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| | In [[mathematics]], especially [[operator theory]], a '''convexoid operator''' is a [[bounded operator|bounded]] [[linear operator]] ''T'' on a complex [[Hilbert space]] ''H'' such that the closure of the [[numerical range]] coincides with the [[convex hull]] of its spectrum. |
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| | An example of such an operator is a [[normal operator]] (or some of its generalization).<!-- It is not known whether a paranormal operator is a convexoid or not, I think. -- Taku. --> |
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| | A closely related operator is a [[spectraloid operator]]: an operator whose [[spectral radius]] coincides with its [[numerical radius]]. In fact, an operator ''T'' is convexoid if and only if <math>T - \lambda</math> is spectraloid for every complex number <math>\lambda</math>.<!-- This result is due to Furuta, I believe -- Taku --> |
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| | == See also == |
| | *[[Aluthge transform]] |
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| | == References == |
| | *T. Furuta. ''[http://www.projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.pja/1195526397 Certain convexoid operators]'' |
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| | [[Category:Operator theory]] |
| | {{mathanalysis-stub}} |
Latest revision as of 11:59, 15 May 2013
In mathematics, especially operator theory, a convexoid operator is a bounded linear operator T on a complex Hilbert space H such that the closure of the numerical range coincides with the convex hull of its spectrum.
An example of such an operator is a normal operator (or some of its generalization).
A closely related operator is a spectraloid operator: an operator whose spectral radius coincides with its numerical radius. In fact, an operator T is convexoid if and only if is spectraloid for every complex number .
See also
References
Template:Mathanalysis-stub