= Spectrum of a polynomial-generated Banach subalgebra
{title2=$\sigma_B(x)=\sigma_A(x)\cup\{\text{bounded components of }\rho_A(x)\}$}
For $B=\overline{\mathbb C[1,x]}$ inside $A$, the displayed <set> is the <full spectrum> of $x$ in $A$. On a bounded complementary component, the <maximum modulus principle> gives $|p(\lambda)|\le\max_{\sigma_A(x)}|p|\le\|p(x)\|$. Evaluation $p(x)\mapsto p(\lambda)$ therefore extends to a <character of an algebra> on $B$, showing $\lambda\in\sigma_B(x)$. Conversely the unbounded resolvent component survives by <resolvent components in a closed unital subalgebra>. In particular, if zero lies in that component, $x^{-1}$ belongs to the <norm> <closure> of <polynomials> in $x$.
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