisms; Weyl fractional calculus; integrated groups 2000 Mathematics subject classification: Primary 44A10; 47D62 Secondary 26A33; 46J25
1. Introduction Let R and R+ be the sets of real and positive real numbers. Widder’s characterization of Laplace transforms of real-valued bounded functions states that, given r ∈ C (∞) (0, ∞), there then exists f ∈ L∞ (0, ∞) such that r(λ) =
∞
e−λt f (t) dt,
λ > 0,
0
if and only if
|r(n) (λ)| : λ > 0, n ∈ N < ∞, sup λn+1 n!
(see [17]). Vector-valued versions of this result have appeared recently (see [1, 10, 12]). Let A be a Banach algebra (with or without identity). For Ω ⊂ R, a family (rλ )λ∈Ω of elements of A is called a pseudo-resolvent if the equation rλ − rµ = (µ − λ)rλ rµ holds for λ, µ ∈ Ω. Take ω ∈ R+ ∪ {0} and let L1ω (R+ ) be the usual Banach algebra with norm given by ∞ f w := |f (t)|eωt dt < ∞, 0
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Let R ≡ (Rλ )|λ|>ω be a pseudo-resolvent on B(X). It is easy to check that the kernel and the range of Rλ are independent of λ (we denote them by ker(R) and Im(R)). Note that (Rλ )|λ|>ω is the resolvent of a densely defined closed operator (A, D(A)) (i.e. Rλ = (λ − A)−1 ) if and only if ker(R) = {0} and Im(R) = X [8, Proposition III.4.6]. The Hille–Yosida theorem for C0 -groups can be found in [8, p. 79] (definitions of the infinitesimal generator, C0 -groups and properties can be also be found therein). Remark (the Hille–Yosida theorem). Let ω 0 and M > 0. For a linear operator (A, D(A)) on a Banach space X, the following properties are equivalent: (i) (A, D(A)) generates a C0 -group (S(t))t∈R such that S(t) M eω|t| with t ∈ R; (ii) (A, D(A)) is closed, densely defined and, for every λ ∈ R with |λ| > ω, we have λ ∈ ρ(A) and (|λ| − ω)n R(λ, A)n M, for all n ∈ N. Proof . (i) ⇒ (ii). We define Φ : L1ω (R) → B(X), ∞ Φ(F )x = F (t)S(t)x dt, −∞
x ∈ X.
Since Φ(ε−λ ) = R(λ, A) with λ > ω and Φ(ε−λ ) = −R(λ, A) with λ < −ω, we apply Theorem 2.2 to obtain (ii). (ii) ⇒ (i). By Theorem 2.2 there exists Φ : L1ω (R) → B(X) such that Φ(ε−λ ) = R(λ, A) with λ > ω and Φ(ε−λ ) = −R(λ, A) with λ < −ω. Since (nε−n )n>ω is a bounded approximate identity, then RΦ = Im(Rλ ) = X. By Theorem 4.1, there exists a C0 -group on X such that S(t) M eω|t| for t ∈ R. It is straightforward to check that (A, D(A)) is the infinitesimal generator of (S(t))t∈R . Acknowledgements. This work was partly supported by Project MTM2004-03036, DGI-FEDER, of the MCYT, Spain, and Project no. E-64, D. G. Arag´ on, Spain. The author is grateful to the referee for a careful reading of this paper and valuable suggestions, comments and ideas that led to its improvement. References 1.
2. 3. 4. 5.
W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector-valued Laplace transforms and Cauchy problems, Monographs in Mathematics, Volume 96 (Birkh¨ auser, Basel, 2001). M. Balabane and H. A. Emamirad, Smooth distribution group and Schr¨ odinger equation in Lp (R), J. Math. Analysis Applic. 70 (1979), 61–71. A. Bobrowski, On the Yosida approximation and the Widder–Arendt representation theorem, Studia Math. 124 (1997), 281–290. A. Bobrowski, Inversion of the Laplace transform and generation of Abel summable semigroups, J. Funct. Analysis 186 (2001), 1–24. W. Chojnacki, On the equivalence of a theorem of Kisy´ nski and the Hille–Yosida generation theorem, Proc. Am. Math. Soc. 126 (1998), 491–497.
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 136, Number 2, February 2008, Pages 519–528 S 0002-9939(07)09036-3 Article electronically published on October 24, 2007
HYPERCYCLIC AND TOPOLOGICALLY MIXING COSINE FUNCTIONS ON BANACH SPACES ANTONIO BONILLA AND PEDRO J. MIANA (Communicated by N. Tomczak-Jaegermann)
Abstract. Our first aim in this paper is to give sufficient conditions for the hypercyclicity and topological mixing of a strongly continuous cosine function. We apply these results to study the cosine function associated to translation groups. We also prove that every separable infinite dimensional complex Banach space admits a topologically mixing uniformly continuous cosine family.
Introduction A bounded linear operator T ∈ B(X) on a separable complex Banach space X is said to be hypercyclic if there exists an x ∈ X such that {T n x}n∈N is dense in X. In 1969, Rolewicz [12] gave the first example of a hypercyclic operator on a Banach space. He showed that if B is the backward shift on l2 (N), then λB is hypercyclic if and only if |λ| > 1. He also wondered if for every separable infinite dimensional Banach space there exists a hypercyclic operator. This question was independently answered in the affirmative by Ansari [1] and Bernal-Gonz´ alez [5]. Bonet and Peris [7] generalized the result for Fr´echet spaces. It is well known that T ∈ B(X) is hypercyclic if and only if for any pair of nonvoid open sets U, V ⊂ X there exists some positive integer n0 such that (∗)
T n0 U ∩ V = ∅.
