= Solution
If the spectrum has one point, the preceding part gives $T=\lambda I$, and the <linear span> of any nonzero vector is a closed proper <invariant subspace> because $\dim H>1$.
Otherwise choose distinct points $\lambda_1,\lambda_2\in K$ and disjoint nonempty relatively open neighborhoods $U,V\subseteq K$. The open-set result gives $P(U)\ne0$ and $P(V)\ne0$, while
$$
P(U)P(V)=P(U\cap V)=0.
$$
Thus $P(U)\ne I$. Set $L=\operatorname{ran}P(U)$. As the range of an <orthogonal projection>, $L$ is closed, and the preceding observations give $\{0\}\ne L\ne H$. The <Borel functional calculus for a normal operator> gives $TP(U)=P(U)T$, so
$$
\boxed{L=\operatorname{ran}P(U)\text{ is a nontrivial closed invariant subspace of }T.}
$$
In fact both $L$ and its orthogonal complement are invariant.
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