spectral theory expandable to arbitrary polynomials? Given a Banach space $X$ and closed operators $A_i$ ($i \in \left\{0,...,n\right\}$) which have a common domain $D$ that is dense in $X$. An obvious candidate for the title of "generalised resolvent of $A_i$" would be
\begin{equation}
R(A_i,z)=\left(\sum_{i=0}^{n} A_i z^i\right)^{-1}
\end{equation}
if it exists. What main parts of the resolvent formalism can be generalised to this setting?


*

*Any nice "generalised resolvent identities"? Is the "generalised resolvent set" (values of $z$ for which the resolvent exists, is bounded and has dense domain) still open?

*Do there exist generalised Weyl sequences to the boundary points of the generalised resolvent set?

*Suppose $X$ is a Hilbert space. Is there any relation between the "generalised spectrum" and the set
\begin{equation}
Q=\left\{z^{*} \in \mathbb{C} \left|\right. \exists \psi \in X:  z^{*} \text{ is a root of} \sum_{i=0}^{n} \langle\psi,A_i \psi\rangle z^i\right\}
\end{equation}
?
You're welcome to answer with a good literature reference if these (standard?) questions are being addressed there already.
There's also the generalisation of these questions to multivariate polynomials to be kept in mind.
 A: What about the following generalisation (it works, but is it useful?) ? Let's deal with polynomials of the form $P(A,B,z)=A-Bz$ first and let's give a modified definition to the "generalised resolvent set" $\rho(A,B)$:
\begin{equation}
\left\{z\in \mathbb{C}\left|
\right. \left(A-zB\right)^{-1}\text{,}B\left(A-zB \right)^{-1}\text{exist on a dense (common) domain and are both bounded}\right\}
\end{equation}


*

*modified first resolvent identity: $\forall z_1,z_2 \in \rho(A,B)$
\begin{equation}
\left(A-z_2B\right)^{-1}-\left(A-z_1B\right)^{-1}=(z_2-z_1)\left(A-z_2B\right)^{-1}B\left(A-z_1B\right)^{-1}=(z_2-z_1)\left(A-z_1B\right)^{-1}B\left(A-z_2B\right)^{-1},
\end{equation}
of which the (easy) proof is left as an exercise.

*Corollary: additional resolvent identity, $\forall z_1,z_2 \in \rho(A,B)$:
\begin{equation}
B\left(A-z_2B\right)^{-1}-B\left(A-z_1B\right)^{-1}=(z_2-z_1)B\left(A-z_2B\right)^{-1}B\left(A-z_1B\right)^{-1}=(z_2-z_1)B\left(A-z_1B\right)^{-1}B\left(A-z_2B\right)^{-1},
\end{equation}

*Corollary: $\rho(A,B)$ is open. Proof:
We can use both resolvent identities repeatedly to obtain $\forall n\in \mathbb{N}$:
\begin{equation}
\left(A-z_2B\right)^{-1} = \left(A-z_1B\right)^{-1} \sum_{j=0}^n (z_2-z_1)^j \left[B\left(A-z_1B\right)^{-1}\right]^j + (z_2-z_1)^{n+1} \left(A-z_1B\right)^{-1}\left[B\left(A-z_1B\right)^{-1}\right]^{n+1}
\end{equation}
\begin{equation}
B\left(A-z_2B\right)^{-1} = B\left(A-z_1B\right)^{-1} \sum_{j=0}^n (z_2-z_1)^j \left[B\left(A-z_1B\right)^{-1}\right]^j + (z_2-z_1)^{n+1} \left[B\left(A-z_1B\right)^{-1}\right]^{n+2}.
\end{equation}
If $\left|z_2-z_1\right|<R_{z_1}:=\min(\left\|\left(A-z_1B\right)^{-1}\right\|^{-1},\left\|B\left(A-z_1B\right)^{-1}\right\|^{-1})$, these expansions can be continued to a convergent power series. Conversely, it can be checked manually that this power series must converge to $\left(A-z_2B\right)^{-1}$ and $B\left(A-z_2B\right)^{-1}$ respectively if it converges. Hence we deduce that if $z_1 \in S(A,B)$, then $D(z_1,R_{z_1}) \subset S(A,B)$. Also we have
\begin{equation}
R_{z_1}\leq dist(z_1, \sigma(A,B)) \text{   } (\dagger)
\end{equation}
where $\sigma(A,B)=\rho(A,B)^c$


*

*Let's propose the following modified definition of a Weyl sequence: $forall z \in \mathbb{C}$ we have that $(\psi_n)_n$ is a Weyl sequence i.f.f.
\begin{equation}
(A-Bz)\psi_n \rightarrow 0
\end{equation}
while $\forall n$
\begin{equation}
max( \left\|\psi_n\right\| , \left\|B\psi_n\right\| )=1
\end{equation}

*Let $z \in \mathbb{C}$ be a boundary point of $\rho(A,B)$. There exists a Weyl sequence $(\psi_n)_n$ associated to $z$. Proof: Let $(z_n)_n$ be a sequence in $\rho(A,B)$ such that $z_n \rightarrow z$. By $(\dagger)$ we can find a sequence $(\phi_n)_n$ such that
\begin{equation}
\max \left(\frac{\left\|(A-Bz_n)^{-1}\phi_n\right\|}{\left\|\phi_n\right\|}, \frac{\left\|B(A-Bz_n)^{-1}\phi_n\right\|}{\left\|\phi_n\right\|}\right) \rightarrow \infty
\end{equation}
Define $\psi_n := (A-Bz_n)^{-1}\phi_n$ and renormalise such that $max( \left\|\psi_n\right\| , \left\|B\psi_n\right\| )=1$ (which means that $\left\|\phi_n\right\| \rightarrow 0$. Now calculate
\begin{eqnarray}
\left\|(A-Bz)\psi_n\right\| &=& \left\|\phi_n + (z-z_n)B\psi_n\right\| \\
&=& \left\|\phi_n\right\| + |z-z_n|\left\|B\psi_n\right\| \rightarrow 0.
\end{eqnarray}
The next step is to generalise to multivariate polynomials of the form $P(A_0,...,A_n,z_1,...,z_n)=A_0 -z_1 A_1-...-z_n A_n$ where it also easy to guess how the definitions of generalised spectrum and Weyl sequences should generalise. All results above have a suitable generalisation.
