Searcharxiv⌕ Search

arXiv subjects

I. Sabadini

Publications and source records attributed to I. Sabadini.

25 records · Page 2Linked to original sources
arXiv2014

The spectral theorem for unitary operators based on the $S$-spectrum

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 specified 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 noncommutative versions of the Riesz-Dunford functional calculus. The main tools to prove the spectral theorem for unitary operators are the quaternionic version of Herglotz's theorem, which relies on the new notion of $q$-positive measure, and quaternionic spectral measures, which are related to the quaternionic Riesz projectors defined by means of the $S$-resolvent operator and the $S$-spectrum. The results in this paper restore the analogy with the complex case in which the classical notion of spectrum appears in the Riesz-Dunford functional calculus as well as in the spectral theorem.

math.SP↗
arXiv2008

Codimension 3 Arithmetically Gorenstein Subschemes of projective $N$-space

We study the lowest dimensional open case of the question whether every arithmetically Cohen--Macaulay subscheme of $\mathbb{P}^N$ is glicci, that is, whether every zero-scheme in $\mathbb{P}^3$ is glicci. We show that a set of $n \geq 56$ points in general position in $\PP^3$ admits no strictly descending Gorenstein liaison or biliaison. In order to prove this theorem, we establish a number of important results about arithmetically Gorenstein zero-schemes in $\mathbb{P}^3$.

math.AG↗
arXiv2008

Slice monogenic functions

In this paper we offer a definition of monogenicity for functions defined on $\rr^{n+1}$ with values in the Clifford algebra $\rr_n$ following an idea inspired by the recent papers \cite{gs}, \cite{advances}. This new class of monogenic functions contains the polynomials (and, more in general, power series) with coefficients in the Clifford algebra $\rr_n$. We will prove a Cauchy integral formula as well as some of its consequences. Finally, we deal with the zeroes of some polynomials and power series.

math.CV↗
arXiv2008

Non commutative functional calculus: unbounded operators

In a recent work, \cite{cgss}, we developed a functional calculus for bounded operators defined on quaternionic Banach spaces. In this paper we show how the results from \cite{cgss} can be extended to the unbounded case, and we highlight the crucial differences between the two cases. In particular, we deduce a new eigenvalue equation, suitable for the construction of a functional calculus for operators whose spectrum is not necessarily real.

math.SP↗
arXiv2008

A new functional calculus for non-commuting operators

In this paper we use the notion of slice monogenic functions \cite{slicecss} to define a new functional calculus for an $n$-tuple $T$ of not necessarily commuting operators. This calculus is different from the one discussed in \cite{jefferies} and it allows the explicit construction of the eigenvalue equation for the $n$-tuple $T$ based on a new notion of spectrum for $T$. Our functional calculus is consistent with the Riesz-Dunford calculus in the case of a single operator.

math.SP↗
arXiv2007

Non commutative functional calculus: bounded operators

In this paper we develop a functional calculus for bounded operators defined on quaternionic Banach spaces. This calculus is based on the notion of slice-regularity, see \cite{gs}, and the key tools are a new resolvent operator and a new eigenvalue problem.

math.SP↗