The Exponential Function on Banach Algebra

Transcription

1 FORMALIZED MATHEMATICS Volume 12, Number 2, 2004 University of Białystok The Exponential Function on Banach Algebra Yasunari Shidama Shinshu University Nagano Summary. In this article, the basic properties of the exponential function on Banach algebra are described. MML Identifier: LOPBAN 4. The notation and terminology used here are introduced in the following papers: [17],[19],[20],[3],[4],[2],[16],[5],[1],[18],[9],[11],[12],[8],[6],[7],[13],[10], [21],[14],and[15]. For simplicity, we use the following convention: X denotes a Banach algebra, pdenotesarealnumber, w, z, z 1, z 2 denoteelementsof X, k, l, m, ndenote naturalnumbers, s 1, s 2, s 3, s, s denotesequencesof X,and r 1 denotesa sequence of real numbers. Let Xbeanonemptynormedalgebrastructureandlet x, ybeelementsof X.Wesaythat x, yarecommutativeifandonlyif: (Def.1) x y = y x. Letusnotethatthepredicate x, yarecommutativeissymmetric. Next we state a number of propositions: (1) If s 2 isconvergentand s 3 isconvergentandlim(s 2 s 3 ) = 0 X,then lims 2 =lims 3. (2) Forevery zsuchthatforeverynaturalnumber nholds s(n) = zholds lims = z. (3) If sisconvergentand s isconvergent,then s s isconvergent. (4) If sisconvergent,then z sisconvergent. (5) If sisconvergent,then s zisconvergent. 173 c 2004 University of Białystok ISSN

2 174 yasunari shidama (6) If sisconvergent,thenlim(z s) = z lims. (7) If sisconvergent,thenlim(s z) =lims z. (8) If sisconvergentand s isconvergent,thenlim(s s ) =lims lims. (9) α=0 (z s 1)(α)) κ N = z α=0 (s 1)(α)) κ N and α=0 (s 1 z)(α)) κ N = α=0 (s 1)(α)) κ N z. (10) α=0 (s 1)(α)) κ N (k) α=0 s 1 (α)) κ N (k). (11) If for every n such that n m holds s 2 (n) = s 3 (n), then α=0 (s 2)(α)) κ N (m) = α=0 (s 3)(α)) κ N (m). (12) Ifforevery nholds s 1 (n) r 1 (n)and r 1 isconvergentandlimr 1 = 0, then s 1 isconvergentandlims 1 = 0 X. Letusconsider Xandlet zbeanelementof X.Thefunctor zexpseq yieldingasequenceof Xisdefinedasfollows: (Def.2) Forevery nholds zexpseq(n) = 1 n! zn N. The scheme ExNormSpace CASE deals with a non empty Banach algebra A andabinaryfunctorfyieldingapointofa,andstatesthat: Forevery kthereexistsasequence s 1 ofasuchthatforevery n holdsif n k,then s 1 (n) =F(k, n)andif n > k,then s 1 (n) = 0 A for all values of the parameters. Next we state the proposition (13) Forevery ksuchthat 0 < kholds (k 1)! k = k!andforall m, ksuch that k mholds (m k)! ((m + 1) k) = ((m + 1) k)!. Let nbeanaturalnumber.thefunctorcoef nyieldsasequenceofreal numbers and is defined by: n! (Def.3) Foreverynaturalnumber kholdsif k n,then (Coef n)(k) = k! (n k)! andif k > n,then (Coef n)(k) = 0. Let nbeanaturalnumber.thefunctorcoefe nyieldingasequenceofreal numbers is defined by: 1 (Def.4) Foreverynaturalnumber kholdsif k n,then (Coefe n)(k) = k! (n k)! andif k > n,then (Coefe n)(k) = 0. Letusconsider X, s 1.ThefunctorShifts 1 yieldingasequenceof Xisdefined as follows: (Def.5) (Shifts 1 )(0) = 0 X andforeverynaturalnumber kholds (Shifts 1 )(k + 1) = s 1 (k). functorexpan(n,z, w)yieldsasequenceof Xandisdefinedby: (Def.6) Foreverynaturalnumber kholdsif k n,then (Expan(n,z, w))(k) = (Coef n)(k) z k N wn k N andif n < k,then (Expan(n,z,w))(k) = 0 X. functorexpane(n,z,w)yieldsasequenceof Xandisdefinedasfollows:

