ON THE RANGE AND THE KERNEL OF THE ELEMENTARY OPERATORS Pn i=1 AiXBi X (original) (raw)

ON THE RANGE AND THE KERNEL OF THE ELEMENTARY OPERATORS P n=1 AiXBi X

2003

Let B(H) denote the algebra of all bounded linear operators on a separable infinite dimensional complex Hilbert space H into itself. For A = (A 1 , A 2 ...An) and B = (B 1 , B 2 ...Bn) n-tuples in B(H), we define the elementary operator ∆ A,B X : B(H) → B(H) by ∆ A,B = A i XB i − X. In this paper we show that if ∆ A,B = 0 = ∆ * A,B , then T + ∆ A,B (X) I ≥ T I for all X ∈ I (proper bilateral ideal) and for all T ∈ ker(∆ A,B | I).

On the range of elementary operators

Integral Equations and Operator Theory, 2005

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.

On the range closure of an elementary operator

Linear Algebra and its Applications, 2005

Let B(H) denote the algebra of operators on a Hilbert H. Let AB ∈ B(B(H)) and E ∈ B(B(H)) denote the elementary operators AB (X) = AXB − X and E(X) = AXB − CXD. We answer two questions posed by Turnšek [Mh. Math. 132 (2001) 349-354] to prove that: (i) if A, B are contractions, then B(H) = −1 AB (0) ⊕ AB (B(H)) if and only if n AB (B(H)) is closed for some integer n 1; (ii) if A, B, C and D are normal operators such that A commutes with C and B commutes with D, then B(H) = E −1 (0) ⊕ E(B(H)) if and only if 0 ∈ iso σ (E).

On the Numerical Range and Norm of Elementary Operators

Linear and Multilinear Algebra, 2004

W 0 ðR A, B J j Þ where VðÁÞ is the joint spatial numerical range, W 0 ðÁÞ is the algebraic numerical range and J is a norm ideal of BðEÞ: We shall show that this inclusion becomes an equality when R A, B is taken to be a derivation. Also, we deduce that wðU A, B J j Þ ! 2ð ffiffi ffi 2 p À 1ÞwðAÞwðBÞ, for A, B 2 BðEÞ and J is a norm ideal of BðEÞ, where wðÁÞ is the numerical radius. On the other hand, in the particular case when E is a Hilbert space, we shall prove that the lower estimate bound kU A, B jJk ! 2ð ffiffi ffi 2 p À 1ÞkAkkBk holds, if one of the following two conditions is satisfied: (i) J is a standard operator algebra of BðEÞ and A, B 2 J: (ii) J is a norm ideal of BðEÞ and A, B 2 BðEÞ:

The numerical range of elementary operators II

Linear Algebra and its Applications, 2001

For A, B ∈ L(H) (the algebra of all bounded linear operators on the Hilbert space H), it is proved that: (i) the generalized derivation δ A,B is convexoid if and only if A and B are convexoid; (ii) the operators δ A,B and δ A,B |C p (where p 1) have the same numerical range and are equal to W 0 (A) − W 0 (B) (where C p is the Banach space of the p-Schatten class operators on H).

On the Maximal Numerical Range of Elementary Operators

International Journal of Pure and Apllied Mathematics

The notion of the numerical range has been generalized in different directions. One such direction, is the maximal numerical range introduced by Stampfli (1970) to derive an identity for the norm of a derivation on L(H). Unlike the other generalizations, the maximal numerical range has not been largely explored by researchers as many only refer to it in their quest to determine the norm of operators. In this paper we establish how the algebraic maximal numerical range of elementary operators is related to the closed convex hull of the maximal numerical range of the implementing operators A = (A1, A2, ..., An), B = (B1, B2, ..., Bn), on the algebra of bounded linear operators on a Hilbert space H. The results obtained are an extension of the work done by Seddik [2] and Fong [9].

On the Norm of Elementary Operators in a Standard Operator Algebras

2003

Let B(H) and A be a C∗−algebra of all bounded linear operators on a complex Hilbert space H and a complex normed algebra, respectively. For A,B ∈ A, define a basic elementary operator MA,B : A→ A by MA,B(X) = AXB. An elementary operator is a finite sum RA,B = n P i=1 MAi,Bi of the basic ones, where A = (A1, ..., An) and B = (B1, ..., Bn) are two n-tuples of elements of A. If A is a standard operator algebra of B(H), it is proved that: (i) [4] °°MA,B + MB,A°° ≥ 2(√2− 1) kAk kBk , for any A,B ∈ A (ii)[1 ] °°MA,B + MB,A°° ≥ kAk kBk , for A,B ∈ A, such that inf λ∈C kA + λBk = kAk or inf λ∈C kB + λAk = kBk , (iii)[3] °°MA,B + MB,A°° = 2 kAk kBk , if kA + λBk = kAk+kBk , for some unit scalar λ. In this note, we are interested in the general situation where A is a standard operator algebra acting on a normed space. We shall prove that °°RA,B°° ≥ sup f,g∈(A∗)1 ̄̄̄̄ n P i=1 f(Ai)g(Bi) ̄̄̄̄ , for any two n-tuples A = (A1, ..., An) and B = (B1, ...,Bn) of elements of A (where (A∗)1 is the unit...

Elementary Operators and New C∗- Algebras

International Journal of Open Problems in Computer Science and Mathematics, 2014

Let H be a complex Hilbert space and B(H) the algebra of all bounded linear operators on H. In this paper, we study the class of pairs of operators A, B ∈ B(H) that have the following property, AT B = T implies B * T A * = T for all T ∈ C 1 (H) (trace class operators). The main result is the equivalence between this character and the fact that the ultra-weak closure of the range of the elementary operator ∆ A,B defined on B(H) by ∆ A,B (X) = AXB − X is equivalent to the generalized quasiadjoint operators. Some new C * -algebras generated by a pair of operators A, B ∈ B(H) are also presented.

On an elementary operator withM-hyponormal operator entries

Mathematische Nachrichten, 2014

A Hilbert space operator T ∈ L (H) is M-hyponormal if there exists a positive real number M such that (T − μ)(T − μ) * ≤ M 2 (T − μ) * (T − μ) for all μ ∈ σ (T). Let A, B * ∈ L (H) be M-hyponormal and let d AB ∈ L (L (H)) denote either the generalized derivation δ AB (X) = AX − X B or the elementary operator AB = AX B − X. We prove that if A, B * are M-hyponormal, then f (d AB) satisfies the generalized Weyl's theorem and f (d * AB) satisfies the generalized a-Weyl's theorem for every f that is analytic on a neighborhood of σ (d AB).