It is said that an operator is topologically mixing if the condition (∗) holds for every n large enough. A one-parameter family {T (t)}t≥0 ⊂ B(X) of bounded linear operators is a oneparameter semigroup of operators in B(X) if it verifies the following two conditions: (i) T (0) = I, (ii) T (t)T (s) = T (t + s) for all t, s ≥ 0. Received by the editors July 17, 2006. 2000 Mathematics Subject Classification. Primary 47D09, 47A16. Key words and phrases. Hypercyclic operators, topologically mixing operators, cosine functions, translation groups. The first author is supported by MEC and FEDER MTM2005-07347 and MEC (Accion special) MTM2006-26627-E. The second author is supported by Project MTM2004-03036, DGI-FEDER, of the MCYT, Spain, and Project E-64, D. G. Arag´ on, Spain. c 2007 American Mathematical Society
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New identities in the Catalan triangle
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J.M. Gutiérrez a,1 , M.A. Hernández a,1 , P.J. Miana b,2 , N. Romero a,∗,1
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a Departamento de Matemáticas y Computación, Universidad de La Rioja, 26004 Logroño, Spain b Departamento de Matemáticas, Universidad de Zaragoza, 50009 Zaragoza, Spain
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Keywords: Catalan numbers; Combinatorial identities; Binomial coefficients
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that appears in a problem related with the dynamical behavior of a family of iterative methods applied to quadratic polynomials. © 2007 Elsevier Inc. All rights reserved.
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In fact, we show some new identities involving the well-known Catalan numbers, and specially the identity
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In this paper we prove new identities in the Catalan triangle whose (n, p) entry is defined by p 2n Bn,p := , n, p ∈ N, p n. n n−p
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* Corresponding author.
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E-mail addresses:
[email protected] (J.M. Gutiérrez),
[email protected] (M.A. Hernández),
[email protected] (P.J. Miana),
[email protected] (N. Romero). 1 Partly supported by the Ministry Education and Science (MTM 2005-03091) and the University of La Rioja (ATUR-05/43). 2 Partly supported by DGI-FEDER (MTM 2004-03036) and the DGA project (E-64).
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0022-247X/$ – see front matter © 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2007.09.073
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Please cite this article in press as: J.M. Gutiérrez et al., New identities in the Catalan triangle, J. Math. Anal. Appl. (2008), doi:10.1016/j.jmaa.2007.09.073
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H ∞ FUNCTIONAL CALCULUS AND MIKHLIN-TYPE MULTIPLIER CONDITIONS ´ E. GALE ´ AND PEDRO J. MIANA JOSE Abstract. Let T be a sectorial operator. It is known that the existence of a bounded (suitably scaled) H ∞ calculus for T , on every sector containing the positive half-line, is equivalent to the existence of a bounded functional calculus + on the Besov algebra Λα ∞,1 (R ) [4]. Such an algebra includes functions defined by Mikhlin-type conditions and so the Besov calculus can be seen as a result on multipliers for T . In this paper, we use fractional derivation to analyse in detail the relationship between Λα ∞,1 and Banach algebras of Mikhlin-type. As a result, we obtain a new version of the quoted equivalence.
1. Introduction On the basis of the work done by A. McIntosh for Hilbert spaces [12], an H ∞ functional calculus is given for sectorial operators on general Banach spaces in [4]. When the operators under discussion are of type 0 the existence of the (suitably scaled) H ∞ calculus is shown to be equivalent to the existence of a functional + calculus defined on a certain Besov space Λα ∞,1 (R ) [4, Theorem 4.10]. + Every n-differentiable function F on R := (0, ∞) obeying Mikhlin-type conditions like sup tk |F (k) (t)| < ∞ (k = 0, 1, ..., n) t>0
belongs to Λα ∞,1 , if n > α, see [4, p. 73], [5, p. 416]. This reinforces the view of the Besov functional calculus as a theorem about multipliers. We study more closely such a link by using fractional derivation, in Section 2 and Section 3 of this paper. The equivalence between the H ∞ calculus and the Besov calculus is proven in [4, Theorem 4.10] through the Paley-Wiener theorem. We show in Section 4 that the way to go from (bounded) analytic functions to functions in Λα ∞,1 is paved in fact with a formula of Cauchy type for fractional derivatives. In Section 5, we apply the results of previous sections to give a characterization of the (scaled) H ∞ calculus in terms of Mikhlin algebras. On the other hand, the sectorial H ∞ calculus provides us, in general, with operators which are not necessarily bounded [4], [16]. It has been shown in [9] and [8] that these operators can always be regarded as certain generalized multipliers, or regular quasimultipliers in the sense defined by J. Esterle in [7]. It is maybe worth pointing out that, as a consequence of the results in sections 3 and 4, an 2000 Mathematics Subject Classification. Primary 47A60, 47D03, 46J15, 26A33; Secondary 47L60, 47B48, 43A22. Key words and phrases. Functional calculus, fractional calculus, Mikhlin multipliers, analytic semigroups, unbounded operators, quasimultipliers. This research has been partially supported by Projects BFM2001-1793 and MTM2004-03036, MCYTDGI and FEDER, Spain, and Project E12/25, D.G.A., Spain. 1
