STATISTICAL BOUNDED SEQUENCES OF BI-COMPLEX NUMBERS Текст научной статьи по специальности «Математика»

Probl. Anal. Issues Anal. Vol. 12 (30), No 2, 2023, pp. 3-16

DOI: 10.15393/j3.art.2023.13090

3

UDC 517.521

S. Bera, B. Ch. Tripathy

STATISTICAL BOUNDED SEQUENCES OF BI-COMPLEX NUMBERS

Abstract. In this paper, we extend statistical bounded sequences of real or complex numbers to the setting of sequences of bi-complex numbers. We define the statistical bounded sequence space of bi-complex numbers and also define the statistical bounded sequence spaces of ideals and I^. We prove some inclusion relations and provide examples. We establish that is the direct sum of I8 and I^. Also, we prove the decomposition theorem for statistical bounded sequences of bi-complex numbers. Finally, summability properties in the light of J.A. Fridy's work are studied. Key words: natural density, bi-complex, statistical bounded, norm. 2020 Mathematical Subject Classification: 40A35, 40G15,

46A45

1. Introduction. In 1892, Segre [12] introduced the notion of bi-complex numbers that form an algebra isomorphic to the tessarines. Thereafter, Srivastava and Srivastava [13], Wagh [17], Sager and Sa$ir [10], Rochon and Shapiro [9] investigated on sequences of bi-complex numbers. The notion of convergence is one of the main tools of analysis. There are a lot of convergences, e.g., Cesaro, Norlund and Riesz, etc. Out of these, statistical convergence is one of the most important notions, which brought a back through development in sequence spaces. Many researchers (e.g., Buck [3], Salat [11], Fridy [4], Tripathy [16], Altinok et.al [1], Tripathy and Nath [14], and Tripathy and Sen [15]) studied the statistical convergence and statistical bounded sequences of real or complex numbers. Research work on statistical convergence in sequence spaces has been done by Albayrak et al. [2], Kuzhaev [5], Nath et al. [6].

Throughout the paper, C0, C\ and C2 denote the set of real, complex, and bi-complex numbers, respectively.

© Petrozavodsk State University, 2023

2. Definition and preliminaries.

2.1 Bi-complex numbers. Segre [12] defined a bi-complex number as:

£ = Zi + I2Z2 = Xi + ¿1X2 + ¿2^3 +

where z1,z2 e C1 and x1,x2,x3,x4 e C0 and the independent units i1,i2 are such, that i\ = i"^ = — 1 and i1i2 = i2i1. Denote the set of bi-complex numbers C2; it is defined as:

C2 = {£: £ = ¿1 + I2Z2; Z1,Z2 e CK^)},

where C1(i1) = {x1 + i1x2 : x1,x2 e Co}. C2 is a vector space over C1(i1). There are four idempotent elements in C2: they are 0,1,e1 = 1+^Lt2 and e2 = 1~21'2, out of which e1 and e2 are nontrivial, such that e1 + e2 = 1 and e1e2 = 0.

A bi-complex number £ = z1 + i2z2 is said to be singular if and only if

k? + 4\ = 0.

Every bi-complex number £ = z1 + i2z2 can be uniquely expressed as the combination of e1 and e2; namely,

£ = + 12Z2 = (Z1 — ¿1^2)61 + (Z1 + i1Z2)e2 = + ^262,

where = (z1 — i1z2) and = (z1 + i1z2).

(i) The ^-conjugation of a bi-complex number £ = z1 + i2z2 is denoted by and is defined by = z1 + i2z2.

(ii) The ^-conjugation of a bi-complex number £ = z1 + i2z2 is denoted by £ and is defined by £ = z1 — i2z2.

(iii) The ¿^-conjugation of a bi-complex number £ = z1 + i2z2 is denoted by £ and is defined by £ = z1 + i2z2, for all z1,z2 e C1(i1) and z1,z2 are the complex conjugates of z1,z2, respectively.

Each of the three conjugations' moduli are given by

(i) \ek = ve^ (ii) \e k = vee (iii) \e u = VZ.

The bi-complex number £ is invertible if \j1 ^ 0. The Euclidean norm ||.|| on C2 is defined by

neik = = VN2 + N2 = y^2 + \^2\2

where £ = x1 + i1 x2 + i2x3 + i1i2x4 = z1 + i2z2 = ^1e1 + ^2e2 and = z1 — i1z2, ^2 = z1 + i1z2; with this, norm C2 is a Banach space, also C2 is a commutative algebra.

