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Shift Operator in l2 Space
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
2014 (English)Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
Abstract [en]

A Hilbert space H is the abstraction of a nite-dimensional Eu-clidean space. The spectrum of a bounded linear operator A : H !H , denoted (A), is given by all numbers 2 C such that (A􀀀I) isnot invertible. The shift operators are one type of bounded linear op-erators. In this report we prove ve claims regarding the spectrum ofthe shifts. We work in the Hilbert space `2 which consists of all squaresummable sequences, both single sided (x0; x1; x2; :::) and double sided(:::x􀀀1; x0; x1; :::). One of the most general results proved applies tothe weighted unilateral shift S dened for (x0; x1; x2; :::) 2 `2 byS(x0; x1; x2; :::) = (0; 0x0; 1x1; 2x2; :::)where fng is a bounded arbitrary weight sequence with n > 0 forall n 0.

Theorem. Let r(S) be the radius of the smallest disc which contain(S).

Then(S) = f : jj r(S)g:

Place, publisher, year, edition, pages
2014. , 26 p.
National Category
Engineering and Technology
URN: urn:nbn:se:kth:diva-147599OAI: diva2:730944
Available from: 2014-06-30 Created: 2014-06-30 Last updated: 2015-04-21Bibliographically approved

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