Applications of Numerical Functional Analysis in Atomless Finite Measure Spaces

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Quest Journals Journal of Research in Applied Mathematics Volume 3 ~ Issue 6 (2017) pp: 01-04 ISSN(Online) : 2394-0743 ISSN (Print): 2394-0735

www.questjournals.org Research Paper

Applications of Numerical Functional Analysis in Atomless Finite Measure Spaces Yousif Mohammed Modawy Elmahy1 , Montasir Salman ElfadelTyfor 2 Department of MathematicFaculty ofScience &Arts In Al-mandg Al-BAHA University KS.A1 Department of PhysicsFaculty of Science & Arts In Al-mandg Al-BAHA University KS.A2 2 Department of Physics Faculty ofEducationin (Al-Hasahisa), Gezira University Sudan

Received 23Feb, 2017; Accepted 08Mar, 2017 Š The author(s) 2017. Published with open access at www.questjournals.org ABSTRACT: This paper study the application of numerical Functional analysis in physics as isotope, theorems was derived to applied it on an Atomless finite measure space. Keyword:Isotopes, Atomless finite measure space, narrow operator.

I. INTRODUCTION This paper attempts to study the application of numerical Functional analysis in many branch of physics we can found it in theoretical physics in Lagrangian formulation of classical mechanics [1] also we found function analysis in quantum mechanics in Hilbert spaces.The basic and historically first class of spaces studied in functional analysis are complete normed vector spaces over the real or complex numbers [8]. Such spaces are calledBanach spaces [4]. An important example is a Hilbert space, where the norm arises from an inner product [9]. These spaces are of fundamental importance in many areas, including the mathematical formulation of quantum mechanics [10].Most spaces considered in functional analysis have finite dimension [5].

II. FUNCTION ANALYSIS IN ATOMLESS FINITE MEASURE SPACE We start by the applicationon an Atomless finite measure space We did for rank-one operators. Lemma (1) [2] Let (Ί, ÎŁ, Îź) be an atomless finite measure space, Îľ > 0, đ?‘‡ ∈ â„’(L1+Îľ (Îź)) a narrow operator, x1 ∈ L1+Îľ (Îź) a simple function, T x1 = x2 , andΊ = D1 ⊔ â‹Ż ⊔ Dâ„“ any partition and Îľ > 0. Then there exists a partition Ί = A ⊔ B such that (i) x1 A 1+Îľ = x1 B 1+Îľ = 2−1 x1 1+Îľ . 1 (ii) Îź Dj ∊ A = Îź Dj ∊ B = Îź(Dj ) for each j = 1, . . . , â„“. 2 (iii) T x1 A − 2−1 x2 < đ?œ€ and T x1 B − 2−1 x2 < đ?œ€. Proof. Let x1 = m k=1 a k đ?&#x;?C k for some a k ∈ đ?•‚ and Ί = C1 ⊔ ¡ ¡ ¡ ⊔ Cm . For each k = 1, . . . , m and j = 1, . . . , â„“ define setsEk,j = Ck ∊ Dj and, using the definition of narrow operator, choose uk,j ∈ L1+Îľ (Îź) so that u2k,j = đ?&#x;?E k ,j ,

uk,j dÎź = 0, and

a k Tuk,j <

Ί

2Îľ . mâ„“

Then set + Ek,j = t ∈ Ek,j âˆś uk,j (t) ≼ 0 ,

− + Ek,j \ Ek,j

1

+ − Which satisfyÎź Ek,j = Îź Ek,j = Îź Ek,j and define 2

m

â„“

m + Ek,j

A=

and

ℓ − Ek,j .

B=

k=1 j=1

k=1 j=1

Let us show that the partitionΊ = A ⊔ Bhas the desired properties. Indeed, observe that

*Corresponding Author:Yousif Mohammed Modawy Elmahy1 Department of Mathematics1

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Applications Of Numerical Functional Analysis In Atomless Finite Measure Spaces m

x1

1+Îľ

A

â„“

m

=

1+Îľ

ak

Îź

+ Ek,j

k=1 j=1 m

1 = 2

â„“

=

ak

1+Îľ

k=1 1+Îľ

ak

j=1

1 = x 2 1

Îź Ck

k=1

1 Îź Ek,j 2

1+Îľ

And that one obviously has x1 B 1+Îľ = x1 A 1+Îľ , thus (i) is show. + Since Ek,j ⊆ Ek,j ⊆ Dj , for each j0 ∈ {1, . . . , â„“} we have that m

