In an octonionic Hilbert space $H$, the octonionic linearity is taken to fail for the maps induced by the octonionic inner products, and it should be replaced with the octonionic para-linearity. However, to introduce the notion of the octonionic para
-linearity we encounter an insurmountable obstacle. That is, the axiom $$leftlangle pu ,urightrangle=pleftlangle u ,urightrangle$$ for any octonion $p$ and element $uin H$ introduced by Goldstine and Horwitz in 1964 can not be interpreted as a property to be obeyed by the octonionic para-linear maps. In this article, we solve this critical problem by showing that this axiom is in fact non-independent from others. This enables us to initiate the study of octonionic para-linear maps. We can thus establish the octonionic Riesz representation theorem which, up to isomorphism, identifies two octonionic Hilbert spaces with one being the dual of the other. The dual space consists of continuous left almost linear functionals and it becomes a right $O$-module under the multiplication defined in terms of the second associators which measures the failure of $O$-linearity. This right multiplication has an alternative expression $${(fodot p)(x)}=pf(p^{-1}x)p,$$ which is a generalized Moufang identity. Remarkably, the multiplication is compatible with the canonical norm, i.e., $$fsh{fodot p}=fsh{f}abs{p}.$$ Our final conclusion is that para-linearity is the nonassociative counterpart of linearity.
In this paper, we generalize a recent work of Liu et al. from the open unit ball $mathbb B^n$ to more general bounded strongly pseudoconvex domains with $C^2$ boundary. It turns out that part of the main result in this paper is in some certain sense
just a part of results in a work of Bracci and Zaitsev. However, the proofs are significantly different: the argument in this paper involves a simple growth estimate for the Caratheodory metric near the boundary of $C^2$ domains and the well-known Grahams estimate on the boundary behavior of the Caratheodory metric on strongly pseudoconvex domains, while Bracci and Zaitsev use other arguments.
In this paper, we present an alternative and elementary proof of a sharp version of the classical boundary Schwarz lemma by Frolova et al. with initial proof via analytic semigroup approach and Julia-Caratheodory theorem for univalent holomorphic sel
f-mappings of the open unit disk $mathbb Dsubset mathbb C$. Our approach has its extra advantage to get the extremal functions of the inequality in the boundary Schwarz lemma.
Slice regular functions have been extensively studied over the past decade, but much less is known about their boundary behavior. In this paper, we initiate the study of Julia theory for slice regular functions. More specifically, we establish the quaternion
The theory of slice regular functions is nowadays widely studied and has found its elegant applications to a functional calculus for quaternionic linear operators and Schur analysis. However, much less is known about their boundary behaviors. In this
paper, we initiate the study of the boundary Julia theory for quaternions. More precisely, we establish the quaternion
The sharp growth and distortion theorems are established for slice monogenic extensions of univalent functions on the unit disc $mathbb Dsubset mathbb C$ in the setting of Clifford algebras, based on a new convex combination identity. The analogous r
esults are also valid in the quaternionic setting for slice regular functions and we can even prove the Koebe type one-quarter theorem in this case. Our growth and distortion theorems for slice regular (slice monogenic) extensions to higher dimensions of univalent holomorphic functions hold without extra geometric assumptions, in contrast to the setting of several complex variables in which the growth and distortion theorems fail in general and hold only for some subclasses with the starlike or convex assumption.