Remark 1. [7] C2 becomes a modified Banach algebra with respect to this norm in the sense that

llf.fc ^ V2||£||c2.\\v\\c2.

Using the representation of a bi-complex number, the set C2 can be expressed as

C2 = Xid + X2e2,

where Xi = {zi—i\z2: zi,z2 e C\(i\)} and X2 = {zi +i\z2 : Zi, z2 e C\(i\)}.

Suppose that Xi and X2 are normed spaces with the norm || • ||i, || • ||2, respectively. The hyperbolic norm on C2 is given by

W£|ili2 " W^i Wiei + ||^2|2e2.

Throughout this article, we consider

01 = 0 + On;

02 = 0 + 0^ + 0i2 + 0iii2 = 01^ + 01e2; 0h = 0 + 0^«2 = 0ei + 0e2;

62 = (02, 02,...).

2.2. Statistical boundedness.

The concept of statistical convergence depends on the notion of natural density of a set of natural numbers.

A subset E of N is said to have natural density 8(E) if

1 n

S(E)= lim-2 Xe(k),

n—>8 n

k" 1

where xe is the characteristic function on E.

Let (£n) and (rqn) be two sequences, such that £k = rqk for almost all k (in short a.a.k.) if 5({k e N: £k ^ ^k}) = 0.

A sequence of bi-complex numbers £ = (£k) is said to be statistically convergent to e C2 with respect to the Euclidean norm on C2 if, for every e > 0,

5({k e N : ||& — f ||c2 ^ £}) = 0,

It is denoted as stat-lim £k = £*.

If = 02, then the sequence (£k) of bi-complex numbers is said to be statistical null.

A sequence of bi-complex number £ = (£k) is said to be statistically Cauchy with respect to the Euclidean norm on C2 if, for every e > 0, there exists xko e N, such that

S({k e N: ll£k — Cfco ||c2 ^ £}) = 0.

A sequence £ = (£k) of bi-complex numbers is said to be statistically bounded if there exists 0 < M e C0, such that

8({k e N: ||&||c2 ^ M}) = 0.

Throughout the paper, w* and b® denote the sets of all and bounded sequences of bi-complex numbers, respectively.

We list the following classes of sequences, which will be used in this article:

b* := {C " (£k) e w*: there exists a bi-complex number rq such that stat- lim £k = rj}.

k^rn

K := {£ = (Ck) e w* : stat-lim^® £k = 02}. Cfr* := {£ = (£k) e w*: £ is statistically Cauchy}.

b® := {£ = (Ck) e w*: there exists 0 < M e Co: ¿({ra: ||&|| ^ M}) = 0}. I® := {(^1ke1 ),^1 k e X1: (^1k) is statistically bounded}. I® := {(^2 ke2),^2k e X2: (^2 k) is statistically bounded}. J® := {C = (Ck) e w*,£k = ke1 + ^2 ke2: (^1 k) is statistically bounded}. J® := {C = (Ck) e w*,£k = k^1 + ^2 ke2: (^2k) is statistically bounded}.

3. Main Result.

Theorem 1. If a sequence (£k) of bi-complex numbers £k = z1k + ¿2z2k, Vfc e N is statistically bounded, then the sequences (z1n) and (z2n) are also statistically bounded.

Proof. Let (£k) be statistically bounded; then there exists an M, such that s({k : ||c2 ^ M}) = 0, which implies 5({k : ||^1k + ¿2^||c2 ^ M}) = 0 and 8({k : | Zjk\ ^ M} ^ 8({k : ||^1k + ^2k||c2 ^ M}) = 0 for j = 1, 2. Hence, (z1k) and (z2k) are statistically bounded.

Conversely, let (z1k) and (z2k) be statistically bounded. Then, without loss of generality, we can find M > 0, such that

S({k: \z1k\ ^ M}) = 0

and

5({k : \\zlk + %2Z2k||c2 ^ M}) ^ 5({k : ¡zlk| ^ M}) + 5({k : ¡z2k| ^ M}) = 0

(by sub-additivity property). Hence, (£k) is statistically bounded. □

In view of the above theorem, we formulate the following corollaries:

Corollary 1. If a sequence (£k), where Çk = x1k + i1x2k + i2x3k + i1i2x4k of bi-complex numbers, is statistically bounded, then the sequences (xpra), p = 1, 2, 3, 4. of real numbers are also statistically bounded.