3 the exponential function on banach algebra 175 (Def.7) Foreverynaturalnumber kholdsif k n,then (Expane(n, z,w))(k) = (Coefe n)(k) z k N wn k N andif n < k,then (Expane(n,z, w))(k) = 0 X. functoralfa(n,z,w)yieldsasequenceof Xandisdefinedasfollows: (Def.8) Foreverynaturalnumber kholdsif k n,then (Alfa(n,z, w))(k) = zexpseq(k) α=0 wexpseq(α)) κ N(n k) and if n < k, then (Alfa(n, z,w))(k) = 0 X. Letusconsider X,let z, wbeelementsof X,andlet nbeanaturalnumber. ThefunctorConj(n,z, w)yieldsasequenceof Xandisdefinedby: (Def.9) Foreverynaturalnumber kholdsif k n,then (Conj(n,z, w))(k) = zexpseq(k) ( α=0 wexpseq(α)) κ N(n) α=0 wexpseq(α)) κ N(n k))andif n < k,then (Conj(n,z, w))(k) = 0 X. One can prove the following propositions: (14) zexpseq(n + 1) = 1 n+1 z zexpseq(n)and zexpseq(0) =1 X and zexpseq(n) z ExpSeq(n). (15) If 0 < k,then (Shifts 1 )(k) = s 1 (k 1). (16) α=0 (s 1)(α)) κ N (k) = α=0 (Shifts 1)(α)) κ N (k) + s 1 (k). (17) For all z, w such that z, w are commutative holds (z + w) n N = α=0 (Expan(n, z,w))(α)) κ N(n). 1 (18) Expane(n,z, w) = n! Expan(n,z, w). (19) Forall z, wsuchthat z, warecommutativeholds 1 n! (z + w) n N = α=0 (Expane(n,z, w))(α)) κ N(n). (20) 0 X ExpSeqisnorm-summableand (0 X ExpSeq) =1 X. Letusconsider Xandlet zbeanelementof X.Observethat zexpseqis norm-summable. Next we state a number of propositions: (21) zexpseq(0) =1 X and (Expan(0,z, w))(0) =1 X. (22) If l k,then (Alfa(k + 1,z, w))(l) = (Alfa(k, z,w))(l) + (Expane(k + 1,z, w))(l). (23) α=0 (Alfa(k +1, z,w))(α)) κ N(k) = α=0 (Alfa(k,z, w))(α)) κ N(k)+ α=0 (Expane(k + 1,z, w))(α)) κ N(k). (24) zexpseq(k) = (Expane(k, z, w))(k). (25) For all z, w such that z, w are commutative holds α=0 z + wexpseq(α)) κ N (n) = α=0 (Alfa(n, z,w))(α)) κ N(n). (26) Forall z, wsuchthat z, warecommutativeholds α=0 zexpseq(α)) κ N(k) α=0 wexpseq(α)) κ N(k) α=0 z + wexpseq(α)) κ N (k) = α=0 (Conj(k,z, w))(α)) κ N(k). (27) 0 z ExpSeq(n).

4 176 yasunari shidama (28) α=0 zexpseq(α)) κ N(k) α=0 z ExpSeq(α)) κ N(k) and α=0 z ExpSeq(α)) κ N(k) ( z ExpSeq)and α=0 zexpseq(α)) κ N(k) ( z ExpSeq). (29) 1 ( z ExpSeq). (30) α=0 z ExpSeq(α)) κ N(n) = α=0 z ExpSeq(α)) κ N(n)andif n m,then α=0 z ExpSeq(α)) κ N(m) α=0 z ExpSeq(α)) κ N(n) = α=0 z ExpSeq(α)) κ N(m) α=0 z ExpSeq(α)) κ N(n). (31) α=0 Conj(k,z, w) (α)) κ N(n) = α=0 Conj(k,z, w) (α)) κ N(n). (32) Foreveryrealnumber psuchthat p > 0thereexists nsuchthatfor every ksuchthat n kholds α=0 Conj(k,z, w) (α)) κ N(k) < p. (33) Forevery s 1 suchthatforevery kholds s 1 (k) = α=0 (Conj(k,z, w))(α)) κ N(k)holds s 1 isconvergentandlims 1 = 0 X. Let XbeaBanachalgebra.ThefunctorexpXyieldingafunctionfromthe carrierof Xintothecarrierof Xisdefinedby: (Def.10) Foreveryelement zofthecarrierof Xholds (expx)(z) = (zexpseq). Letusconsider X, z.thefunctorexpzyieldsanelementof Xandisdefined by: (Def.11) expz = (expx)(z). One can prove the following propositions: (34) Forevery zholdsexpz = (zexpseq). (35) Letgiven z 1, z 2.Suppose z 1, z 2 arecommutative.thenexp(z 1 + z 2 ) = expz 1 expz 2 andexp(z 2 +z 1 ) =expz 2 expz 1 andexp(z 1 +z 2 ) =exp(z 2 + z 1 )andexpz 1,expz 2 arecommutative. (36) Forall z 1, z 2 suchthat z 1, z 2 arecommutativeholds z 1 expz 2 =expz 2 z 1. (37) exp(0 X ) =1 X. (38) expz exp( z) =1 X andexp( z) expz =1 X. (39) expzisinvertibleand (expz) 1 =exp( z)andexp( z)isinvertible and (exp( z)) 1 =expz. (40) Forevery zandforallrealnumbers s, tholds s z, t zarecommutative. (41) Letgiven zand s, tberealnumbers.thenexp(s z) exp(t z) = exp((s+t) z)andexp(t z) exp(s z) =exp((t+s) z)andexp((s+t) z) = exp((t + s) z)andexp(s z),exp(t z)arecommutative. References [1] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41 46, [2] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91 96, [3] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55 65, [4] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1): ,

5 the exponential function on banach algebra 177 [5] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35 40, [6] Jarosław Kotowicz. Convergent sequences and the limit of sequences. Formalized Mathematics, 1(2): , [7] Jarosław Kotowicz. Monotone real sequences. Subsequences. Formalized Mathematics, 1(3): , [8] Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2): , [9] Eugeniusz Kusak, Wojciech Leończuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2): , [10] Takaya Nishiyama and Yasuho Mizuhara. Binary arithmetics. Formalized Mathematics, 4(1):83 86, [11] Jan Popiołek. Some properties of functions modul and signum. Formalized Mathematics, 1(2): , [12] Jan Popiołek. Real normed space. Formalized Mathematics, 2(1): , [13] Konrad Raczkowski and Andrzej Nędzusiak. Series. Formalized Mathematics, 2(4): , [14] Yasunari Shidama. The Banach algebra of bounded linear operators. Formalized Mathematics, 12(2): , [15] Yasunari Shidama. The series on Banach algebra. Formalized Mathematics, 12(2): , [16] Andrzej Trybulec. Subsets of complex numbers. To appear in Formalized Mathematics. [17] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9 11, [18] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2): , [19] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67 71, [20] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73 83, [21] Yuguang Yang and Yasunari Shidama. Trigonometric functions and existence of circle ratio. Formalized Mathematics, 7(2): , Received February 13, 2004