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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, .A / D .A/. D { W 2 .A/ } / , (iii) if A is invertible, then A is invertible and .A / 1 D .A 1 / . Note: Here .A/ denotes the spectrum of A 2 L .H/ ; that is, the set of 2 C such that A 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) nd 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) nd 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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This note was uploaded on 02/01/2011 for the course MATH 171 taught by Professor R during the Fall '09 term at Stanford.
 Fall '09
 R
 Math

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