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→Minimal normal extension: +def of minimal extension |
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Thus one is interested in the normal extension that is, in some sense, smallest. More precisely, a normal operator ''B'' acting on a Hilbert space ''K'' is said to be a '''minimal extension''' of a subnormal ''A'' if '' K' '' ⊂ ''K'' is a reducing subspace of ''B'' and ''H'' ⊂ '' K' '', then ''K' '' = ''K''. (A subspace is a reducing subspace of ''B'' if it is invariant under both ''B'' and ''B*''.)
One can show that if two operators ''B''<sub>1</sub> and ''B''<sub>2</sub> are minimal extensions on ''K''<sub>1</sub> and ''K''<sub>2</sub>, respectively, then there exists an unitary operator
:<math>U: K_1 \rightarrow K_2.</math>
Also, the following interwining relationship holds:
:<math>U B_1 = B_2 U.\,</math>
[[Category:Operator theory]]
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