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hw6 - j 2 C prove that ± j 0 as j 1 5 Suppose H;K;L are...

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Mathematics Department Stanford University Math 175 Homework 6 1. Suppose X is a normed space and T W X ! X is a linear operator such that T. { x 2 X W k x k ± 1 } / is contained in a compact subset of X , and let I W X ! X be the identity map on X . Prove that the null space N of I C T (i.e. { x 2 X W x C T.x/ D 0 } ) is finite dimensional. Hint: Prove that the closed unit ball in the null space is compact. 2. If X is a normed space and S is a finite dimensional subspace, prove that each point x 2 X has at least one closest point e x 2 S ; that is 9 e x 2 S such that k x ² e x k D min z 2 S k x ² z k . 3. If T , N are as in Q.1 and if e x is a closest point in N to x (as in Q.2 with S D N ), prove: (i) There is a fixed constant C such that k x ² e x k ± C k .I C T /.x/ k for each x 2 X . Hint: Otherwise this fails with C D k , k D 1 ; 2 ;::: , hence 9 a sequence x k with k .I C T /.x k / k < 1 k k x k ² e x k k . (ii) .I C T /.X/ is a closed linear subspace of X . Hint: Consider x k with x k C T.x k / ! y 2 X , and use (i). 4. Let T W H ! H be a linear operator on the Hilbert space H and suppose T. { x 2 H W k x k ± 1 } / is contained in a compact subset of H . If e 1 ;e 2 ;::: is an orthonormal sequence in H and T.e j / D ± j e j for each j D 1 ; 2 ;::: (where ±
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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, 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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