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I'm trying to find a factorization for $f(z) = z - c$ in inner and outer functions.

So, first I try to find for what $c$, $f$ is cyclic for the unilateral shift $M_z$. There is a theorem that states that if $A$ is a bounded operator on a Hilbert space $H$ then $x \in H$ is cyclic for $A$ iff zero is the only element that is orthogonal to all $A^n x$ for $n = 0,1,2,\ldots$.

Let $M_z$ be the multiplication operator by $z$ ("the unilateral shift on $H^2$").

So, if $g_n(z) = M_z^n (z - c) = -cz^n + z^{n + 1}$ I find that the power series coefficients of $g_n$ (which is given by $g_n(z) = \sum_m a_{n,m} z^m$) are $a_{n,n} = -c$ and $a_{n,n+1} = 1$. If I insert this in the inner product (where $h(z) = \sum \overline{b_n} z^n$), $(g_n, h) = 0$ for all $n = 0, 1, 2, \ldots$. I obtain that $b_{n + 1} - c b_n = 0$ for $n \geq 0$. This gives that $b_n = C \cdot c^n$. But $\sum c^n z^n = 0$ only if $c = 0$. This would imply that $f(z) = z$ is cyclic but a cyclic vector cannot have a zero in the unit disk! What is wrong here?

If I would find this $c$, then I would find a factorization if the $c$ makes $f$ cyclic because then the inner function is $1$ and the outer function $f$. How to do it if $f$ is not cyclic?

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    I'm confused by parts of your question. For example, $g_n$ seems to have more than one definition, and the second one I think has some indexing problems.2010-10-12
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    Meyer: Oops $n$ should be $m$. Basically I have an expression for $g_n$ and I need to have the power series expansion, that is the second $g_n$, then I compare coefficients, that is the $a_{n,n}$ and $a_{n,n+1}$.2010-10-12

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What is wrong is the reasoning where you wrote "But $\sum c^n z^n = 0$ only if $c=0$." What you need to know is for which choices of $c$ does the condition $b_{n+1}=c b_n$ for all $n$, with $b_0,b_1,\ldots$ being a square summable sequence, force all of the $b_n$'s to be zero. If $c=0$, this is not the case; just take $h(z)=1$. The coefficients are not forced to be zero when $|c|<1$, where you can take $h(z)=1 + cz +c^2z^2 +c^3z^3 +\cdots$. But the condition $b_{n+1}=c b_n$ is impossible for $|c|\geq 1$ unless $h=0$, so your criterion is $|c|\geq1$.

When $f$ is not cyclic, for inner part you can take the holomorphic automorphism of the unit disk that swaps $c$ and 0, and what remains will be an invertible element of $H^\infty$, hence cyclic.

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    Oh, how silly of me. Thank you.2010-10-12