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On the Normality of Operators Revista Colombiana de Matem¶aticas Volumen 39 (2005), p¶aginas87{95 On the normality of operators Salah Mecheri King Saud University, Saudi Arabia Abstract. In this paper we will investigate the normality in (WN) and (Y) classes. Keywords and phrases. Normal operators, Hilbert space, hermitian operators. 2000 Mathematics Subject Classi¯cation. Primary: 47A15. Secondary: 47B20, 47A63. Resumen. En este art¶³culo nosotros investigaremos la normalidad en clases (WN) y (Y). 1. Introduction Let H be a complex Hilbert space with inner product h;i and let B(H) be the algebra of all bounded linear operators on H. For A in B(H) the adjoint of A ¤ ¤ 1 is denoted by A . For any operator A in B(H) set, as usual, j A j= (A A) 2 and [A¤;A] = A¤A ¡ AA¤ = j A j2 ¡ j A¤ j2 (the self commutator of A), and consider the following standard de¯nitions: A is hyponormal if j A¤ j2 · j A j2 (i.e., if [A¤;A] is nonnegative or, equivalently, if kA¤xk · kAxk for every x in H), normal if A¤A = AA¤, quasinormal if A¤A commutes with A, and m-hyponormal if there exists a positive number m, such that m2(A ¡ ¸I)¤(A ¡ ¸I) ¡ (A ¡ ¸I)(A ¡ ¸I)¤ ¸ 0; for all ¸ 2 C: Let (N); (QN); (H); and (m ¡ H) denote the classes constituting of normal, quasinormal, hyponormal, and m-hyponormal operators. Then (N) ½ (QN) ½ (H) ½ (m ¡ H): An operator T in B(H) is said to be hermitian if T = T ¤. It is well known that hermitian operators can be characterized in the following way: an operator T in B(H) is hermitian if and only if hT x; xi is real. In [3] the authors gave an other characterization involving inequalities. We denote by (WN) the class of operators in B(H) satisfying the following inequality (Re T )2 · jT j2 ; 87 88 SALAH MECHERI (T +T ¤) (T ¡T ¤) where Re T = 2 is the real part of T and we will write ImT = 2 for the imaginary part of T . This class has been introduced by Fong and Istratescu [3] who conjectured: If T 2 (WN) and σ(T )(the spectrum of T ) is real, then T is hermitian. It is known that if in addition T is hyponormal or T + T ¤ commutes with TT ¤ or if H is ¯nite-dimensionel the conjecture holds [3, Corollary 2.2]. Notice that T is hermitian if and only if (Re T )2 ¸ jT j2 : (1:1) By reversing the inequality (1.1) we obtain the class (WN): This class contains the class of hyponormal operators. Indeed, T hyponormal implies that (Re T )2 + (ImT )2 · jT j2 : Since (ImT )2 is a positive operator, (Re T )2 · jT j2 ; that is, T 2 (WN). In this note we will investigate the normality in (WN). A is said to be of class Y® for ® ¸ 1 if there exists a positive number k® such that ¤ ¤ ® 2 ¤ jAA ¡ A Aj · k®(A ¡ ¸) (A ¡ ¸) for all ¸ 2 C : It is known that Y® ½ Y¯ if 1 · ® · ¯. Let Y = [1·®Y®. We remark that a class Y1 operator A is M-hyponormal, i.e., there exists a positive number M such that (A ¡ ¸I)(A ¡ ¸I)¤ · M 2(A ¡ ¸I)¤(A ¡ ¸I) for all ¸ 2 C ; and M-hyponormal operators are class Y2 (see[13]). w 0 In [15] Weber shows that every compact operator in R(±A) \ fAg is 0 quasinilpotent, where fAg denotes the commutant of A, ±A stands for the derivation operator generated by A, i.e., ±A(X) = AX ¡ XA, R(±A) is the w range (image) of ±A and R(±A) is its weak closure. In [7] we showed that We- ber's result is a consequence of a more general result. A reasonable conjecture is the following: w ¤ 0 (S) Let T be a compact operator. If T 2 R(±A) \ fA g , then T is quasinilpotent, where fA¤g0 is the commutant of the adjoint of A. In [7], [8] it is proved that if A is normal, subnormal, dominant or m- hyponormal, then the conjecture holds. In this paper we shall see that the conjecture holds for an interesting class of operators which includes hyponor- mal operators. It is clear that the class (Y) contains the class of normal oper- ators. In the following we will denote the spectrum, the point spectrum, the approximate reduced spectrum, and the approximate spectrum of an operator A 2 