Math 5410 Preliminary Exam
Jan 2013
Name Signature
Do all 5 problems.
1. (a) Find the Greenís Function G(x;y )for operator Awhere
Ay =y00 +y
with y0(0) = y(1) = 0:
(b) Find the operator norm of the Greenís operator from L2to L2:
2. Let Tbe a bounded operator on a Hilbert space Hand f'n:n2Ngbe an orthonormal system of H:
(a) Show 'n*0weakly.
(b) Using part (a) or otherwise, show that if Tis compact, then kT'
nk!0:
(c) If P1
n=1 kT'
nk2<1;then Tis compact.
3. Find k2 1
4r ekrin the sense of distributional derivatives. Here r=px2+y2+z2:
4. Let f(x)= 1
1+x2,fn(x)=nf (nx);8x2R; n =1;2:
(a) Give the deÖnition of the limit of a sequence of distributions on C1
0(R):
(b) Find the limit of ffng1
1as a sequence of distributions:
5. Let Hbe a Hilbert space and Tbe a bounded linear operator on H:
(a) Give the deÖnition of the adjoint operator T:Show that Tis well deÖned and bounded.
(b) Given H=L2([0;1]) ;Önd an example of self-adjoint operator on H:
(c) Show that eigenvalues of self-adjoint operators must be real.
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