Int. J. Contemp. Math. Sciences, Vol. 5, 2010, no. 12, 579 - 589
Composite Convolution Operators on 2(Z) Shallu Sharma, B. S. Komal1 and Sunil Kumar Sharma Department of Mathematics, University of Jammu Jammu - 180006, India Abstract The compact, Hermitian composite convolution operators are characterized in this paper. It is shown that the set of all convolution operators on 2 (Z) is a maximal abelian subalgebra of B(2 (Z)).
Mathematics Subject Classification: 47B99, 47B38 Keywords: Convolution product, Hermitian operator, isometry, maximal subalgebra, adjoint of an operator, compact operator
1. Introduction: For p = 1,2, let p (Z) denote the space of p-th summable sequences of complex numbers. If p = 2, then 2 (Z) is Hilbert space under ∞ the inner product f, g = fn gn and for p = 1, 1 (Z) is a Banach space n=−∞
under the norm ||x|| =
∞
|xn |. If φ ∈ 1 (Z), f ∈ 2 (Z), then we form the
n=−∞
convolution product f ∗ φ which is defined by (f ∗ φ)(m) =
∞ n=−∞
f (n)φ(m−n).
If T : Z → Z is a mapping such that the transformation CT,φ : 2 (Z) → 2 (Z) defined by (CT,φ f ) = (f ∗ φ)oT is bounded. We shall call CT,φ a composite convolution operator induced by the pair (φ, T ). In case T(z) = z for all z ∈ Z, we write CT,φ = Cφ which is known as a convolution operator. In this paper we initiate the study of composite Convolution Operators. The Hermitian, isometric composite convolution Operators are characterized. We 1
[email protected]
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also prove that the set of all convolution operators is a maximal abelian subalgebra of B(2 (Z)), the Banach algebra of all bounded linear operator on 2 (Z). The adjoint of a composite convolution operator is obtained. It is shown that there doesnot exist any non-zero compact convolution operator. For literature concerning composite Operators and convolution operators, we refer to Singh and Komal [11], Komal and Gupta [5], Komal and Sharma [6] Kumar [7], Nordgren [8], Ridge [9], Singh, Gupta and Komal [10].
2. Bounded Convolution Operators on 2(Z): In this section,
we study convolution operators on 2 (Z). In this paper we take φ(n, m) = φ(n − m). The function φ can also be treated as a function of Z.
Theorem 2.1: Let φ ∈ 2 (Z × Z). Then Cφ : 2 (Z) → 2 (Z) is a bounded operator.
Proof: For f ∈ 2 (Z), consider
||Cφ f ||
2
= = = ≤
∞ n=−∞ ∞ n=−∞ ∞
|(Cφ f )(n)|2 |(f ∗ φ)(n)|2 |
∞
f (m)φ(n − m)|2
n=−∞ m=−∞ ∞ ∞
|f (m)|
n=−∞ m=−∞ ∞ 2
= ||f ||
2
2
∞
|φ(n − m)|2
m=−∞ ∞
|φ(n − m)|2
n=−∞ m=−∞ 2
= ||φ|| ||f || Hence Cφ is a bounded operator.
Example 2.2: Let Φ : Z × Z → C be defined by Φ(n, m) = φ(n − m) =
1, 0,
if m = n elsewhere
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Then ||Cφ f ||
2
= =
∞
|
∞
f (m)φ(n − m)|2
n=−∞ m=−∞ ∞ 2
|f (n)| = ||f ||2
n=−∞
Therefore Cφ is a bounded operator.
Theorem 2.3: Let Cφ ∈ B(2 (Z)). Then Cφ is Hermitian if and only if
φ(m − n) = φ(n − m). Proof: Suppose φ(m − n) = φ(n − m). That is, φ = φ∗ . For f, g ∈ 2 (Z), we have
Cφ f, g = = = = =
∞
(Cφ f )(n)g(n)
n=−∞ ∞
(
∞
f (m)φ(n − m))g(n)
n=−∞ m=−∞ ∞
f (m)
∞
m=−∞ ∞
n=−∞ ∞
n=−∞ ∞
m=−∞
f (m)
φ(n − m)g(n) φ(m − n) g(n)
f (m)(g ∗ φ)(m)
n=−∞
= f, Cφ g Hence Cφ is Hermitian. Conversely, suppose that Cφ is Hermitian. Then Cφ = Cφ∗ Now (Cφ en )(m) = φ(m − n)
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and (Cφ∗ en )(n) = Cφ∗ en , em = en , Cφ em = Cφ em , en = (Cφ em )(n) = φ(n − m) = φ∗ (m − n) Hence φ = φ∗
Example 2.4 : Let φ : Z × Z → C be defined by φ(n, m) =
1 , (n−m)2
1,
for m = n for m = n
Then φ ∈ 2 (Z × Z) and φ(n, m) = φ(m − n) Therefore φ(n, m) = φ(m − n) ∀ n, m ∈ Z Hence Cφ is Hermitian.
