In this talk we will highlight recent results for spectral representations of bounded and unbounded normal operators on a Hilbert space over the quaternions. Obtaining a spectral representation for a normal operator on a Hilbert space over the quaternions is motivated by an abstract formulation of quantum mechanics that was first suggested by Birkhoff and von Neumann in the 1936. Due to the noncommutativity of the quaternions, there are several possible notions of spectrum of an operator. We shall see that the recently introduced (circa 2007) notion of S-spectrum is the proper notion of spectrum for a spectral representation of a normal operator.

 

This talk is based on joint work with Daniel Alpay and Fabrizio Colombo.

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