A boundary Nevanlinna-Pick interpolation problem is posed and solved in the quaternionic setting. Given nonnegative real numbers $kappa_1, ldots, kappa_N$, quaternions $p_1, ldots, p_N$ all of modulus $1$, so that the $2$-spheres determined by each p
oint do not intersect and $p_u eq 1$ for $u = 1,ldots, N$, and quaternions $s_1, ldots, s_N$, we wish to find a slice hyperholomorphic Schur function $s$ so that $$lim_{substack{rrightarrow 1 rin(0,1)}} s(r p_u) = s_uquad {rm for} quad u=1,ldots, N,$$ and $$lim_{substack{rrightarrow 1 rin(0,1)}}frac{1-s(rp_u)overline{s_u}}{1-r}lekappa_u,quad {rm for} quad u=1,ldots, N.$$ Our arguments relies on the theory of slice hyperholomorphic functions and reproducing kernel Hilbert spaces.
The quaternionic spectral theorem has already been considered in the literature, see e.g. [22], [31], [32], however, except for the finite dimensional case in which the notion of spectrum is associated to an eigenvalue problem, see [21], it is not sp
ecified which notion of spectrum underlies the theorem. In this paper we prove the quaternionic spectral theorem for unitary operators using the $S$-spectrum. In the case of quaternionic matrices, the $S$-spectrum coincides with the right-spectrum and so our result recovers the well known theorem for matrices. The notion of $S$-spectrum is relatively new, see [17], and has been used for quaternionic linear operators, as well as for $n$-tuples of not necessarily commuting operators, to define and study a noncommutati
A classical theorem of Herglotz states that a function $nmapsto r(n)$ from $mathbb Z$ into $mathbb C^{stimes s}$ is positive definite if and only there exists a $mathbb C^{stimes s}$-valued positive measure $dmu$ on $[0,2pi]$ such that $r(n)=int_0^{2
pi}e^{int}dmu(t)$for $nin mathbb Z$. We prove a quaternionic analogue of this result when the function is allowed to have a number of negative squares. A key tool in the argument is the theory of slice hyperholomorphic functions, and the representation of such functions which have a positive real part in the unit ball of the quaternions. We study in great detail the case of positive definite functions.
