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I have some doubt regarding annihilator of an matrix. Please help me to understand the subject.

Let $T$ be a n by n Complex matrix. Let $m(x)$ be the minimal polynomial for $T$. Now suppose $f(T)=0$ for some $f$-holomorphic function on a neighborhood of the spectrum of $T$. Here $f(T)$ is defined using Riesz Functional Calculus for $T$. Now the question is Can we say that $f(x)=m(x)g(x)$ for some $g$-holomorphic function in a neighborhood of the spectrum of $T$? If it is then how?

We know that set of all polynomial which annihilate $T$ form a Principal Ideal. But I don't know whether the set of all holomorphic function on $\sigma(T)$ which annihilate $T$ forms a Principal ideal or not.

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