A general scheme is proposed for constructing vector coherent states, in analogy with the well-known canonical coherent states, and their deformed versions, when these latter are expressed as infinite series in powers of a complex variable z. In the present scheme, the variable z is replaced by matrix valued functions over appropriate domains. As particular examples, we analyze the quaternionic extensions of the canonical coherent states and the Gilmore–Perelomov and Barut–Girardello coherent states arising from representations of SU(1,1). Possible physical applications are indicated.

Coherent states, similar to the canonical coherent states of the harmonic oscillator, labeled by quaternions are established on the right and left quaternionic Hilbert spaces. On the left quaternionic Hilbert space reproducing kernels are established. As was claimed by Adler and Millard (J. Math. Phys. 1997 38 2117-26) it is proved that these coherent states cannot be realized as a group action on a quaternionic Hilbert space. As an extension of the complex Hermite polynomials, quaternionic Hermite polynomials are defined and these polynomials are associated with an operator as eigenfunctions.

We define quaternionic Hermite polynomials by analogy with two families of complex Hermite polynomials. As in the complex case, these polynomials consatitute orthogonal families of vectors in ambient quaternionic L 2 -spaces. Using these polynomials, we then define regular and anti-regular subspaces of these L 2 -spaces, the associated reproducing kernels and the ensuing quaternionic coherent states.

In a right quaternionic Hilbert space, for a bounded right linear operator, the Kato S-spectrum is introduced and studied to a certain extent. In particular, it is shown that the Kato S-spectrum is a non-empty compact subset of the S-spectrum and it contains the boundary of the S-spectrum. Using right-slice regular functions, local Sspectrum, at a point of a right quaternionic Hilbert space, and the local spectral subsets are introduced and studied. The S-surjectivity spectrum and its connections to the Kato S-spectrum, approximate S-point spectrum and local S-spectrum are investigated. The generalized Kato S-spectrum is introduced and it is shown that the generalized Kato S-spectrum is a compact subset of the S-spectrum.

Parallel to the quantization of the complex plane, using the canonical coherent states of a right quaternionic Hilbert space, quaternion field of quaternionic quantum mechanics is quantized. Associated upper symbols, lower symbols and related quantities are analyzed. Quaternionic version of the harmonic oscillator and Weyl-Heisenberg algebra are also obtained.