# Functional calculus for vector-valued holomorphic functions?

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

• I do not know offhand of any references for a functional calculus for operator-valued functions (or even what that should mean exactly, since one would have to decide which tensor product norm to use) but F.-H. Vasilescu (ams.org/mathscinet-getitem?mr=0530689) gave a simplified treatment of Taylor's functional calculus based on a Martinelli-type integral formula (very much like the one variable formula you've stated). A very nice, readable account of these topics may be found in Raul Curto's article ams.org/mathscinet-getitem?mr=0976843 – Mike Jury Jun 25 '12 at 16:11
• There is a multivariate version of Agler's hereditary functional calculus, see for example the work of Ambrozie, Englis and Muller (ams.org/mathscinet-getitem?mr=1911848). – Mike Jury Jun 25 '12 at 16:37
• I did not think seriously about your question so this is just an idea, but: the smooth calculus for bounded operators is quite easy to construct by hand, first for polynomials, then by uniform completion. Why not do the same for an ntuple, starting with multivariate polynomials? in Reed-Simon you can find a short account of the theory – Piero D'Ancona Jun 25 '12 at 17:40
• Reduction to the scalar case is a standard technique that might work here. For any vectors $v,w \in C$ we have a 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. – Nik Weaver Jun 25 '12 at 19:41
• You can find a construction of an operator-valued holomorphic functional calculus for families of commuting sectorial operators in Kalton/Weis, $H^\infty$-calculus and sums of closed operators (MR1866491). This functional calculus includes the case where the operators in question are bounded and commute with each other as a special case. – user8707 Jun 25 '12 at 22:15