IWOTA 2016

The Spectral Theorem for $d$-tuples of strongly commuting unbounded normal operators on a quaternionic Hilbert space

Crow 206

In this talk we shall show that corresponding to a $d$-tuple of strongly commuting normal operators $(T_1, \ldots, T_d)$ on a quaternionic Hilbert space $\mathcal{H}$, there exists a spectral measure $E$ on the Borel sets of $\mathbb{C}_+^d := \{ (\lambda_1, \ldots \lambda_d) \in \mathbb{C}^d: { \rm Im}\, \lambda_n \geq 0 \}$ such that $$\langle T_n x, y \rangle = \int_{\mathbb{C}_+^d} {\rm Re}(\lambda_n) d\langle E(\lambda_1, \ldots, \lambda_d) x, y \rangle + \int_{\mathbb{C}_+^d} {\rm Im}(\lambda_n) d\langle J_n E(\lambda_1, \ldots, \lambda_d)x, y \rangle,$$ where $x \in \mathcal{D}(T_n)$ and $J_n$ is some anti self-adjoint and unitary operator on $\mathcal{H}$ for $n=1,\ldots,d$. The study of spectral theory for operators on a quaternionic Hilbert spaces is motivated by a general formulation of quantum mechanics formulated by Birkhoff and von Neumann. We shall also touch upon this connection. This talk is partially based on joint work with D. Alpay and F. Colombo.