On the essential numerical range of a generalized derivation

Magajna, Bojan

doi:10.1090/s0002-9939-1987-0866435-1

Cited by 3 publications

(5 citation statements)

References 19 publications

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“…One may see [2,3,6,7,13,18,20,21,26,27,30,32,36,37] for many interesting results on W e (A) and W e (A).…”

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confidence: 99%

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The joint essential numerical range of operators: convexity and related results

Poon

2009

Studia Math.

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Abstract. Let W (A) and We(A) be the joint numerical range and the joint essential numerical range of an m-tuple of self-adjoint operators A = (A1, . . . , Am) acting on an infinite-dimensional Hilbert space. It is shown that We(A) is always convex and admits many equivalent formulations. In particular, for any fixed i ∈ {1, . . . , m}, We(A) can be obtained as the intersection of all sets of the formwhere F = F * has finite rank. Moreover, the closure cl(W (A)) of W (A) is always starshaped with the elements in We(A) as star centers. Although cl(W (A)) is usually not convex, an analog of the separation theorem is obtained, namely, for any element d / ∈ cl(W (A)), there is a linear functional f such that f (d) > sup{f (a) : a ∈ cl(W (Ã))}, whereÃ is obtained from A by perturbing one of the components Ai by a finite rank self-adjoint operator. Other results on W (A) and We(A) extending those on a single operator are obtained.

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“…One may see [2,3,6,7,13,18,20,21,26,27,30,32,36,37] for many interesting results on W e (A) and W e (A).…”

mentioning

confidence: 99%

“…This might be the reason why the convexity of W e (A) is seldom discussed for m > 3. In fact, some researchers have studied different geometrical properties of W e (A) under the assumption that W e (A) is convex, and some have examined W e (A) for different classes of operators without discussing their convexity; for example, see [6,26,27,30,32].…”

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confidence: 99%

The joint essential numerical range of operators: convexity and related results

Poon

2009

Studia Math.

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“…The famous results on the norms of inner derivation and the generalized derivation as obtained by Stampfli [6] using maximal numerical range have ever since provided a crucial lead in defining of norms of elementary operators.We recall the works of Kyle [9] who examines the relationship between the numerical range of an inner derivation, and that of its implementing element. In his paper, Magajna [2] gives the essential numerical range of the the generalized derivation defined on the Hilbert-Schmidt class in terms of the numerical and the essential numerical ranges of the implementing operators. Shaw [10] in particular, established that the algebra numerical range of a generalized derivation restricted to a norm ideal J is equal to the difference of the algebra numerical ranges of the implementing operators provided that J contains all finite rank operators and is suitably normed .…”

Section: Norm Idealsmentioning

confidence: 99%

On the numerical range of a generalized derivation

Runji¹,

Agure²,

Nyamwala³

2017

imf

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“…The essential numerical range of the restriction of a generalized derivation to the class of Hilbert-Schmidt has been computed in [7], by B. Magajna. He has shown that…”

Section: Theorem 22 Let H K Be Two Separable Hilbert Spaces Andmentioning

confidence: 99%

“…Up to now, their spectra and their essential spectra have been characterized; see [4,5,3]. In [7], B. Magajna has determined the essential numerical range of the restriction of a generalized derivation to the class of Hilbert-Schmidt.…”

Section: Introduction Let L(h)mentioning

confidence: 99%