A Commutator Formula for Subnormal Tuples of Operators

Abstract

Let \({\mathbb{S}=(S_1,\ldots,S_k)}\) be a pure subnormal k-tuple of operators with minimal normal extension \({\mathbb{N}}\) and defect space \({\mathcal{M}}\). Let \({\Lambda_j=(S_j^*|_\mathcal{M})^*}\). We prove $$[f(\mathbb{S}^*)h(\mathbb{S}),S_j]|_\mathcal{M}= \frac{1}{2\pi i}\int_{sp(\mathbb{N})}f(\bar{u})h(u)(u_j-\Lambda_j)e(du),$$ where \({e(\cdot)=P_\mathcal{M}E(\cdot)|_\mathcal{M}}\), \({E(\cdot)}\) is the spectral measure of \({\mathbb{N}}\), \({P_\mathcal{M}}\) is the projection to \({\mathcal{M}}\), f is any analytic function on \({\times_{j=1}^k\sigma(S_j^*)}\) and h is any analytic function on \({\times_{j=1}^k\sigma(S_j)}\). If \({ \mathrm{dim}\mathcal{M} < \infty}\), then this commutator equals to $$\frac{1}{2\pi i}\int_{A}\mu_j(u_j)df(\bar{u})\wedge dh(u),$$ where \({A=\{(\bar{u}_1,\ldots,\bar{u}_k):u=(u_1,\ldots,u_k) \, \mathrm{is \, in \, the \, joint \, point \, spectrum \, of} \, \mathbb{S}^*\}}\), and \({\mu_j(\cdot)}\) is the mosaic of S j .