Math 5410 Preliminary Exam
Jan 2012
Name Signature
Do all 5 problems.
1. (a) Find the Green’s Function G(x;y )for operator Awhere
Ay =y00 +y
with y0(0) = y(1) = 0:
(b) De…ne T:L2(0;1) !L2(0;1) such that for any f2L2(0;1)
T f (x) = Z1
0
G(x; y)f(y)dy:
Explain what spectral theorem is and why it is applicable.
(c) Show that kTk= max fjj:is an eigenvalue of Tg.
(d) Compute kTk:
2. Let Tbe a compact operator on a Hilbert space Hand f'n:n2Ngbe an orthonormal system of H:
(a) Show 'n*0weakly. Explain why this gives an example of weakly convergent sequence which is
not strongly convergent.
(b) Using part (a). or otherwise, show kT'nk ! 0
3. (a) Let nbe a sequence of complex numbers. Then operator Sde…ned by Sf =P1
n=1 nhf; 'ni'n
is compact i¤ limn!1 n= 0:
(a) Let fbe an operator on a Banach space X; give the de…nition of fbeing Fréchet di¤erentiable
at a point x2X:
(b) De…ne f:C[0;1] ! C[0;1] by [f(x)](t) = x(t) + R1
0(x(st))2ds: Compute f0(x):
4. Let f(x) = ex2,fn(x) = nf (nx);8x2R; n = 1;2 :
(a) Given the de…nition of the limit of a sequence of distributions in R:
(b) Find the limit of ffng1
1as a sequence of distributions. You may use the fact that R1
1 ex2dx =
p:
5. (a) Give the de…nition of a compact linear operator from a Banach space Xto itself.
(b) Given X=L2([0;1]) ;…nd an example of compact linear opeator on Xand explain why
(c) Given X=L2([0;1]) ;…nd an example of NON compact linear opeator on Xand explain why.
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