Composite Convolution Operators on l 2 (Z)

Transcription

1 Int. J. Contemp. Math. Sciences, Vol. 5, 2010, no. 12, Composite Convolution Operators on l 2 (Z) Shallu Sharma, B. S. Komal 1 and Sunil Kumar Sharma Department of Mathematics, University of Jammu Jammu , India Abstract The compact, Hermitian composite convolution operators are characterized in this paper. It is shown that the set of all convolution operators on l 2 (Z) is a maximal abelian subalgebra of B(l 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 l p (Z) denote the space of p-th summable sequences of complex numbers. If p 2, then l 2 (Z) is Hilbert space under the inner product f,g f n g n and for p 1, l 1 (Z) is a Banach space under the norm x n n x n. If φ l 1 (Z),f l 2 (Z), then we form the 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 C T,φ : l 2 (Z) l 2 (Z) defined by (C T,φ f)(f φ)ot is bounded. We shall call C T,φ a composite convolution operator induced by the pair (φ, T ). In case T(z) z for all z Z, we write C T,φ C φ 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 bskomal2@yahoo.co.in

2 580 S. Sharma, B. S. Komal and S. K. Sharma also prove that the set of all convolution operators is a maximal abelian subalgebra of B(l 2 (Z)), the Banach algebra of all bounded linear operator on l 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 l 2 (Z): In this section, we study convolution operators on l 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 l 2 (Z Z). Then C φ : l 2 (Z) l 2 (Z) is a bounded operator. Proof: For f l 2 (Z), consider C φ f 2 (C φ f)(n) 2 n n n (f φ)(n) 2 f(m)φ(n m) 2 n f 2 φ 2 f 2 f(m) 2 n φ(n m) 2 φ(n m) 2 Hence C φ is a bounded operator. Example 2.2: Let Φ : Z Z C be defined by Φ(n, m) φ(n m) { 1, if m n 0, elsewhere

3 Composite convolution operators 581 Then C φ f 2 f(m)φ(n m) 2 n n f(n) 2 f 2 Therefore C φ is a bounded operator. Theorem 2.3: Let C φ B(l 2 (Z)). Then C φ 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 l 2 (Z), we have C φ f,g (C φ f)(n)g(n) n n ( n n f,c φ g f(m)φ(n m))g(n) f(m) f(m) φ(n m)g(n) n f(m)(g φ)(m) φ(m n) g(n) Hence C φ is Hermitian. Conversely, suppose that C φ is Hermitian. Then Now C φ C φ (C φ e n )(m) φ(m n)

4 582 S. Sharma, B. S. Komal and S. K. Sharma and (Cφe n )(n) Cφe n,e m e n,c φ e m C φ e m,e n (C φ e m )(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 for m n 1, for m n Then φ l 2 (Z Z) and φ(n, m) φ(m n) Therefore φ(n, m) φ(m n) n, m Z Hence C φ is Hermitian. Theorem 2.5: Let C φ (l 2 (Z)). Then C φ is compact if and only if φ 0. Proof: Suppose C φ is compact. We show that φ 0. Ifφ(p, q) 0 for some p, q Z C φ e n 2 (C φ e n )(m) 2 (e n φ)(m) 2 φ(m n) 2 φ(p q) 2 for every n (1) But e n 0 weakly. From (1) we can conclude that C φ e n does not converge to zero strongly. This contradicts our supposition. Hence φ 0. Conversely, if φ 0, then C φ 0 and therefore it is compact.

5 Composite convolution operators 583 Theorem 2.6: Let S {C φ : C φ B(l 2 (Z)}. Then S is maximal abelian subalgebra of B(l 2 (Z)). Proof: Let C φ and C Ψ be two convolution operators on l 2 (Z). Then and (C φ + C ψ )f C φ f + C ψ f f φ + f ψ f (φ + ψ) C φ + ψ(f) C φ+ψ (f). (αc φ (f)) α(c φ f) α(f φ) f (αφ) C αφ f. Moreover C ψ C φ C ψ φ. Hence S is an algebra. Next we prove that if S is maximal abelian subalgebra, suppose A commutes with C φ for every φ. Then for every n Z. Ae n A(e n e 0 ) A(e 0 e n ) AC en e 0 C en Ae 0 C en ψ ψ e n e n ψ C ψ e n. This shows that A C ψ

