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Roll No. ........................
(c) Zero
(d) None of these
Total No. of Questions: 11]
[Total No. of Printed Pages: 09]
(iv)
If T : H → H and S : H → H are bounded linear operators and T is compact and S * S ≤ T * T then :
(a) T is compact
(b) S is compact
(c) Neither (a) nor (b) true
(d) None of these
OP-504
M.Sc. (Reg./Pvt./ATKT)
Examination, 2024
(Fourth Semester)
MATHEMATICS
Theory of Linear Operators-II
Paper-V
Time : 3 Hours]
[Maximum Marks : 85/100
(v)
Let (Pn) be a monotone increasing sequence of projections Pn defined on a Hilbert space H, then P projects H onto :
(a)
$$ \bigcap_{n=1}^\infty N(P_n) $$
(b)
$$ \bigcup_{n=1}^\infty N(P_n) $$
(c)
$$ \bigcap_{n=1}^\infty P_n(H) $$
(d)
$$ \bigcup_{n=1}^\infty P_n(H) $$
Note : Attempt the questions all Sections as directed.
Section A
(Objective Type Questions)
3×5=15
Note : Attempt all questions. All questions carry equal marks.
Section B
(Short Answer Type Questions)
5×5=25
Note : Attempt all questions. All questions carry equal marks.
1.
Choose the correct answer :
(i)
T : β → β defined by Tx =
$$ \left( \frac{x_1}{1}, \frac{x_2}{2}, \frac{x_3}{3}, \dots \right) $$
is :
(a) compact
(b) non-compact
(c) either (a) or (b)
(d) neither (a) nor (b)
(ii)
If T : X → X is a compact linear operator on a normed space X and let λ ≠ 0. The operator equations Tx - λx = 0 and T*f - λ*f = 0 have :
(a) different linearly independent solutions
(b) different linearly dependent solutions
(c) some number of linearly independent solutions
(d) some number of linearly dependent solutions
(iii)
For a bounded self-adjoint linear operator on a complex Hilbert space eigenvectors corresponding to different eigenvalues are :
(a) Orthogonal
(b) Same
2.
If T : X → X is a compact linear operator on a normed space X and let λ = 0.. Then prove that there exists a smallest integer q (depending on λ) such that from r = q on, the ranges Tλr (X) are all equal and if q > 0, the inclusions
$$ T_{\lambda}^{q}(X) \supset T_{\lambda}^{q+1}(X) \dots \supset T_{\lambda}^{r}(X) $$
are all proper.
Or
If T : X → X is a compact linear operator on a normed space X and let λ ≠ 0, r > 0, then show that X can be represented in the form X = N(Tλr) ⊕ Tλr(X).
3.
Let T : X → X be a compact linear operator on a normed space X, prove that if T has non-zero spectral values, every one of them must be an eigenvalue of T.
4.
Explain the following :
(a) Compact integral operator
(b) Neumann series.
Or
Prove that the residual spectrum σr(T) of a bounded self-adjoint linear operator T on a complex Hilbert space H is empty.
Or
Explain the following :
(a) Hilbert adjoint operator
(b) Hermitian operator.
5.
Write a short note on positive operator.
Or
Write a short note on square roots of a positive operator.
6.
Prove that for any Projection P on a Hilbert space H.
(a)
$$ \langle Px, x \rangle = ||Px||^2 $$
(b)
$$ P \ge 0 $$
(c)
$$ ||P|| \le 1; ||P|| = 1 \text{ if } P(H) \ne \{0\} $$
Or
If (Pn) is a monotone increasing sequence of projections Pn defined on a Hilbert space H. Then prove that (Pn) is strongly operator convergent and the limit operator P is a projection defined on H.
8.
Prove that for a given linearly independent set {f1,...,fm} in the dual space X' of a normed space X, there are elements z1,...,zm in X such that :
$$ f_j(z_k) = \delta_{jk} = \begin{cases} 0 & (j \neq k) \\ 1 & (j = k) \end{cases} $$
where j, k = 1, ..., m.
Or
If T : X → X is a compact linear operator on a normed space X and let λ ≠ 0. Then prove that equation Tx - λx = y has a solution x for every y ∈ X if and only if the homogeneous equation Tx - λx = 0 has only the trivial solution x = 0. In this case the solution of Tx - λx = y is unique and Tλ has a bounded inverse.
Section C
(Long Answer Type Questions)
5×9=45
Note : Attempt all questions. All question carry equal marks.
7.
If T : X → X is a compact linear operator on a normed space X and let λ ≠ 0. Then prove that Tf - λf = g (g ∈ X') given has a solution f if and only if g is such that g(x) = 0 for all x ∈ X which satisfy Tx - λx = 0. Hence if Tf - λf = 0 has only the trivial solution x = 0, then Tf - λf = g with any given g ∈ X is solvable.
11.
Prove that for any projections on a Hilbert space H the following two statements holds :
(a) P = P1P2 is a projection on H if and only if the projections P1 and P2 commute. Then P projects H onto Y = Y1 ∩ Y2 where Yj = Pj (H), for j = 1, 2.
(b) Two closed subspaces Y and V of H are orthogonal if and only if the corresponding projections satisfy PY PV = 0.
Or
Let P1 and P2 be projections on a Hilbert space H. Then prove that :
(a) The sum P = P1 + P2 is a projection on H if and only if Y1 = P1 (H) and Y2 = P2 (H) are orthogonal.
(b) If P = P1 + P2 is a projection, P projects H onto Y = Y1 ⊕ Y2.
9.
Prove that for any bounded self-adjoint linear operator T on a complex Hilbert-space M we have
$$ ||T|| = \max_{||x||=1} | \langle Tx, x \rangle | $$
$$ | \langle m, m \rangle | = \sup_{||x||=1} | \langle Tx, x \rangle | $$
Or
If H is a complex Hilbert space and T : H → H is a bounded self adjoint linear operator and H = {0}.
Then prove that
$$ m = \inf_{||x||=1} \langle Tx, x \rangle $$
and $$ \sup_{||x||=1} \langle Tx, x \rangle $$
are spectral values of T.
10.
Let (Tn) be a sequence of bounded self adjoint linear operators on a Complex Hilbert space H. Such that T1 ≤ T2 ≤ ... ≤ Tn ≤ ... ≤ K where K is a bounded self adjoint linear operator on H. Suppose that any Tj commutes with K and with every Tm. Then prove that (Tn) is strongly operator convergent and the limit operator is linear, bounded and self adjoint and satisfy T ≤ K.
Or
Prove that every positive bounded self adjoint linear operator T : H → H on a complex Hilbert space M has a positive square root A, which is unique.