I'm reading some functional analysis and it's after lunch so I can't think. Can someone please help with this quick question. I have the following:
Let $T\in B(H)$ be a self adjoint operator and $\varepsilon>0$. Let $p$ be the spectral projection of $T$ corresponding to the Borel subset $[-\varepsilon,\varepsilon]$ (that is, the indicator function of $[-\varepsilon,\varepsilon]$ applied to $T$ by Borel functional calculus). Then the following operator $$S = (1-p)T+\varepsilon p$$ has a self adjoint bounded inverse.
Why? Is there an obvious inverse I am missing, or is this some bounded inverse theorem (or one of the other functional analysis theorems) business.
I was about to post an answer, when I saw that what I had to say had been said in a comment. Some readers may wonder what it means for an operator to "correspond to" a function. That's a medium-long story - my favorite version is not the most popular one, decided to mark this community-wiki and post a complete answer for readers who don't know that stuff. Note again that the proof is contained in a comment above, of which this is just an exegesis.
A suitable version of the Spectral Theorem says that wlog $H=L^2(\mu)$ and $T$ is given by pointwise multiplication by $m\in L^\infty(\mu)$: $$(Tf)(x)=m(x)f(x).$$In this context $p$ is given by multiplcation by $$k_\epsilon=\chi_{[-\epsilon,\epsilon]}\circ m,$$and $S$ is multiplication by $$M=(1-k_\epsilon)m+\epsilon k_\epsilon.$$Considering a total of two cases shows that $1/M$ is bounded. (Also $1/M$ is real-valued since $m$ is; that's the "self-adjoint" part.)