Some Properties of *- n-class Q Operators

Transcription

1 Volume 117 No , ISSN: printed version; ISSN: on-line version url: ijpam.eu Some Properties of *- n-class Q Operators D.Senthilkumar 1 and S.Parvatham 2 1,2 Post Graduate and Research Department of Mathematics, Government Arts College, Coimbatore senthilsenkumhari@gmail.com 2 parvathasathish@gmail.com Abstract In this paper, we introduce a new class of operators, which we call the class of *-n-class Q operators. This class of operators contains the class of * paranormal operators. We prove some basic properties and a structure theorem for *-n-class Q operators. We also give the matrix representation for this class of operators. AMS Subject Classification: 47A05, 47A11, 47B47. Key Words and Phrases:Class Q operators, spectrum, approximate point spectrum. 1 Introduction and preliminaries Let H be an infinite dimensional separable complex Hilbert space. Let BH be the algebra of all bounded linear operators acting on H. Let T be an operator on H. An operator T is called class Q [2], if T 2 T 2 2T T + I 0, equivalently T Q if T x T 2 x 2 + x 2 for every x H. It was also proved that T is paranormal if and only if λt is in class Q for all λ > 0 and every paranormal operator is a normaloid of class Q. Also he showed that the restriction of T to an invariant subspace is again a class Q operator. Devika, Suresh [1], introduced a new class of operators which we call the quasi class Q operators and it is defined as for T BH, T 2 x T x 2 + T x 2, for every x H. A k-quasi class Q operator is defined as follows []. An operator T is said to be k-quasi class Q operator if 129

2 T k+1 x T k+2 x 2 + T k x 2, for every x H and k is a natural number. D.Senthilkumar,prasad T in [4], has defined the new class of operators, which we call M-class Q operators. An operator T is called M-class Q if for a fixed real number M 1 T satisfies M 2 T 2 T 2 2T T + I 0, or equivalently T x M 2 T 2 x 2 + x 2 for every x H and for a fixed real number M 1. In [6], Youngoh Yang and Cheoul jun Kim introduced a class Q operators. if T 2 T 2 2T T + I 0, then T is called class Q operators. He also proved that if T is class Q if and only if T x T 2 x 2 + x 2 for every x H. Authors have introduced classes of quasi class Q, k-quasi class Q, M-class Q and studied properties of these classes of operators. If T BH, we shall write NT and RT for the null space and the range of T, respectively. Also, let σt and σ a T denote the spectrum and the approximate point spectrum of T, respectively. Let σ p T,πT, ET denotes the point spectrum of T, the set of poles of the resolvent of T, the set of all eigenvalues of T which are isolated in σt, respectively. The spectrum σt of T is the set of λsuch that T λ is not invertible on all of the Hilbert space, where the λ s are complex numbers and I is the identity operator. In this paper, we introduce a new class of operators, which we call the class of *-n-class Q operators. This class of operators contains the class of * paranormal operators. We prove some basic properties and a structure theorem for *-n-class Q operators. We also give the matrix representation for this class of operators and we prove that the restriction of *-n-class Q operator T to an invariant subspace is again *-n-class Q operator. 2 Main results In this section we have defined and give some properties of *-n - class Q operators. Definition 1. An operator T BH is said to be *-n-class Q operator if for every positive integer n 2 and for every x H T x n T 1+n x 2 + n x 2. Theorem 2. For each positive integer n 2, T is *-n-class Q operator if and only if T T 1+n 1 + nt T + ni 0. Proof. Since T is *-n-class Q operator, then T x n T 1+n x 2 + n x 2 10

3 for every x H and for any positive integer n 0. Then T 1+n x 2 + n x n T x 2 T 1+n x n T x 2 + n x 2 0 T 1+n x, T 1+n x 1 + n T x, T x + n x, x 0 T T 1+n 1 + nt T + nx, x 0 T T 1+n 1 + nt T + ni 0 Theorem. If T BH is *- class Q operator then T is *-n-class Q operator. Proof. Let T BH is *-class Q operator,then T 2 T 2 2T T I. We use induction principle, when n = 1, the result is true. When n = 2, T T T T + 2I T 2T T IT T T Now we assume the result is true for n = k 1. Then for n = k, T 1+k T 1+k 1 + kt T + ki 0 Corollary 4. operator. Corollary 5. operator. If T BH is *-n class Q operator then T is *-n + 1-class Q If T BH is *-n class Q operator then αt is also *-n-class Q Theorem 6. Let T BH. If λ 1 2 T is an operator of *-n-class Q, then T is *-n-paranormal operator for all λ > 0. Proof. Since λ 1 2 T is an operator of *-n-class Q then λ 1 2 T λ 1 2 T 1+n 1 + nλ 1 2 T λ 1 2 T + ni 0 By multiplying λ 1+n and letting λ = µ, we have T is *-n-paranormal operator for all λ > 0. By simple calculation we get the following results. Theorem 7. If *-n-class Q operator T doubly commutes with an isometric operator S, then TS is an operator of *-n-class Q. Theorem 8. If a *-n-class Q operator T BH is unitarily equivalent to operator S, then S is an operator of *-n-class Q. Theorem 9. If T BH is of *-n class Q operator for a positive integer n, λ T2 0 λ σ p T and T is of the form T = on H = kert λ rant λ, 0 T then 1. T 2 = 0 and 2. T is *-n-class Q operator. λ T2 Proof. Let T = on H = kert λ rant λ. Without the loss of 0 T generality assume that λ = 1, then by Theorem 2, T T 1+n 1+nT T +ni 0. 11

