Tamil Nadu Open University (TNOU) 2006 M.Phil Mathematics Spectral Theory - Question Paper
7. (a) State and prove the spectral theorem for compact self-adjoint operators.
(b) Let A be a self-adjoint operator on H, u e c ([mA, mA ]) and U(A) eBL(H). Prove that
| U (A) | = sup {| it (?) |: feS(A)}.
M.Phil. DEGREE EXAMINATION -JUNE 2006
SPECTRAL THEORY
Time : 3 hours Maximum marks : 75
Answer any FIVE questions.
Each question carries 15 marks.
1. (a) Let {Un : n = 1, 2, 3,--} be an orthonormal set in a Hilbert Space H. For a sequence (Kn) of scalars, prove that the following are equivalent.
(i) There exists xe H such that for n = l,2,3,- (x,Un ) = Kn.
(ii) ]T | Kn |2 < oo and
n=l
oo
(iii) Kn Un converges in H.
n=1
(b) Let {xn : n = 1, 2, 3 } be an orthogonal set
00
in H. Then prove that x converges in H if and only
n=l
if z
n=l
2. (a) State and prove unique Hahn-Banach extension theorem.
(b) If (xn) is a sequence in H such that ((xn,y)) converges for every y e H then prove that there is a unique xeH such that (xn) converges weakly to x in H.
3. (a) Let A be an operator on a Hilbert space H. Suppose there is an operator B on H such that (A(x), y) = {x, B(y)) for all x, y e H. Then prove that
A is bounded and B = A'.
(b) Let Ae BL(H). Then prove that
(i) A is unitary if and only if || A(x) || = || x || for all x H and A is onto.
(ii) A is normal if and only if | A(x) | = | A\x) | for all x e H .
4. (a) Prove that Ae BL (H) is invertible ii BL(H) iff A is bounded below and the range of A ii dense in H.
(b) Let K = C . If Ae BL (H), then prove that
(i) | A | < 2 Ra and
(ii) Ra2 < (RA)2 .
5. (a) Let K = C and A be a normal operator Prove that || A || = RA =rA.
(b) Let H be an n-dimensional Hilbert Space over K, and Ae BL (H). Prove that there is ac orthonormal basis for H consisting of eigen vectors of A iff A is normal in K - C and iff A is self-adjoint in K = R.
6. (a) Let A be a non-zero compact self-adjoint operator on a real or complex Hilbert space. || A || or
-1| A || is an eigen value of A. Prove that there are only a finite number of linearly independent eigen vectors corresponding to this eigen value.
(b) Prove that a compact self-adjoint operator on a Hilbert space is positive iff all of its eigen values are non-negative.
3 MPL-637
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