Extended tensor products and an operator-valued spectral mapping theorem

We introduce the notion of an extended tensor product of Banach spaces $X$ and $Y$. It is defined as a triple consisting of a Banach space $X\boxtimes Y$ and two full subalgebras $\mathbf B_0(X)$ and $\mathbf B_0(Y)$ of the algebras $\mathbf B(X)$ and $\mathbf B(Y)$ of all bounded linear operators on $X$ and $Y$ respectively. It is assumed that $X\boxtimes Y$ is an extension of the ordinary tensor product $X\otimes Y$, and the functionals on $X^*\otimes Y^*$ and operators on $\mathbf B_0(X)\otimes\mathbf B_0(Y)$ have a canonical extension from $X\otimes Y$ to $X\boxtimes Y$. Every pseudo-resolvent $\mathbf B_0(Y)$ generates a functional calculus that sends analytic $\mathbf B_0(X)$-valued functions in a neighbourhood of the singular set of the pseudo-resolvent to operators on $X\boxtimes Y$. We prove an analogue of the spectral mapping theorem for such a functional calculus.