4

I'm working on the following problem in Conway's Functional Analysis. Here $\phi$ is a bounded measurable function on $(X, \Omega, \mu)$. enter image description here I was able to answer the first part of the problem but I am stuck on the second. My first idea was to look at the spectrum, as injectivity + closed range $\implies$ surjectivity. However, I haven't figured out the case when $\phi$ is zero on a set of positive measure. One sufficient condition I came up with is for $X \setminus ker(\phi)$ to be a closed set. I don't know if this is also a necessary condition and I would really appreciate some help!

vxnture
  • 728
  • 3
    Although OP does not expressly specify the domain or codomain of $M_{\phi}$ in the body of this question, a glance at Conway's text confirms that these spaces are both what one might expect, i.e., the Hilbert space $L^2(X,\Omega,\mu)$, making this question a duplicate of (part of) this 2012 question, for which an answer and an elaboration of that answer have already been posted. – leslie townes Dec 11 '24 at 09:25

0 Answers0