Corollary 2. If a sequence (£k), where Çk = ^1ke1 + ^2ke2 of bi-complex numbers, is statistically bounded, then the sequences (y1k) and (^2k) are statistically bounded.

Result 1. The inclusion relations

are strict; this follows from the following example:

Example 1. Consider a sequences (£k) and (rqk) of bi-complex numbers defined by

From the above example, it can be observed that (£k) R b*, but (£k) e b8.

Result 2. b8 c 68.

The converse parts are not true. Let us consider a sequence (£k) of bi-complex numbers, defined by

rk2i1 + k2i2, if k = n2,n e N;

(i) b* c 68 (Ü)c6* c 68

Pn + fc2z2 + fcï1ï2, if fc = n3,n e N;

Ck = ' ii — ¿2, 0,

if k = n2 + 1;

otherwise.

e1 — e2, if k = n2 + 1;

<

e1 + e2, if k = n2 + 2;

^e1e2, otherwise.

We observe that (&.) e 68, but (&.) R

Result 3.

(1

(2) I2

I8 i8

b* b*

(3) 1® 3 b*

(4) 1

b*

The inclusions are strict; this follows from the following examples:

Example 2. Let us consider a sequence (£k) of bi-complex numbers, defined by

Ck = ei + ^2fce2,@k e N

where

Vik = i

kii, if k = n3,n e N;

ii, if k = n3 + 1;

ei + e2, if k = n3 + 2;

eie2, otherwise.

and

^2 k = <

if k = n3,n e N; if fc = n3 + 1; if k = n3 + 2; otherwise.

A/ki1,

k2n, — (e1 + ^2)k

In the above example, it can be observed that (£k) is in J® but not in b®.

Theorem 2. The space b® is a linear space over C1(i1).

Proof. Let (£k), (rqk) e b®. Therefore, there exists M > 0, such that

8({k e N: ||&||c2 ^ M}) = 0, 5({k e N: ||c2 ^ M}) = 0.

Then (£k + rqk) e b® follows from the following inclusion relation:

{k e N: ||&+Vk||c2 ^ 2M} c {k e N: ||&||c2 ^ M}u{fc e N: ||c2 ^ M}.

For (£k) e b® and a e C1 (z1), similarly, it can be shown that (a^k) e b®. Therefore, the space b® is a linear space over C1(i1). □

Lemma 1. The spaces I®, I®, J® and J® are linear spaces over C1(i1).

Lemma 2. The space 6* is a commutative algebra with the identity 1 = 1 + 0«1 + 0i2 + 0i1i2 under coordinate-wise addition, real scalar multiplication, and term by term multiplication.

Proof. We know that C2 is a commutative algebra (linear space that is a commutative ring) with the identity 1 = 1 + 0«1 + 0i2 + 0i1i2 and b* c C2. Since b* is a linear space over C1(i1) and a commutative ring with the product defined on b*, such that

(a£k • Vk) = (Ck • oti]k), @(Ck), (Vk) e b* and Va e C^n). Hence, we see that 6* is a commutative algebra. □ In view of Remark 1, we have the following lemma:

Lemma 3. The space b* is a modified Banach algebra with respect to the norm HfW = inf W£kWc2= (Ck) e b*. Proof. We have the following inequality:

||£ • nH ^ V2||£|||M, for all e . (1)

From the definition of Banach algebra and using the eq.(1), we can easily prove that 6* is a modified Banach algebra with respect to the norm || • ||. □

Theorem 3. The spaces I8 and I8 are commutative Banach algebras.

Proof. Let e I8 be an arbitrary Cauchy sequence in I8. Then ^' is Cauchy sequence in b*8. Since 68 is complete, there exists rq e 68, such that

— V

||^p — ^||c2 = 0, as p — 8

inf — ^||c2 = 0, as p —> 8

inf ||^1pe1 + ^2pe2 — ^1e1 — ^2&2Hc2 = 0, as p — 8

inf ||^1p — 11 — 0, inf ||^2p — ^21|2 — 0, as p — 8.

Since ^' e I8, so ^'2p = 01 and, hence, ^2 = 01. So that rq e I8. Thus, I8 is a commutative Banach algebra and the identity element of I8 is (e1). Similarly, we can prove that I28 is a commutative Banach algebra with the identity element of I8 is (e2). □

Corollary 3. The spaces I8 and I8 are Gelfand algebras. Theorem 4. If a = (ak) e I8 and b = (bk) e I8, then

(1) ei • a e I®.

