Shift Operator in l2 Space
Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
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 (AI) 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(:::x1; 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.
Engineering and Technology
IdentifiersURN: urn:nbn:se:kth:diva-147599OAI: oai:DiVA.org:kth-147599DiVA: diva2:730944
Shimorin, Serguei, Univ.-lektor