4 Now, T 1+n = T = 1 n 0 T 1+n and 1 0 n T T T 1+n 1 n j=0 = T 2T n j n n n + T 1+n T So, T T 1+n 1 + nt T + ni 0 gives A B B 0 C Where A = n1 + T 2 T2 + n, B = n 1 + nt 2 T and C = n n + T T 1+n 1 + nt T + n A B But,we know that, If A is a matrix of the form B 0 if and only if C A 0, C 0 and B = A 1 2 W C 1 2 for some contraction W. Therefore n1 + T 2 T2 + n 0, which implies that 1 + nt 2 T2 0. This gives T 2 = 0, since n is a positive integer. Also T is *-n-class Q operator. Corollary 10. If T BH is of *-n class Q operator for a positive integer n, λ 0 then T is of the form T = on H = kert λ rant λ, where T 0 T is *-n-class Q operator and kert λ = {0}. Theorem 11. If T BH is a *-n-class Q operator for a positive integer n, T does not have dense range and T has the following 2 2 matrix representation T1 T T = 2 on H = rant ker T 0 T, if and only if T 1 is also *-n-class Q operator on rant and T = 0.Further more σt = σt 1 {0} where σt denotes the spectrum of T. Proof. Let T BH be *-n class Q operator and P be an orthogonal projection onto rant. Then T 1 = T P = P T P. By Theorem 2 we have that P T T 1+n 1 + nt T + nip 0 T 1 T 1+n nt 1 T 1 + ni 1 + nt 2 T n T Therefore T 1 is *-n-class Q operator on rant. Also for any x = x 1, x 2 H, T k x 2, x 2 = T k I P x, I P x = I P x, T k I P x = 0 12

5 This implies T = 0 Since σt τ = σt 1 σt where τ is the union of the holes in σt, which happens to be a subset of σt 1 σt [by corollary 7, 11]. σt = 0 and σt 1 σt has no interior points we have σt = σt 1 {0}. T1 T Suppose that T = 2 on H = rant ker T 0 T where T 1 is *-n-class Q operator on rant and T = 0 T 1+n = T T 1+n = T 1+n 1 n 0 T 1+n and T T 1 0 = n T T1 T 1+n 1 T1 n n T1 1+n n n + T 1+n T Since T = 0 and T is *-n class Q operator, T T 1+n 1 + nt T + ni = Hence T is *-n class Q operator. T 1 T1 1+n 1 + nt 1 T1 + T 2 T2 + n Theorem 12. Let M be a closed T -invariant subspace of H. Then the restriction T M of a *-n class Q operator T to M is *-n class Q operator. Proof. By Theorem 11, T M is also *-n class Q operator. Theorem 1. If T is *-n-class Q operator and λ 0, then T x = λx implies T x = λx for every unit vector x in H. Proof. Suppose T is *-n class Q operator and since T x = λx, we have λ n λ 21+n + n λ 2, then 1 1+n λ 21+n + n = λ 2 Hence T x 2 λ 2.Also T λ x, T λ x λ 2 2 λ 2 + λ 2 = 0. Hence, T x = λx Corollary 14. Let T is *-n class Q operator and λ, µ be distinct eigen values of T. If x and y are the corresponding eigen vectors of λ and µ respectively, then x, y = 0. 0 Corollary 15. λ C. If T is *-n class Q operator then βt λ αt λ for all Corollary 16. For T BH, let T is *-n class Q operator, 1. If λ σ a T and T λx m 0 for unit vectors x m then T λ x m 0 2. Let λ and µ λ µ be in σ a T. If T λx m 0 and T µy m 0 for unit vectors x m and y m then x m, y m 0. 1

6 Theorem 17. Let T be a regular *- n class Q operator, then the approximate point spectrum lies in the disc σ ap T {λ C : 1+n 1 2 T 1 T n 2 +n T λ T Proof. Suppose T is regular n class Q operator, then for every unit vector x in H, we have x 2 = T 1 T x 2 T 1 2 T x 2 T n T 1+n x 2 + n x 2 T n T n 2 T x 2 + n T 1 2 T x 2 Hence T x n x 2 T 1 2 T n 2 + n T 1 2 Now assume that λ σ ap T. Then there exists a sequence { x m }, x m = 1 such that T λx m 0 when m we have Now when m, λ References T x m λx m T x m λ x m T λ 1 + n 1/2 λ T 1 T n 2 + n T 1 2 1/2 1+n 1/2 T 1 T n 2 +n T 1 2 1/2 [1] A. Devika, G. Suresh, Some properties of quasi class Q operators, International Journal of Applied Mathematics and Statistical Sciences IJAMSS, 2, 201 no. 1, [2] B. P. Duggal, C. S. Kubrusly and N. Levan, Contractions of class Q and invariant subspaces, Bull. Korean Math. Soc , no. 1, [] V.R. Hamiti, on k-quasi class Q operators,bulletin of Mathematical Analysis and Applications,6 2014, no., 1 7. [4] D. Senthil Kumar, T. Prasad, M class Q composition operators, Scientia Magna., , no. 1, [5] D. Senthilkumar, P. Maheswari Naik And D. Kiruthika, Quasi class Q* composition operators, International J. of Math. Sci. and Engg. Appls. IJMSEA,5 2011, no. 4, 1 9. [6] Y. Yang and Cheoul Jun Kim, Contractions of class Q*, Far East. J.Math.Sci.FJMS, , no.,

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