DEPARTMENT OF MATHEMATICS University of Utah Ph.D. PRELIMINARY EXAMINATION IN ANALYSIS August 2014

Consider \(L^2(\mathbb Z):=H\). Recall two topologies on the set of bounded linear operators on \(H\): The norm topology which is metric topology given by \[ d(A,B)=\sup _{v\in H:\lVert v\rVert =1}\lVert Av-Bv\rVert . \] The strong operator topology which is given by \(A_i\to A_\infty \) if for every \(v\in H\) we have \[ \lim _{i\to \infty }\lVert A_iv-A_\infty v\rVert =0. \]

(a) Show that if \(A_i\) converges to \(A_\infty \) in the norm topology then it converges to \(A_\infty \) in the strong operator topology.

(b) Give an example of \(A_i\) which converges to \(A_\infty \) in the strong operator topology but not the norm topology.

(c) Let \(A_i\) be the operator defined by \(A_iv=w\) where \(w_j=v_j\) if \(j\ne i\) and \(w_i=iv_i\). Show \(A_1,\ldots \) is a sequence of operators that does not converge in either topology.

Let \((X,\mathcal B,\mu )\) be a measure space. \(U\in \mathcal B\) is called an atom if \(\mu (U)>0\) and \(\mu (V)=0\) for all \(V\), a proper subset of \(U\). If \((X,\mathcal B,\mu )\) is \(\sigma \)-finite measure space and \(f:X\to Y\) then \(f_*(\mu )\) has at most countably many atoms.

Let \(f_i:[0,1]\to \mathbb R\) be a sequence of functions that converge pointwise to \(f_\infty \).

(a) Show that if the \(f_i\) are uniformly bounded then \[ \lim _{i\to \infty }\int f_i\,d\lambda =\int f_\infty \,d\lambda . \]

(b) What if \(\lVert f_i\rVert _1\le 1\) for all \(i\)?

(c) What if \(\lVert f_i\rVert _2\le 1\) for all \(i\)?

Let \(f\in L^1(\lambda ,[0,1])\) have the property that for any measurable set \(A\) with \(\lambda (A)=\frac {1}{\pi }\) we have \(\int _A f\,d\lambda =0\). Show that \(f=0\) almost everywhere.

Let \(A\subset \mathbb R\) and \(A+A\) denote \(\{a_1+a_2:\ a_1,a_2\in A\}\). Show that if \(\lambda (A^c)=0\) then \(A+A=\mathbb R\).

Let \(f\) be a nonzero entire function. Assume that the set \(Z(f)=\{z\in \mathbb C:\ f(z)=0\}\) is unbounded. Show that \(f\) had an essential singularity at \(\infty \).

Let \(f\) be an a holomorphic function defined on an open set \(U\subset \mathbb C\) and \(a\in U\). Assume that \(f'(a)\) is nonzero. Show that for sufficiently small circles \(C\) centered at \(a\) \[ \frac {2\pi i}{f'(a)}=\int _C \frac {dz}{f(z)-f(a)} \]

Let \(f\) be an entire function and \(n\) a positive integer. Show that there exists an entire function \(g\) such that \(f=g^n\) if and only if the order of each zero of \(f\) is divisible by \(n\).

Show that all the zeroes of \(f(z)=z^4+6z+3\) lie inside the circle \(|z|=2\).

Evaluate \[ \int _C \frac {1}{(z+i)(z-1)}\,dz \] where \(C\) is pictured on the next page:

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