(2) e2 • a = 02.

(3) ei • b = 02.

(4) e2 • b e I®.

Proof. Let a = (ak) = kei) e I® and b = (bk) = (^2ke2) e I®.

(1) a = (ai, a2, a3,...)

i.e., ei • a = (aiei, a2ei, a3e\,...) = (ai, a2, a3,...) = a e I®.

(2) e2 • a = (^e2, a2e2, a;^,...) = (O2, O2, O2,...) = 62 .

(3) Similar to (2).

(4) b = (&i,&2,&3,...)

i.e., e2 • b = (e2^,e2&2,e2&3,...) = (&i,&2,&3,...) = b e I®. □ Result 4.

(1) I® Y I® = &®.

(2) J® Y J® = b®.

(3) I® XI® = e2.

(4) J® x J® *

Result 5. If £ = (£fc) e and rf = (e^uik) e I®,^" = (e2^2 k) e I®, then

a i "

ç = ^ + V .

Result 6. 6® = I® © I®.

Corollary 4. b®/I® is isomorphic to I®.

We formulate the following theorem without demo.

Theorem 5. If £ = (£k) e J® x J®, where £ = ei^i + e2^2, then a e I® and b e I®, a = ei^i, b = e2^2.

Definition 1. Let us define a relation „ on b® as follows: For £ = (^k^ = (Vk) e ,

C „ V ô \\£ - ¿i»2 =

It can be easily verified that it is equivalence relation on b®.

Now,

Wf — vW hi2 =0h

e1 ||^1k — ^1k||1 + e2|^2k — ¿2k 2 = 02 = e10 + e20 e1 ||^1k — ¿1k||1 = e10 = 0 and e2|^2k — ¿2k||2 = e20 = 0.

Since, ||e11iii2 = e1 and ||e21= e2. So we can write /j1 „ ¿1 and ^2 „ ¿2, where e I8 and e I8. The equivalence class [£] on b* is

[£] = {< : e „ cu

K] = [P1] + [M

Theorem 6. Let £ = (£k) and iq = () e b*8 and let B = {k: ^ ^k}. Then 8(B) = 0 if 'q e [£].

Proof. Since rq e [£],

Wf — vW h%2 =0h

||(^1k e1 + ^2k&2) — (¿1k e1 + ¿2 k e2|| iii2 = 0h W^1k — ¿1k || 1^1 + ||^2k — ¿2k W2e2 = 0e1 + 0e2

II 1 II 111 ^ 11

W^1k — ^1kW1 = 0 and ||^2k — ^2kW2 = 0.

Now,

VI i 12 I 2 ||2

^1k—^Wi; ^2k—^2k W2 ^ 4) - 0.

Therefore,

$ ({k: WCk — Vk||c2 > e}) = 0.

□

Lemma 4. Let £ = (£k) e 68 and if £ e I8 y I8, then £ is singular statistically bounded.

Proof. Here £ is statistically bounded. So, we only need to prove that for all k e N, is singular.

Let £ e I8 yI8; then either £ = (^1 ke1),^1k e X1, or £ = (^2 ke2),^2k e X2. Since ei are singular and k e Xi, so, for all k e N, ei are also singular, where i = 1, 2. □

Definition 1. A sequence £ = (£k) e b8 is convergent to £* in || • ||ili2 if

— C } ¿112 = 0h

Definition 2. A sequence £ = (£k) e b8 is called Cauchy sequence in

ll^fc — Cfco }¿112 = 0h,

l 1 if

or,

Cfc „ Cko ■

Theorem 7. If a bounded sequence £ = (£k),£k = k + e2^2k is statistically Cauchy, then £ is a Cauchy sequence in || • ||2.

Proof. Let £ = (£k) be statistically Cauchy; then, for each e > 0, there exists n0 e N, such that

5({k: ||& - Cno||c2 ^ 4) = 0. 5({fc: ||^ifc - ^ko ||i ^ ^}) = 0

and

S({k: ||^2k - ^2ko 2 ^ e2}) = 0.

Which implies that £j are statistical upper bounds of the sequences (||^jk—^jko||j and, hence, the statistical least upper bounds of (||^jk-^jko||j are £j■ Since £j are arbitrary, so, the statistical least upper bounds of (|^jk — Vjko ||j are zero.

