Good afternoon,

I would like to ask a question on the functional calculus of several commuting operators. If someone knows some good/standard references, could you please tell me about them.

Firstly, if we have a bounded linear operator on a Hilbert space $T\in \mathcal{B}(\mathcal{H}),$ and a holomorphic function $f\colon \sigma(T) \to \mathcal{B}(\mathcal{C})$ on a neighborhood of the spectrum of $T,$ where $\mathcal{C}$ is another complex Hilbert space, we can define the functional calculus of $T$ for $f$ as follows :

$$f(T) = \frac{1}{2\pi i}\int_{\gamma} f(z)\otimes (z-T)^{-1}dz \in \mathcal{B}(\mathcal{C}\otimes \mathcal{H}),$$ where $\gamma$ is a contour which rounds the spectrum $\sigma(T).$

*So how we can define a functional calculus of several commuting operators for vector(operator)-valued holomorphic functions?*

In the case of scalar-valued functions, it is the work of Taylor (1970) : http://www.ams.org/mathscinet-getitem?mr=0271741. I have not read all the details of the paper, because in the paper, the author uses a lot of sheaf theory which I don't know well.

So **my question** is : firstly, can the definition of functional calculus of Taylor be generalized to the case of vector/operator-valued holomorphic functions? Secondly, does there any reference which presents these results?

Any help is appreciated. Thanks in advance.

Duc Anh

slice map$\sigma_{v,w}$ from $B(C \otimes H)$ to $B(H)$ which takes $A \otimes B$ to $\langle Av,w\rangle B$. The idea would be to define $\sigma_{v,w}(f(T_1,\ldots, T_n))$ using functional calculus for scalar valued functions, for each $v,w \in C$, and then let $f(T_1,\ldots, T_n)$ be the operator in $B(C\otimes H)$ that has all of those slices. $\endgroup$ – Nik Weaver Jun 25 '12 at 19:41