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.
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.