Decomposition of a scalar operator into a product of unitary operators with two points in spectrum

✍ Scribed by Sergio Albeverio; Slavik Rabanovich

• Publisher: Elsevier Science
• Year: 2010
• Tongue: English
• Weight: 220 KB
• Volume: 433
• Category: Article
• ISSN: 0024-3795
• DOI: 10.1016/j.laa.2010.04.039

✦ Synopsis

We consider products of unitary operators with at most two points in their spectra, 1 and e iα . We prove that the scalar operator e iγ I is a product of k such operators if α(1 + 1/(k -3)) γ α(k -1 -1/(k -3)) for k 5. Also we prove that for e iα / = -1, only a countable number of scalar operators can be decomposed in a product of four operators from the mentioned class. As a corollary we show that every unitary operator on an infinite-dimensional space is a product of finitely many such operators.