Subnormal operator: Difference between revisions

Revision as of 21:39, 25 October 2021

Revision as of 18:31, 26 October 2021

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Given a subnormal operator ''A'', its normal extension ''B'' is not unique. For example, let ''A'' be the unilateral shift, on ''l''²('''N'''). One normal extension is the bilateral shift ''B'' on ''l''²('''Z''') defined by

:<math>B (\cdotsldots, a_{-1}, {\hat a_0}, a_1, \cdotsldots) = (\cdotsldots, {\hat a_{-1}}, a_0, a_1, \cdotsldots),</math>

where ˆ denotes the zero-th position. ''B'' can be expressed in terms of the operator matrix

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:<math>

B' (\cdotsldots, a_{-2}, a_{-1}, {\hat a_0}, a_1, a_2, \cdotsldots) = (\cdotsldots, - a_{-2}, {\hat a_{-1}}, a_0, a_1, a_2, \cdotsldots).

</math>