Can the level set of the resolvent norm be constant on a set of positive measure?

Question

Thanks to Shargorodsky we know that the resolvent norm can be constant on open sets for unbounded operators on a separable Hilbert space. However this fails for bounded operators on Hilbert spaces. My question is the following. Suppose I am in the nicest case possible, a separable Hilbert space, and I have a bounded operator $T$. Can the level sets of the resolvent norm have positive measure?

Here the resolvent is defined as $R(z,T)=(T-zI)^{-1}$ and I am interested in $$ \{z:\left\|R(z,T)\right\|^{-1}=\epsilon\} $$ for some positive $\epsilon$ (the above norm is the operator norm).

Here's a link to Shargorodsky's paper: https://scholar.google.co.uk/scholar?hl=en&q=On+the+level+sets+of+the+resolvent+norm+of+a+linear+operator%2C&btnG=&as_sdt=1%2C5&as_sdtp=

Answer 1

It may be worth noting that the reciprocal of the resolvent norm is subharmonic. Although I am not sure how to convert this to an answer.