B(H) by σ(A), σp(A), σar(A), and σa(A) respectively (see [7]). In this paper we will investigate the normality in (Y). ON THE NORMALITY OF OPERATORS 89 2. Preliminaries Let us list some spectral properties of operators in (WN). Proposition 2.1. [3] Suppose that T 2 B(H) is in (WN). Then we have (i) If ¸ is a real number, then T ¡ ¸ is also in (WN). (ii) k(T ¡ ¸)¤xk · 3 k(T ¡ ¸)xk ; for all x 2 H and all real number ¸: (iii) If M is an invariant supspace of T , then T jM is also in (WN); if furthermore, T jM is hermitian, then M reduces T . (iv) If ¸ is real eigenvalue of T , then the eigenspace ker(T ¡ ¸) reduces T: (v) If ¸ is real and (T ¡ ¸)nx = 0 for some n ¸ 1, then (T ¡ ¸)x = 0: Remark 2.1. As a consequence of the previous proposition, if (¸n)n2N is a countable sequence of real eigenvalues of T in (WN), then H = M © M ? and T jM is hermitian where M = ©n¸0 ker(T ¡ ¸nI): 3. Normality in (Y)-classes Let us begin by the following Berberian techniques [2]: Let H be a complex Hilbert space. Then there exists an Hilbert space H± ⊃ H, and an isometric *{homomorphism ' : B(H) 7! B(H±)(A 7! A±) preserving order, i.e., for all A; B 2 B(H) and for all ®; ¯ 2 C we have: (1) '(A¤) = '(A)¤; (2) '(®A + ¯B) = ®'(A) + ¯'(B); (3) '(IH) = IH± ; (4) '(AB) = '(A)'(B); (5) k '(A) k=k A k; (6) '(A) · '(B) if A · B; (7) σ('(A)) = σ(A); σa(A) = σa('(A)) = σp('(A)); (8) if A is a positive operator, then '(A®) = j '(A) j® for all ® > 0. Lemma 3.1. If S 2 (Y), then '(S) 2 (Y): Proof. If S 2 (Y), then there exists ® ¸ 1 and k® > 0 such that ¤ ¤ ® 2 ¤ j TT ¡ T T j · k®(T ¡ ¸I) (T ¡ ¸I); for all ¸ 2 C: It follows from the properties of the map ' that ¤ ¤ ® 2 ¤ '(j TT ¡ T T j ) · '(k®(T ¡ ¸I) (T ¡ ¸I)); for all ¸ 2 C: By the condition (8) above we have '(j TT ¤ ¡ T ¤T j®) = j '(TT ¤ ¡ T ¤T ) j® ; for all ® > 0. Therefore ¤ ¤ ® 2 ¤ j'(T )'(T ) ¡ '(T )'(T )j · '(k®(T ¡ ¸I) (T ¡ ¸I)); for all ¸ 2 C: Hence '(T ) 2 (Y). ¤X 90 SALAH MECHERI Now we will present some spectral properties of the class (Y). Theorem 3.1. Let S 2 (Y). ¤ (i) If ¸ 2 σp(S); then ¸ 2 σp(S ); furthermore if ¸ 6= ¹, then M¸ (the proper subspace associated with ¸) is orthogonal to M¹; ¤ (ii) If ¸ 2 σa(S); then ¸ 2 σa(S ); (iii) SS¤ ¡ S¤S is not invertible; (iv) If M is an invariant subspace for S and S jM is normal, then M reduces S; (v) If there exists a reducing subspace M , then S jM 2 (Y). Proof. For (i) and (iv) see [13]. (ii) Let ¹ 2 σa(S) from the condition (7) above, we have σa(S) = σa('(S)) = σp('(S): Therefore ¹ 2 σp('(S)). By applying Lemma 3.1 and the above con- dition (i), we get ¤ ¤ ¹ 2 σp('(S) ) = σp('(S )): ¤ Hence ¹ 2 σa('(S )). (iii) Let S 2 (Y). Then there exists an integer n ¸ 1 and kn > 0 such that ¤ ¤ 2n¡1 2 k j SS ¡ S S j x k· kn k (S ¡ ¸I)x k for all x 2 H; and for all ¸ 2 C: It is known that σa(S) 6= Á. If ¸ 2 σa(S), then there exists a normed ¤ sequence (xn) in H such that k (S ¡ ¸I)xn k! 0. Then (SS ¡ ¤ ¤ ¤ S S)xn ! 0 and so, (SS ¡ S S) is not invertible. ¤X Let A denote a complex Banach Algebra with identity e. A state on A is a functional f 2 A¤ such that f(e) = 1 = kfk. For x 2 A let W0(x) = ff(x): f is a state on Ag be the numerical range of x [14]. W0(x) is a compact convex set containing coσ(x) ( the convex hull of the spectrum of x ) [1]. For the case A = B(H), if A 2 B(H) then W0(A) = W (A), where W (A) = f(Ah; h): h 2 H; khk = 1g is the special numerical range of A. An element a 2 A is ¯nite if 0 2 W0(ax ¡ xa) for each x in A; F(A) (or F) denotes the set of all ¯nite elements of A. It is known that F contains every normal, hyponormal and dominant operator (see [7], [14]). In [10] the author initiated the study of a more general class of ¯nite operators called generalized pair of ¯nite operators de¯ned by GF = f(A; B) 2 B(H) £ B(H): kAX ¡ XB ¡ Ik ¸ 1; 8X 2 B(H)g : ON THE NORMALITY OF OPERATORS 91 Now we will prove that the class (Y) of operators is included in the class of ¯nite operators.
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