Unformatted text preview: j 2 C ), prove that ± j ! 0 as j ! 1 . 5. Suppose H;K;L are Hilbert spaces and A W H ! K , B W K ! L are bounded linear operators. Prove: (i) .BA/ ± D A ± B ± , (ii) if K D H D a complex Hilbert space, ².A ± / D ².A/. D { ± W ± 2 ².A/ } / , (iii) if A is invertible, then A ± is invertible and .A ± / ² 1 D .A ² 1 / ± . Note: Here ².A/ denotes the spectrum of A 2 L .H/ ; that is, the set of ± 2 C such that A ² ±I is not invertible. 6. If S W ` 2 ! ` 2 is the shift operator ( x D .x 1 ;x 2 ;:::/ 7! . ;x 1 ;x 2 ;:::/ ) (i) ﬁnd the adjoint S ± W ` 2 ! ` 2 , (ii) prove that the spectrum of S ± is the closed unit disk { z 2 C W j z j ± 1 } , and (iii) ﬁnd the spectrum of S . Hint for (ii): Start by showing that every ± 2 C with j ± j < 1 is an eigenvalue of S ± . 7, 8, 9: Problems 7.17, 7.18, 7.20...
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 Fall '09
 R
 Math, Linear Algebra, Vector Space, Hilbert space, 2 L, Q.1

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