By Prof. Fabrizio Colombo Politecnico di Milano, Irene Sabadini, Daniele C. Struppa
<i>This booklet provides a sensible calculus for <i>n</i>-tuples of no longer inevitably commuting linear operators. particularly, a practical calculus for quaternionic linear operators is built. those calculi are in line with a brand new thought of hyperholomorphicity for services with values in a Clifford algebra: the so-called slice monogenic features that are rigorously defined within the publication. relating to features with values within the algebra of quaternions those capabilities are named slice typical functions.</i>
<p>Except for the appendix and the advent all effects are new and seem for the 1st time geared up in a monograph. the fabric has been conscientiously ready to be as self-contained as attainable. The meant viewers involves researchers, graduate and postgraduate scholars attracted to operator concept, spectral conception, hypercomplex research, and mathematical physics.</p>
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Extra resources for Noncommutative Functional Calculus: Theory and Applications of Slice Hyperholomorphic Functions
1. Let x, s ∈ Rn+1 . We call S −1 (s, x) := xn s−1−n n≥0 the noncommutative Cauchy kernel series. 2. The noncommutative Cauchy kernel series is convergent for |x| < |s|. 3. Let x, s ∈ Rn+1 be such that xs = sx. Then, the function S(s, x) = −(x − s)−1 (x2 − 2Re[s]x + |s|2 ), is the inverse of the noncommutative Cauchy kernel series. Proof. Let us verify that −(x − s)−1 (x2 − 2Re[s]x + |s|2 ) xn s−1−n = 1. n≥0 54 Chapter 2. Slice monogenic functions We therefore obtain xn s−1−n = s + x − 2 Re[s].
We deﬁne the notion of I-derivative by means of the operator: ∂I := 1 2 ∂ ∂ −I ∂u ∂v For consistency, we will denote by ∂ I the operator . 1 2 ∂ ∂u ∂ + I ∂v . Using the notation we have just introduced, the condition of left s-monogenicity will be expressed, in short, by ∂ I f = 0. Right s-monogenicity will be expressed, with an abuse of notation, by f ∂ I = 0. 5. It is easy to verify that the (left) s-monogenic functions on U ⊆ Rn+1 form a right Rn -module. In fact it is trivial that if f, g ∈ M(U ), then for every I ∈ S one has ∂ I fI = ∂ I gI = 0, thus ∂ I (f + g)I = 0.
A|=0 We now need to show that the functions FA are holomorphic. Since f is smonogenic we have that its restriction to CI satisﬁes ∂ ∂ +I ∂u ∂v fI (u + Iv) = 0 and so ∂ ∂ +I (fA + gA I)IA ∂u ∂v ∂ ∂ ∂ ∂ = fA + I fA + gA I − gA = 0. ∂u ∂v ∂u ∂v Since the imaginary units commute with any real-valued function, we obtain the system: ⎧ ∂ ∂ ⎪ ⎪ fA − gA = 0, ⎨ ∂u ∂v ⎪ ∂ ∂ ⎪ ⎩ fA + gA = 0 ∂v ∂u for all multi-indices A. Therefore all the functions FA = fA + gA I satisfy the standard Cauchy–Riemann system and so they are holomorphic.
Noncommutative Functional Calculus: Theory and Applications of Slice Hyperholomorphic Functions by Prof. Fabrizio Colombo Politecnico di Milano, Irene Sabadini, Daniele C. Struppa