â„“

m

Dj 0 ∊ A =

Dj 0 ∊

+ Ek,j

k=1 j=1

And hence

m

Îź Dj 0 ∊ A =

Îź k=1

Analogously it is show that Îź Dj ∊ B = observe that

1 2

x1

A

− x1

k=1

m

k=1

1 Îź C k ∊ Dj 0 = Îź D j 0 . 2

for every j ∈ {1, . . . , ℓ}which finishes (ii). To show (iii)

â„“

m

=

B

1 = 2

Îź Ek,j 0

Îź Dj

m

k=1

m

1 = 2

+ Ek,j 0

+ Ek,j

=

a k đ?&#x;?E + − đ?&#x;? k ,j

E− k ,j

â„“

=

k=1 j=1

a k uk,j k=1 j=1

And hence, m

T x1

A

− x1

B

â„“

≤

a k T uk,j < 2đ?œ€. k=1 j=1

Therefore, one has that T x1

Analogously, one obtains T x1

B

1 1 − x2 = 2 T x1 A − T x1 A − T x1 2 2 1 = T x1 A − x1 B < đ?œ€. 2 1 − x2 < đ?œ€ finishing the proof of (iii). A

B

2

Lemma (2) [3] Let (Ί, ÎŁ, Îź) be an atomless finite measure space, Îľ > 0, let T ∈ â„’(L1+Îľ (Îź)) be a narrow operator, and letx1 , x2 ∈ L1+Îľ (Îź) be simple functions such that T x1 = x2 . Then for each n ∈ â„• and Îľ > 0 there exists a partition Ί = A1 ⊔¡ ¡ ¡âŠ” A2n such that for each k = 1, . . . , 2n one has 1+Îľ (a) x1 A k = 2− n x1 1+Îľ . 1+Îľ

(b) x2 A k = 2−n x2 (c) T x1 A k − 2−n x2 < đ?œ€.

1+Îľ

.

Proof. Let x2 = â„“j=1 bj đ?&#x;?D j for some b j ∈ đ?•‚ and Ί = D1 ⊔¡ ¡ ¡âŠ” Dâ„“ . We proceed by induction on n.Suppose first that n = 1 and use Lemma (1) to find a partition Ί = A ⊔ B satisfying properties (i)–(iii). Then (i) and (iii) mean (a) and (c) for A1 = A, A2 = B. Besides, observe that ( b) follows from (ii): â„“

x1

1+Îľ A1 1+Îľ

=

â„“

bj j=1 −1

1+Îľ

Îź A 1 ∊ Dj =

bj

1+Îľ

j=1

1 1 Îź Dj = x 2 2 2

1+Îľ

1+Îľ

And analogously x2 A 2 = 2 x2 . For the induction step suppose that the statement of the lemma is true for n ∈ â„• and find a partition Ί = A1 ⊔ ¡ ¡ ¡âŠ” A2n such that for every k = 1, . . . , 2n the following hold: 1+Îľ 1+Îľ x1 A k = 2−n x1 1+Îľ , x1 A k = 2−n x2 1+Îľ , and T x1 A k − 2−n x2 < đ?œ€. (1) n Then, for each k = 1, . . . , 2 use Lemma (1) for x1 A k instead of x1 , T x1 A k instead of x2 , the decomposition 2− n

â„“

Ί =

Dj ∊ A k k=1 j=1

*Corresponding Author:Yousif Mohammed Modawy Elmahy1

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Applications Of Numerical Functional Analysis In Atomless Finite Measure Spaces Îľ

Instead of Ί = D1 ⊔ ¡ ¡ ¡ ⊔ Dâ„“ and instead of Îľ, and find a partition Ί = A(k) ⊔ B(k) satisfyingproperties 2 (i)–(iii). Namely, for each k = 1, . . . , 2n we have that: 1+Îľ 1+Îľ 1+Îľ (i) x1 (A k ∊ A(k)) = x1 (A k ∊ B(k)) = 2−1 x1 A k . (ii) Îź Dj ∊ Ak ∊ A k

1

= Îź Dj ∊ A k ∊ B k

= Îź Dj ∊ Ak For each j = 1, . . . , â„“. 2

Îľ

Îľ

(iii) T x1 (A k ∊ A(k)) − 2−1 T x1 A k < and T x1 (A k ∊ B(k)) − 2−1 T x1 A k < . 2 2 Let us show that the partition Ί = A 1 ∊ A 1 ⊔ ¡ ¡ ¡ ⊔ A 2 n ∊ A 2n ⊔ A 1 ∊ B 1 ⊔ ¡ ¡ ¡ ⊔ A 2 n ∊ B 2n Has the desired properties for n + 1: Property (a): using (i) and (1), one obtains 1+Îľ

1+Îľ

x1 A k ∊ A k = x1 A k ∊ B k = 2−1 x1 Property (b): for each k = 1, . . . , 2n use (ii) and (1) to obtain

1+Îľ Ak

= 2− n+1 x1

1+Îľ

.