Theorem 2.5: Let Cφ ∈ (2 (Z)). Then Cφ is compact if and only if φ = 0. Proof: Suppose Cφ is compact. We show that φ = 0. If φ(p, q) = 0 for some
p, q ∈ Z
||Cφ en ||
2
= = =
∞ m=−∞ ∞ m=−∞ ∞
|(Cφ en )(m)|2 |(en ∗ φ)(m)|2 |φ(m − n)|2
m=−∞
≥ |φ(p − q)|2 for every n
(1)
But en → 0 weakly. From (1) we can conclude that Cφ en does not converge to zero strongly. This contradicts our supposition. Hence φ = 0. Conversely, if φ = 0, then Cφ = 0 and therefore it is compact.
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Theorem 2.6: Let S = {Cφ : Cφ ∈ B(2 (Z)}. Then S is maximal abelian
subalgebra of B(2 (Z)). Proof: Let Cφ and CΨ be two convolution operators on 2 (Z). Then (Cφ + Cψ )f = Cφ f + Cψ f = f ∗φ+f ∗ψ = f ∗ (φ + ψ) = Cφ + ψ(f ) = Cφ+ψ (f ). and (αCφ (f )) = α(Cφ f ) = α(f ∗ φ) = f ∗ (αφ) = Cαφ f. Moreover Cψ Cφ = Cψ ∗ φ .
Hence S is an algebra. Next we prove that if S is maximal abelian subalgebra, suppose A commutes with Cφ for every φ. Then for every n ∈ Z. Aen = A(en ∗ e0 ) = A(e0 ∗ en ) = ACen e0 = Cen Ae0 = Ce n ψ = ψ ∗ en = en ∗ ψ = Cψ en . This shows that A = Cψ
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S. Sharma, B. S. Komal and S. K. Sharma
Hence S is maximal abelian subalgebra.
3.Bounded Composite Convolution Operators : The main purpose of this section is to study composite convolution operators.
Theorem 3.1: Let T : N → N be a mapping and φ ∈ 1 (Z). then
CT,φ : 2 (Z) → 2 (Z) is a bounded operator if there exist M > 0 such that f0 (n) ≤ M for all n ∈ Z. Proof: For f ∈ 2 (Z), consider ||CT,φ f ||
2
= = =
∞
|(CT,φ f )(m)|2
m=−∞ ∞ m=−∞ ∞
|(f ∗ φ)T (m)|2
|(f ∗ φ)T (m)|2
m=−∞ P ∈T −1 (m)
=
∞
f0 (m)|(φ ∗ f )(m)|2
m=−∞
= ≤ =
∞
f0 (m)|
m=−∞ ∞ m=−∞ ∞ m=−∞
f0 (m)[
∞
φ(n)f (m − n)|2
n=−∞ ∞
n=−∞ ∞
f0 (m)[
|φ(n)f (m − n)|]2 |(fm (n)|λn ]2
n=−∞
where fm (n) = f (m − n) and λn = |φ(n)| so that
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Composite convolution operators
λ(Z) = ||CT,φ f ||
2
= ≤ =
∞
|φ(n)| < ∞
n=−∞ ∞
f0 (m)[
m=−∞ ∞ m=−∞ ∞
∞
|I(n)fm (n)|λn ]2 , where I(n) = 1 for all n.
n=−∞ ∞
f0 (m)[
n=−∞ ∞
f0 (m)[
m=−∞
= ||φ||1
and
∞
∞
2
|fm (n)| λn
|I(n)|2 λn ],
n=−∞ ∞
|fm (n)|2 |φ(n)|
n=−∞ ∞
|φ(n)|]
n=−∞
f0 (m)|fm (n)|2 |φ(n)|
m=−∞ n=−∞
= ||φ||1 = ||φ||1 = ||φ||1
∞
∞
m=−∞ n=−∞ ∞ ∞ n=−∞ m=−∞ ∞
|φ(n)|
n=−∞ ∞
≤ M||φ||1 =
f0 (m)|f (m − n)|2 |φ(n)| f0 (m)|f (m − n)|2 |φ(n)| ∞
f0 (m)|f (m − n)|2
m=−∞
|φ(n)|||f ||22
n=−∞ 2 M||φ||1 ||f ||22
Hence ||CT,φ f ||2 ≤ K||f ||2 where K 2 = M||φ||2 This proves that CT,φ is a bounded operator. For g ∈ 2 (Z), φ ∈ 1 (Z), we define (Ag)(n) =
∞ m=−∞
g(m)φ(T (m) − n).
