Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-24T10:52:43.326Z Has data issue: false hasContentIssue false
  • English
  • Français

On numerical ranges of generalized derivations and related properties

Part of: General theory of linear operators Special classes of linear operators

Published online by Cambridge University Press:  09 April 2009

Sen-Yen Shaw
Sen-Yen Shaw
Affiliation:
Department of Mathematics National Central University Chung-Li, Taiwan 320 Republic of, China
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper is concerned with the numerical range and some related properties of the operator Δ/ S: T → AT – TB(T∈S), where A, B are (bounded linear) operators on the normed linear spaces X and Y. respectively, and S is a linear subspace of the space ℒ (Y, X) of all operators from Y to X. S is assumed to contain all finite operators, to be invariant under Δ, and to be suitably normed (not necessarily with the operator norm). Then the algebra numerical range of Δ/ S is equal to the difference of the algebra numerical ranges of A and B. When X = Y and S = ℒ (X), Δ is Hermitian (resp. normal) in ℒ (ℒ(X)) if and only if A–λ and B–λ are Hermitian (resp. normal) in ℒ(X)for some scalar λ;if X: = H is a Hilbert space and if S is a C *-algebra or a minimal norm ideal in ℒ(H)then any Hermitian (resp. normal) operator in S is of the form Δ/ S for some Hermitian (resp. normal) operators A and B. AT = TB implies A*T = TB* are hyponormal operators on the Hilbert spaces H1 and H2, respectively, and T is a Hilbert-Schmidt operator from H2 to H1.

Keywords

algebra numerical rangederivationHermitian and normal operatorshyponormal operatornormed operator idealSchatten p-classC*-algebraHlibert-Schmidt operator.

MSC classification

Secondary: 47A12: Numerical range, numerical radius 47B15: Hermitian and normal operators (spectral measures, functional calculus, etc.) 47B47: Commutators, derivations, elementary operators, etc.
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 36 , Issue 1 , February 1984 , pp. 134 - 142
Copyright
Copyright © Australian Mathematical Society 1984

References

[1]Anderson, J. and Foias, C., ‘Properties which normal operators share with normal derivations and related operators’, Pacific J. Math. 61 (1975), 313–325.CrossRefGoogle Scholar
[2]Aubin, J. P., Applied functional analysis (Wiley-Interscience, New York, 1979).Google Scholar
[3]Berberian, S. K., ‘Extensions of a theorem of Fuglede and putnam’, Proc. Amer. Math. Soc. 71 (1978), 113–114.CrossRefGoogle Scholar
[4]Bonsall, F. F. and Duncan, J., Numerical ranges of operators on normal spaces and of elements of normed algebras (London Math. Soc. Lecture Note Series 2, Cambridge Univ. Press, 1971).CrossRefGoogle Scholar
[5]Fialkow, L., ‘A note on the operator X → AX – XB’, Trans. Amer. Math. Soc. 23 (1978), 147–168.Google Scholar
[6]Fong, C. K., ‘Normal operators on Banach spaces’, Glasgow Math. J. 20 (1979), 163–168.CrossRefGoogle Scholar
[7]Kadison, R. V., ‘Derivations of operator algebras’, Ann. of. Math. 83 (1966), 280–293.CrossRefGoogle Scholar
[8]Kyle, J., ‘Numerical ranges of derivations’, Proc. Edinburgh Math. Soc. 21 (1978), 33–39.CrossRefGoogle Scholar
[9]Pietsch, A., Operator ideals (North-Holland Mathematical library vol. 2, 1980).Google Scholar
[10]Rudin, W., Functional analysis (McGraw-Hill, New York, 1973).Google Scholar
[11]Sakai, S., ‘Derivations of W*-algebaras’, Ann. of Math. 83 (1966), 273–279.CrossRefGoogle Scholar
[12]Sinclair, A. M., ‘Jordan homomorphisms and derivations on semi-simple Banach algebras’, Proc. Amer. Math. Soc. 24 (1970), 209–214.Google Scholar
[13]Sourour, A. R., ‘Isometries of norm ideals of compact operators’, preprint.Google Scholar