â„“ 1+Îľ

x2

(A k ∊ A(k))

=

bj

1+Îľ

Îź Dj ∊ Ak ∊ A(k)

j=1

1 = 2

â„“ 1+Îľ

bj j=1

1 Îź Dj ∊ A k 2

1 1+đ?œ€ = đ?‘Ľ2 đ??´đ?‘˜ = 2− đ?‘›+1 đ?‘Ľ2 2 1+đ?œ€ And analogously đ?‘Ľ2 (đ??´đ?‘˜ ∊ đ??ľ) = 2− đ?‘›+1 đ?‘Ľ2 1+đ?œ€ . đ?‘› Property (c): for each đ?‘˜ = 1, . . . , 2 use (iii) and (1) to write

1+đ?œ€

1 − 2− đ?‘› +1 đ?‘Ľ2 ≤ đ?‘‡ đ?‘Ľ1 đ??´đ?‘˜ ∊đ??´ đ?‘˜ − 2−1 đ?‘‡ đ?‘Ľ1 đ??´đ?‘˜ + đ?‘‡ đ?‘Ľ1 2 đ?œ€ đ?œ€ < + = đ?œ€. 2 2 And analogously đ?‘‡ đ?‘Ľ1 (đ??´đ?‘˜ ∊ đ??ľ(đ?‘˜)) − 2− đ?‘›+1 đ?‘Ľ2 < đ?œ€, which completes the proof. đ?‘‡ đ?‘Ľ1

(đ??´đ?‘˜ ∊ đ??´(đ?‘˜))

đ??´đ?‘˜

− 2−đ?‘› đ?‘Ľ2

III .APPLICATIONS OF NUMERICAL FUNCTIONAL IN ATOMLESS FINITE MEASURE SPACE Lemma (3) [6] Let (đ?›ş, đ?›´, đ?œ‡) be an atomless finite measure space, đ?œ€ > 0, let đ?‘‡ ∈ â„’(đ??ż1+đ?œ€ (đ?œ‡)) be a narrow operator, and let đ?‘Ľ1 , đ?‘Ľ2 ∈ đ??ż1+đ?œ€ (đ?œ‡) be simple functions such that đ?‘‡đ?‘Ľ1 = đ?‘Ľ2 . Then for each đ?‘› ∈ â„•, each number đ?‘— đ?œ†of the formđ?œ† = đ?‘› where đ?‘— ∈ {1, . . . , 2đ?‘› − 1}and each đ?œ€ > 0there exists a partition đ?›ş = đ??´ ⊔ đ??ľ such that: 2 (A) đ?‘Ľ1 đ??´ 1+đ?œ€ = đ?œ† đ?‘Ľ1 1+đ?œ€ . (B) đ?‘Ľ2 đ??ľ 1+đ?œ€ = (1 − đ?œ†) đ?‘Ľ2 1+đ?œ€ . (C) đ?‘‡ đ?‘Ľ1 đ??´ − đ?œ†đ?‘Ľ1 < đ?œ€. Proof. Use Lemma (2) to choose a partition đ?›ş = đ??´1 ⊔ ¡ ¡ ¡ ⊔ đ??´2đ?‘› Satisfying properties (a)–(c) with đ?œ€/đ?‘— instead of đ?œ€. Then, setting 2đ?‘›