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Theorem 3.2: Let CT,φ ∈ 2 (Z). Then
∗ =A CT,φ
Proof: For f, g ∈ 2 (Z), consider
CT,φ f, g = = =
∞
(CT,φ f )(m)g(m)
m=−∞ ∞ m=−∞ ∞
(f ∗ φ)(T (m))g(m) ∞
f (n)φ(T (m) − n))g(m)
m=−∞ n=−∞
=
∞
∞
f (n)φ(T (m) − n))g(m)
n=−∞ m=−∞
= =
∞ n=−∞ ∞
f (n)(
∞
φ(T (m) − n))g(m))
m=−∞
f (n)(Ag)(n)
n=−∞
= f, Ag
∗ = A. This proves that CT,φ
Theorem 3.3: Let CT,φ ∈ B(2 (Z)) and φ(m) = δ0m. Then CT,φ is an isometry if and only if T is invertible.
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Proof: Suppose the condition is true. Then ||CT,φ f ||2 = = =
∞ m=−∞ ∞ m=−∞ ∞
|(CT,φ f )(m)|2 |(f ∗ φ)(T (m))|2
|(f ∗ φ)(T (p))|2
m=−∞ p∈T −1 (m)
=
∞
|(f ∗ φ)(m)|2
m=−∞ p∈T −1 (m)
= = = =
∞ m=−∞ ∞ m=−∞ ∞
f0 (m)|(f ∗ φ)(m)|2 |(f ∗ φ)(m)|2 ∞
|f (p)φ(m − p)|2
m=−∞ p=−∞ ∞
|f (m)|2
m=−∞ 2
= ||f ||
Hence CT,φ is an isometry. Conversely, if T is not invertible, then either T is not injective or T is not surjective. For n ∈ Z − T (Z), we have ∞
||CT,φ en ||2 = = =
m=−∞ ∞ m=−∞ ∞
f0 (m)|(en ∗ φ).(m)|2 f0 (m)|(
∞
en (p)φ(m − p))|2
p=−∞
f0 (m)|φ(m − n)|2
m=−∞
= f0 (n)|φ(0)|2 = 0 But ||en || = 1, so that CT,φ is not isometry. Similarly if T is not injective then a simple computation shows that
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||CT,φ en ||2 = f0 (n) > 1 and ||en || = 1 This again proves that CT,φ is not isometry.
References 1. Carlson, J.W. : “Weighted composition operators on 2 ,” Dissertation, Purdue University, Wert Lafayette, Indian (1985). 2. Gupta and Komal, B.S.: “Composition integral operator on L2 (μ)” Pitmann Lecture Notes in Mathematics Series 377, 92-99, (1977). 3. Halmos, P.R., : “ A Hilbert space problem book,” Springer Verlag, New york, (1974). 4. Kaninska, A. and Musieelak, J., : “ On Convolution Operator in Orlicz spaces,” Revirta Mathematica de la, Universidad Computense de Madrid, Vol. 2, 151-158, (1989). 5. Komal, B.S. and Gupta, D.K.,: “ Normal composition operators, ” Acta. Sci. Math. (Szeged) 47 (1984), 445-448. 6. Komal, B.S. and Sharma T.K., “Composition operator on p , 0 < p < 1,” Jammu University Reviews Vol. 1994, 41-48. 7. Kumar : “ Composition operators on L2 (λ),” Thesis University of Jammu, (1978). 8. Nordgren, E.A. : “ Composition operators on Hilbert spaces ” Lecture notes in Maths, 693, Springer Verlag, New York, (1978), 37-63.
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9. Ridge, W.C.:“ Composition operators,” Thesis Indiana, University, (1969).
10. Singh R.K., Gupta, D.K. and Komal, B.S.:“ Some results on Composition operators on 2 , Internat. J. Math. and Math. Soc. 2(1979), 29-34. 11. Singh, R.K. and Komal, B.S.: “Composition operators on p and its adjoint,” Proc. Amer. Math. Soc. 70 (1978), 21-25.
12. Stepanov, V.D., :“ On Convolution Integral operators”, Soviet Math. Dokal, 19, No. 6, 1978.
13. Stepanov., V.D. : “ On boundedness and compactness of a class of Convolution operators,” Soviet Math. Dokal. 41, 446-470 (1990).