6 584 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 l 1 (Z). then C T,φ : l 2 (Z) l 2 (Z) is a bounded operator if there exist M > 0 such that f 0 (n) M for all n Z. Proof: For f l 2 (Z), consider C T,φ f 2 (C T,φ f)(m) 2 (f φ)t (m) 2 P T 1 (m) (f φ)t (m) 2 f 0 (m) (φ f)(m) 2 f 0 (m) φ(n)f(m n) 2 f 0 (m)[ f 0 (m)[ n n n φ(n)f(m n) ] 2 (f m (n) λ n ] 2 where f m (n) f(m n) and λ n φ(n) so that

7 Composite convolution operators 585 λ(z) φ(n) < and C T,φ f 2 n f 0 (m)[ I(n)f m (n) λ n ] 2, where I(n) 1 for all n. f 0 (m)[ f 0 (m)[ φ 1 n n n n f m (n) 2 λ n I(n) 2 λ n ], n f m (n) 2 φ(n) n f 0 (m) f m (n) 2 φ(n) φ(n) ] φ 1 f 0 (m) f(m n) 2 φ(n) φ 1 φ 1 n n n φ(n) f 0 (m) f(m n) 2 φ(n) f 0 (m) f(m n) 2 M φ 1 φ(n) f 2 2 n M φ 2 1 f 2 2 Hence C T,φ f 2 K f 2 where K 2 M φ 2 This proves that C T,φ is a bounded operator. For g l 2 (Z),φ l 1 (Z), we define (Ag)(n) g(m)φ(t (m) n).

8 586 S. Sharma, B. S. Komal and S. K. Sharma Theorem 3.2: Let C T,φ l 2 (Z). Then C T,φ A Proof: For f,g l 2 (Z), consider C T,φ f,g (C T,φ f)(m)g(m) n (f φ)(t (m))g(m) f(n)φ(t (m) n))g(m) f(n)φ(t (m) n))g(m) n n n f,ag f(n)( φ(t (m) n))g(m)) f(n)(ag)(n) This proves that C T,φ A. Theorem 3.3: Let C T,φ B(l 2 (Z)) and φ(m) δ m 0. isometry if and only if T is invertible. Then C T,φ is an

9 Composite convolution operators 587 Proof: Suppose the condition is true. Then C T,φ f 2 (C T,φ f)(m) 2 (f φ)(t (m)) 2 p T 1 (m) p T 1 (m) p f 2 (f φ)(t (p)) 2 (f φ)(m) 2 f 0 (m) (f φ)(m) 2 (f φ)(m) 2 f(m) 2 f(p)φ(m p) 2 Hence C T,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 C T,φ e n 2 f 0 (m) (e n φ).(m) 2 f 0 (m) ( f 0 (n) φ(0) 2 0 p f 0 (m) φ(m n) 2 e n (p)φ(m p)) 2 But e n 1, so that C T,φ is not isometry. Similarly if T is not injective then a simple computation shows that

10 588 S. Sharma, B. S. Komal and S. K. Sharma and C T,φ e n 2 f 0 (n) > 1 e n 1 This again proves that C T,φ is not isometry. References 1. Carlson, J.W. : Weighted composition operators on l 2, Dissertation, Purdue University, Wert Lafayette, Indian (1985). 2. Gupta and Komal, B.S.: Composition integral operator on L 2 (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, , (1989). 5. Komal, B.S. and Gupta, D.K.,: Normal composition operators, Acta. Sci. Math. (Szeged) 47 (1984), Komal, B.S. and Sharma T.K., Composition operator on l p, 0 <p<1, Jammu University Reviews Vol. 1994, Kumar : Composition operators on L 2 (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),

11 Composite convolution operators 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 l 2, Internat. J. Math. and Math. Soc. 2(1979), Singh, R.K. and Komal, B.S.: Composition operators on l p and its adjoint, Proc. Amer. Math. Soc. 70 (1978), Stepanov, V.D., : On Convolution Integral operators, Soviet Math. Dokal, 19, No. 6, Stepanov., V.D. : On boundedness and compactness of a class of Convolution operators, Soviet Math. Dokal. 41, (1990).