On normal derivations

Anderson, Joel

doi:10.1090/s0002-9939-1973-0312313-6

Cited by 93 publications

(52 citation statements)

References 6 publications

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“…For any operator A in B(H) set, as usual, |A| = (A * A) 1 2 and [A * , A] = A * A − AA * = |A| 2 − |A * | 2 (the self-commutator of A), and consider the following standard definitions: A is hyponormal if |A * | 2 ≤ |A| 2 (i.e., if [A * , A] is nonnegative or, equivalently, if A * x ≤ Ax for every x in H), p-hyponormal (for some 0 < p ≤ 1) if |A * | 2p ≤ |A| 2p , quasihyponormal if 0 ≤ A * [A * , A]A, and paranormal if Ax 2 ≤ A 2 x x for every x in H. Let U denote the class of 142 Duggal, Jeon and Kubrusly IEOT operators A satisfying the absolute value condition |A| 2 ≤ |A 2 |, and let H (1), H(p), Q (1) and K denote, respectively, the classes consisting of hyponormal, phyponormal, quasihyponormal and paranormal operators. Then…”

Section: Introductionmentioning

confidence: 99%

Contractions Satisfying the Absolute Value Property $$ |A|^2 \leq |A^2| $$

Duggal

Kubrusly

2004

Integral Equations and Operator Theory

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Let B(H) denote the algebra of operators on a complex Hilbert space H, and let U denote the class of operators A ∈ B(H) which satisfy the absolute value condition |A| 2 ≤ |A 2 |. It is proved that if A ∈ U is a contraction, then either A has a nontrivial invariant subspace or A is a proper contraction and the nonnegative operator D = |A 2 |−|A| 2 is strongly stable. A Putnam-Fuglede type commutativity theorem is proved for contractions A in U , and it is shown that if normal subspaces of A ∈ U are reducing, then every compact operator in the intersection of the weak closure of the range of the derivation δ A (X) = AX − XA with the commutant of A * is quasinilpotent.

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Section: Introductionmentioning

confidence: 99%

Contractions Satisfying the Absolute Value Property $$ |A|^2 \leq |A^2| $$

Duggal

Kubrusly

2004

Integral Equations and Operator Theory

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“…(1) The assumption of m-torsion freeness on R in Corollary 3.5 is essential. For instance, let F be a field of characteristic p > 0, and let R def.…”

Section: Proposition 34 a Derivation δ Of A Semiprime Ring R Is X-imentioning

confidence: 99%

An Identity With Generalized Derivations

Lee

Zhou

2009

J. Algebra Appl.

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Communicated by M. FerreroLet R be a prime ring that is not commutative and such that R ∼ = M 2 (GF(2)), let D, G be two generalized derivations of R, and let m, n be two fixed positive integers. Then D(x m )x n = x n G(x m ) for all x ∈ R iff the following two conditions hold: (1) There exists w ∈ Q, the symmetric Martindale quotient ring of R, such that D(x) = xw and G(x) = wx for all x ∈ R; (2) either w ∈ C, or x m and x n are C-dependent for all x ∈ R. We also consider the situation for the semiprime case. . Downloaded from www.worldscientific.com by PENNSYLVANIA STATE UNIVERSITY on 03/15/15. For personal use only.= bx − xb is a derivation of R, which is called the inner derivation induced by the element b. In [4] Brešar introduced the algebraic definition of generalized derivations. A map G : R → R is called a generalized derivation if there exists a derivation δ : R → R such thatfor all x, y ∈ R. The derivation δ is uniquely determined by the generalized derivation G. We call δ the derivation associated with G. For a, b ∈ R, the map x ∈ R → ax + xb defines a generalized derivation, which is called an inner generalized derivation. It is well-known that each derivation (resp. generalized derivation) of R can be uniquely extended to a derivation (resp. generalized derivation) of U . A derivation (resp. generalized derivation) of R is called X-inner if its extension to U is inner. Otherwise, it is called X-outer. Inner generalized derivations have been extensively studied in the theory of operator algebras (see [1, 4, 2, 22, etc.]). The study of generalized derivations from the viewpoint of pure algebra was given in [5, 13, 18, etc.]. In this paper we will prove the following Theorem 1.1. Let R be a prime ring that is not commutative and such that R ∼ = M 2 (GF(2)), let D, G be two generalized derivations of R, and let m, n be two fixed positive integers. Then D(x m )x n = x n G(x m ) for all x ∈ R iff the following two conditions hold: (1) There exists w ∈ Q such that D(x) = xw and G(x) = wx for all x ∈ R; (2) either w ∈ C, or x m and x n are C-dependent for all x ∈ R. Remarks.(1) We notice that the exceptional case in Theorem 1.1 indeed occurs. For instance, let w = 0 1 1 1 ∈ M 2 (GF(2)). Then x 3 wx = xwx 3 for all x ∈ M 2 (GF(2)), but w 3 and w are independent over GF(2).(2) We must characterize prime rings R satisfying the property (*): There exist distinct positive integers n, m such that x m and x n are C-dependent for all x ∈ R. A complete description is given as follows:Let R be a prime ring, not commutative, with extended centroid C. Then ( * ) holds iff C is a finite field and R ∼ = M s (C) for some integer s > 1.Proof. "⇒" Since (*) holds, x n yx m = x m yx n for all x, y ∈ R. Thus R is a prime PI-ring (as n = m). By Posner's Theorem [14, p. 57], Z(R), the center of R, is nonzero, and RC is a finite-dimensional central simple C-algebra, where C is equal to the quotient field of Z(R). Clearly, x m and x n are C-dependent for all x ∈ RC. The Wedderburn-Artin Theorem asserts that RC ∼ = M s (D), where...

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“…is closed under taking adjoints (ii) (A, B) ∈ GF 0 H (iii) T + ∆ A,B (X) 1 T 1 , for all T ∈ ker ∆ A,B | C1 and for all X ∈ C 1 .…”

Section: Theorem 5 ([5]) Letmentioning

confidence: 99%