Talk:Unitary operator

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Unitary Operator

I don't agree with the last remark about the spectrum in the section "Properties". I think that :

• the property "isometry" implies only that the eigenvalues of U are on the unit circle, but does not say anything about the rest of the spectrum.
• The spectrum of a Unitary is indeed on the unit circle, but the the "isometry" property is not enough, one needs the "surjectivity".

As an example, consider the Hilbert space l^2, with basis e_0, e_1, and consider the (one-side) shift operator U, defined by U(e_n)=e_{n+1}

This is an isometry, but the spectrum is the full closed unit disc, and has actually no eigenvalue.

Nicolas

i think you're right. why don't you go ahead and change it. Mct mht 10:50, 17 May 2006 (UTC)

[edit] Merging

Rather than merging with "unitarity", I think there is a case for merging the "unitary transformation" page into the "unitary operator" page as the underlying concept is the same and the distinction, if there is one, can likely be explained with a sentence in a single article. As an engineer, my view is that the existing unitary operator page is easier to understand. I got into this topic because I came across the term "unitary transform" in a journal article and needed a definition.

Also, might I suggest geometric rotation as a useful example? --Markgforbes 01:17, 10 June 2006 (UTC)

[edit] unitary representations

it seems that some material on projective unitary representations was removed. it is relevant stuff and, as long as it is correct information, surely it can be fitted in the article somewhere. i suggest that the stuff be added back, by either the original contributor or someone else. Mct mht 04:12, 10 June 2006 (UTC)

Be my guest, if you have the knowledge and time to do a good job. I cut that stuff off because it was integrating poorly. Oleg Alexandrov (talk) 15:49, 10 June 2006 (UTC)

[edit] Definition and equivalence

I think there is no reason to assume that the operator is bounded. If it preserves the product, it must be bounded. Further, maybe it would be more clear to write that the operator is surjective as "it has dense range" (but it is the same in this case). Note, that under this definition, the shifting operator on l² is not unitarian.

the right shift T is an isometry but not unitary. T*T = I but TT* ≠ I. Mct mht 02:52, 26 July 2006 (UTC)

it was me who replaced surjective by dense range, since it is, in principle, a weaker requirement. an operator is invertible iff it's bounded below and has dense range. therefore an isometry is invertible, therefore unitary, iff it has dense range. Mct mht 03:03, 26 July 2006 (UTC)