Designs for calculating the spectral multiplicity of orthogonal sums

Let $A$ and $B$ be operators in spaces $X$ and $Y$ respectively and suppose that $B$ has a “rich” system of sets $\Delta$, $\Delta\subset\mathbb C$ with $Y(\Delta)$ dense in $Y$, where $Y(\Delta)=\{y\in Y:\|p(B)y\|\leqslant C_y\sup_\Delta|p|\text{ for any complex polynomial }p\}$. Then $\mu_{A\oplus B}=\max(\mu_A, \mu_B)=\mu_A$ ($\mu_A$ denotes the spectral multiplicity of an operator $A$ i. e. the number $\min\{\dim L:\operatorname{span}(A^nL:n\geqslant0)=X\}$). For example, if $B$ is a Toeplitz operator $T\bar g$ with $g\in H^\infty$, $g\not\equiv\mathrm{const}$ and if, moreover, $g(\mathbb D)\setminus\text {\{polynomially convex hull of the spectrum of }A\}\ne\varnothing$ then $\mu_{A\oplus T\bar g}=\mu_A$. To the contrary, if $A=T_f$ with $f\in H^\infty$ and $g(\mathbb D)\subset f(\mathbb D)$ then (under some additional regularity assumptions on $f$) we have $\mu_{Tf\oplus Tg}=\mu_{Tf}+\mu_{Tg}$. We give also some examples of univalent and essentially univalent functions $f$ $(f\in H^\infty)$ with $\mu_{Tf}>1$.