Let $H$ be a Hilbert space, $X$ a topological space, and $\{A_t\}_{t\in X}$ a continuous family of bounded, invertible operators on $H$. Continuous here in the sense that the corresponding map $X\rightarrow B(H)$ is continuous (with respect to the uniform topology on $B(H)$). Then the family of inverses $\{A^{-1}_t\}_{t\in X}$ is also continuous.

Question: what kind of generalizations are there for families of unbounded operators?

To be more specific, suppose that $\{A_t\}_{t\in X}$ is a family of unbounded, self-adjoint operators on $H$ whose domains $D_t$ are (possibly) varying in $H$, but with $D:=\bigcap_t D_t$ dense in $H$. If each $A_t$ has a bounded inverse $A_t^{-1}$, I would like to know if there are situations (i.e. conditions we have to impose) where we can conclude that $\{A^{-1}_t\}_{t\in X}$ is a continuous family of operators. One of the issues (which is part of the question) is what the right notion of continuity would be in this situation. I am aware of some partial answers:

  1. If all the $A_t$ have the same domain, i.e. $D=D_t$ for all $t$, and if the matrix coefficients $\langle A_t u, v\rangle$ are continuous functions of $t\in X$, for every $u\in D$ and $v\in H$, then the answer is yes (this is proved e.g. in: "Kriegl, Michor; Differentiable perturbation of unbounded operators. Math. Ann. 327 (2003), no. 1, 191–201. 47A55")

  2. Example where uniform continuity seems to fail: $H:=L^2(\mathbb{R})$, $A_t:=\mathcal{F}^{-1} (1+|\xi|^2)^{t/2}\mathcal{F}:H\rightarrow H$, where $\mathcal{F}$ denotes the Fourier transform. Then $\{A\}_{t\in [0,1]}$ is a family of unbounded, self-adjoint operators on $H$, with domains given by the Sobolev spaces $D_t=L^2_t$. The inverses are $A_t^{-1}=A_{-t}$, which are bounded on $L^2$, but $\{A_{-t}\}_{t\in [0,1]}$ is not continuous in the uniform topology (it is continuous for the strong topology).

The application I have in mind is this; given a smooth family of pseudo-convex complex structures on a compact manifold with boundary such that the corresponding $\overline{\partial}$-Laplacians are invertible on $L^2(\Lambda^{p,q}T^\ast)$ for some $p,q$, then is the family of inverses uniformly continuous? (In the case without boundary the answer is yes)

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    $\begingroup$ The condition you're interested in is usually called norm-resolvent convergence (or continuity rather, I guess), and the literature on this must be near infinite (though it's more than one typically has in common situations in spectral theory; strong resolvent convergence is probably a more frequently useful notion). $\endgroup$ – Christian Remling Jul 12 '17 at 18:04

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