Department of Mathematics

The equation AX = XB implies A * X = XB * when A and B are normal operators is known as the familiar Fuglede-Putnam theorem. In this paper, the hypothesis on A and B can be relaxed by using a Hilbert-Schmidt operator X: Let A be a (p, k)-quasihyponormal operator and B * be an invertible (p, k)-quasihyponormal operator such that AX = XB for a Hilbert Schmidt operators X, then A * X = XB * . As a consequence of this result, we obtain that the range of the generalized derivation induced by this class of operators is orthogonal to its kernel.

Let A be a bounded linear operator acting on a Hilbert space H. The B-Weyl spectrum of A is the set σBw(A) of all λ ∈ C such that A−λI is not a B-Fredholm operator of index 0. Let E(A) be the set of all isolated eigenvalues of A. Recently in [6] Berkani showed that if A is a hyponormal operator, then A satisfies generalized Weyl's theorem σ Bw (A) = σ(A) \ E(A), and the B-Weyl spectrum σ Bw (A) of A satisfies the spectral mapping theorem. In [51], H. Weyl proved that weyl's theorem holds for hermitian operators. Weyl's theorem has been extended from hermitian operators to hyponormal and Toeplitz operators [12], and to several classes of operators including semi-normal operators ([9], [10]). Recently W. Y. Lee [35] showed that Weyl's theorem holds for algebraically hyponormal operators. R. Curto and Y. M. Han [14] have extended Lee's results to algebraically paranormal operators. In [19] the authors showed that Weyl's theorem holds for algebraically p-hyponormal operators. As Berkani has shown in , if the generalized Weyl's theorem holds for A, then so does Weyl's theorem. In this paper all the above results are generalized by proving that generalized Weyl's theorem holds for the case where A is an algebraically (p, k)-quasihyponormal or an algebarically paranormal operator which includes all the above mentioned operators.

We say operators A, B on Hilbert space satisfy Fuglede-Putnam theorem if AX = XB for some X implies A * X = XB * . We show that if either (1) A is p-hyponormal and B * is a class Y operator or (2) A is a class Y operator and B * is p-hyponormal, then A, B satisfy Fuglede-Putnam theorem.

Let B(H) denote the algebra of all bounded linear operators on a separable infinite dimensional complex Hilbert space H into itself. Let A = (A1, A2, .., An) and B = (B1, B2, .., Bn) be n-tuples in B(H), we define the elementary operator EA,B : B(H) → B(H) by EA,B(X) = È n i=1 AiXBi.

Let H denote a complex Hilbert space, L(H ) the algebra of all bounded linear operators on H and C 1 (H), the trace class operators. We study the pairs of operators A, B /L (H) with the property that AT 0/TB and T /C 1 (H ) implies B *T 0/TA *. The main result is that this property is equivalent to the self-adjointness of the ultraweak closure of the range of a generalised derivation. *

In this Paper we obtain some sufficient and some necessary conditions that the identity be in the closure of the range of an inner derivation. We obtain some results concerning the intersection of the closure of the range of the inner derivation induced by .4 and the commutant of A. 0 1998 Published by Elsevier Science Inc. All rights reserved.

An operator T is said to be paranormal if ||T 2 x|| ≥ ||T x|| 2 holds for every unit vector x. Several extensions of paranormal operators are considered until now, for example absolute-k-paranormal and p-paranormal introduced in [10], , respectively. Yamazaki and Yanagida introduced the class of absolute-(p, r)-paranormal operators as a further generalization of the classes of both absolute-k-paranormal and p-paranormal operators. An operator T ∈ B(H) is called absolute-(p, r)-paranormal operator if |||T | p |T * | r x|| r ≥ |||T * | r x|| p+r for every unit vector x ∈ H and for positive real numbers p > 0 and r > 0.

JIPAM. Another Version of Anderson's Inequality in the Ideal of all Compact Operators, Authors: Mecheri Salah,, ... This article was printed from JIPAM http://jipam.vu.edu.au The URL for this article is: http://jipam.vu.edu.au/article.php?sid=563.

King Saud University College of Science Department of Mathematics POBox2455, Riyadh 11451 Saudi Arabia. EMail: [email protected] EMail: [email protected] ... Some Variants of Anderson's Inequality in C1−Classes ... J. Ineq. Pure and Appl. Math. 4(1) Art. 24, 2003 ...

Let H be a complex Hilbert space and L(H) the algebra of all bounded linear operators on H. For A, B ∈ L(H), let δA,B : L(H) ↦→ L(H) be the generalized derivation defined by δA,B(X) = AX −XB. In this paper we will present some gener-alized finite operators and we discuss new ...

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 Classification. Primary: 47A15. Secondary: 47B20, 47A63. ... Resumen. En este artıculo ...

We prove that this holds in the case where A and B satisfy the Fuglede– Putnam theorem. Finally, we apply the obtained results to double operator integrals. ... 2000 Mathematics Subject Classification: Primary: 47B47, 47B10, 47B21, 47B49. Secondary: 47A30, 47A13, 47A60. ...