Hence, ||£k — Cko||¿^2 " eiH^ik — ko||i + e2||^2k — ^2ko¡2 = 0h,j = 1,2. □

Corollary 5. If a sequence £ = (£k),£k = k + e2^2k is statistically convergent, then £ is a Cauchy sequence in || • ||2.

Theorem 8. Let g = (£k) be statistically convergent to £ * .If ( = ((k) e [£ ], then ( is statistically convergent to £ * in || • || 2.

Proof. Since £ is statistically convergent to £*, so

c e[es ne — c||^ = 0.

Now

?

iC — C !¿l¿2 ^ — C !¿l¿2 — Ci¿l¿2 = 0h.

Hence, ( is statistically convergent to £ * in || • || ixi 2. □

Tripathy [16] proved the decomposition theorem for statistically bounded sequences of real numbers.

The following theorem is the decomposition theorem for sequences of bi-complex numbers.

Theorem 9. If a sequence £ = (£k) of bi-complex numbers is statistically bounded, then there exists a bounded sequence rq = (rqk) of bi-complex numbers and a statistically null sequence ( = ((k) of bi-complex numbers, such that £ = rq + (.

Proof. Let £ = (£k), where Çk = ^1ke1 + ke2, be a statistically bounded sequence. Then 8 (B) = 0, where B = {k : ||£k ||c2 ^ M}. Define the sequences rq = (rqk) and ( = ((k) as follows:

'^k, if k e Bc; e1e2, otherwise.

^ ■ eie2, if k e Bc;

, otherwise.

From the above construction of "q and (, we have

£ = V + C

where rq e bCX) and ( e 6q. □

Following Lemma 1.1 of Salat [11], we state the following result without proof:

Proposition 1. A sequence (£k) of bi-complex numbers is statistically bounded if and only if there exists a set K = {k1 < k2 < ...} c N, such that 8(K) = 1 and (Çkn ) is bounded.

4. Summability properties.We are going to use the idea by Fridy [4].

Lemma 5. Let us consider a sequence £ = (£k) of bi-complex numbers, such that |£k|^ 01 for infinitely many k; then there exists a sequence rq = (rj) e , such that

I:

J]îk Vk

...... = 8.

k-1

Proof. Consider an increasing sequence (nk) of natural numbers, such that

nk ^ k2 and \Cnk |¿1 ^ 01.

Let us consider a sequence rq = (rqk) defined by

Vk =

, if k = ni ,j e N; e\ — e2, if k = + 1, j e N; ei + e2, otherwise.

Now, {fc: ||^k|| ^ 2} c {n: n = fc2, k e N}.

Thus, 5(fc: fe|| ^ 2}) c 8({n: n = fc2, fc e N}) = 0 and

.

k=1

□

Let T = (tn,k) be any summability matrix. Let £ = (£k) e w*; then £ is called a T bounded sequence if

T

T (£) = fn,k&) e b°

k-l

The set of all T bounded sequences is denoted by

bT8 = {£ = (&) e w* : T(£)e b8}.

Theorem 10. There is no row finite matrix T = (tn,k), such that b, contains b*.

Proof. Let T = (tn,k) be any row finite summability matrix. Choose \tni k' |^ 0i. Choose kx ^ k , such that

\tni,k" \h ^ 0i and \im,k \h = 0i for all k ^ k[.

We can select an increasing sequence of rows and columns, such that for each r

\t'nr,kr \i1 ^ 0,kr ^ r

and

tnr,k = 0, for all k > kr. Define the sequence £ = (£k) as

1

Ck =

tnr ,kr k2, (—1) k,

r ¿-0 tnr,kiCkJ , if — fcr;

if k " kr—i;

otherwise.

Then £ is not a T bounded sequence. But for any sufficiently large M > 0,

we have

{k: ||fk||c2 ^ M} c {kr ,kr-1,r e N} c {r2: r e N} y {r2 - 1: r e N}.

Hence, £ e . □

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Received January 08, 2023.

In revised form, May 12, 2023.

Accepted May 21, 2023.

Published online June 10, 2023.

Department of Mathematics, Tripura University Suryamaninagar, Agartala-799022, Tripura(W), India

Subhajit Bera

E-mail: berasubhajit0@gmail.com

Binod Chandra Tripathy E-mail: tripathybc@gmail.com