đ?‘—

đ??´=

đ??´đ?‘˜ đ?‘Žđ?‘›đ?‘‘đ??ľ = đ?‘˜=1

đ??´đ?‘˜ , đ?‘˜=đ?‘— +1

One obtains đ?‘—

đ?‘Ľ1

đ??´

1+đ?œ€

=

đ??ľ

1+đ?œ€

đ?‘Ľ1

đ??´đ?‘˜

đ?‘˜=1

2đ?‘›

đ?‘Ľ2

đ?‘— 1+đ?œ€

=

đ?‘Ľ2

2−đ?‘› đ?‘Ľ1

=

1+đ?œ€

= đ?œ† đ?‘Ľ1

1+đ?œ€

,

đ?‘˜=1

2đ?‘› 1+đ?œ€ đ??´đ?‘˜

2−đ?‘› đ?‘Ľ2

=

đ?‘˜=đ?‘— +1

1+đ?œ€

= 1 + đ?œ† đ?‘Ľ1

1+đ?œ€

,

đ?‘˜=đ?‘— +1

and đ?‘—

� �1

đ??´

− đ?œ†đ?‘Ľ2 =

đ?‘—

đ?‘‡ đ?‘Ľ1 đ?‘˜=1

đ??´đ?‘˜

−

đ?‘—

2

−đ?‘›

�2 ≤

đ?‘˜=1

đ?‘‡ đ?‘Ľ1 đ?‘˜=1

đ??´đ?‘˜

đ?œ€ − 2−đ?‘› đ?‘Ľ2 < đ?‘— = đ?œ€ đ?‘—

As desired. By using anther way (in general case), about above lemma:

*Corresponding Author:Yousif Mohammed Modawy Elmahy1

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Applications Of Numerical Functional Analysis In Atomless Finite Measure Spaces Lemma (4) [6]Let (đ?›ş, đ?›´, đ?œ‡) be an Atomless finite measure space, đ?œ€ > 0 let đ?‘‡ ∈ â„’(đ??ż1+đ?œ€ (đ?œ‡)) be a narrow operator, and let đ?‘Ľđ?‘› , đ?‘Ľđ?‘›+1 ∈ đ??ż1+đ?œ€ (đ?œ‡) be of simple functions such that đ?‘‡đ?‘Ľđ?‘› = đ?‘Ľđ?‘› +1 . Then for each đ?‘› ∈ â„•, đ?‘— each number đ?œ†of the form đ?œ† = đ?‘› where đ?‘— ∈ {1, . . . , 2đ?‘› − 1}and each đ?œ€ > 0 there exists a partition đ?›ş = đ??´ ⊔ 2 đ??ľ such that: (i) đ?‘Ľđ?‘› đ??´ 1+đ?œ€ = đ?œ† đ?‘Ľđ?‘› 1+đ?œ€ . (ii) đ?‘Ľđ?‘› +1 đ??ľ 1+đ?œ€ = (1 − đ?œ†) đ?‘Ľđ?‘›+1 1+đ?œ€ . (iii) đ?‘‡ đ?‘Ľđ?‘› đ??´ − đ?œ†đ?‘Ľđ?‘›+1 < đ?œ€. Proof. Use Lemma (2) to choose a partition đ?›ş = đ??´1 ⊔ ¡ ¡ ¡ ⊔ đ??´2đ?‘› satisfying properties (i)–(iii) with đ?œ€/đ?‘— instead of đ?œ€. Then, setting 2n

đ?‘—

đ??´=

đ??´đ?‘˜ đ?‘Žđ?‘›đ?‘‘đ??ľ = đ?‘˜=1

Ak , đ?‘˜=đ?‘— +1

We have j

xn

1+Îľ

A

=

xn+1

B

xn

Ak

=

2−n xn

=

k=1

2n 1+Îľ

j 1+Îľ

1+Îľ

= Îť xn

1+Îľ

,

k=1 2n

xn+1

1+Îľ Ak

k=j+1

2−n xn+1

=

1+Îľ

= 1 + Îť xn

1+Îľ

,

k=j+1

and j

T xn

A

− Îťxn+1 =

j

T xn k=1

j

≤

T xn k=1

Ak

Ak

2−n xn

− k=1

Îľ − 2−n xn+1 < đ?‘— = Îľ. j

as desired.

REFERENCE [1]https://www.quora.com/What-are-common-uses-of-functional-analysis-in-theoretical-physics [2] Miguel MartĂ­n a,∗, Javier MerĂ­a, Mikhail Popov , On the numerical radius of operators in Lebesgue spaces [3] M. MartĂ­n, J. MerĂ­, M. Popov, on the numerical radius of operators in Lebesgue spaces^✊, J. of functional analysis, 261 (2011) 149168. [4] Hall, B.C. (2013), Quantum Theory for Mathematicians, Springer, p. 147 [5] Rudin, Walter (1991). Functional analysis. McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. [6] M. MartĂ­n, J. MerĂ­, M. Popov, on the numerical radius of operators in Lebesgue spaces^✊, J. of functional analysis, 261 (2011) 149168. [7] M. MartĂ­n, J. MerĂ­, M. Popov, on the numerical radius of operators in Lebesgue spaces^✊, J. of functional analysis, 261 (2011) 149168. [8] https://en.wikipedia.org/wiki/Functional_analysis. [9]https://en.wikipedia.org/wiki/Functional_analysis. [10] Same as reference [8]

*Corresponding Author:Yousif Mohammed Modawy Elmahy1

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