Let H be a separable Hilbert space over the complex field. The class
S := {N|M : N is normal on H and M is an invariant subspace for Ng of subnormal operators. This notion was introduced by Halmos. The minimal normal extension Ň of a subnormal operator S was introduced by
σ (S) and then Bram proved that
Halmos. Halmos proved that σ(Ň)
(S) is obtained by filling certain number of holes in the spectrum (Ň) of the minimal normal extension Ň of a subnormal operator S.
Let σ (S) := σ (Ň) be the spectrum of the minimal normal extension Ň of S; which is called the normal spectrum of a subnormal operator S: This notion is due to Abrahamse and Douglas. We give several well-known characterization of subnormality. Let C* (S1) and C* (S2) be the C*- algebras generated by S1 and S2 respectively, where S1 and S2 are bounded operators on H:
Next we give a characterization for subnormality which is purely C - algebraic. We also establish an intrinsic characterization of the normal spectrum for a subnormal operator, which enables us to answer the fol-lowing two questions.
Let II be a *- representation from C* (S1) onto C* (S2) such that II(S1) = S2.
If S1 is subnormal, then does it follow that S2 is subnormal? What is the relation between σ (S1) and σ (S2)?
The first question was asked by Bram and second was asked by Abrahamse and Douglas. Answers to these questions were given by Bunce and Deddens.
| Identifer | oai:union.ndltd.org:IISc/oai:etd.ncsi.iisc.ernet.in:2005/3289 |
| Date | January 2013 |
| Creators | Kumar, Sumit |
| Contributors | Misra, Gadadhar |
| Source Sets | India Institute of Science |
| Language | en_US |
| Detected Language | English |
| Type | Thesis |
| Relation | G25645 |
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