A new functional calculus for noncommuting operators

By Fabrizio Colombo, Irene Sabadini, Daniele C. Struppa. 2007

108 citations recorded by the source corpus.

Abstract In this paper we use the notion of slice monogenic functions [F. Colombo, I. Sabadini, D.C. Struppa, Slice monogenic functions, Israel J. Math., in press] 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 [B. Jefferies, Spectral Properties of Noncommuting Operators, Lecture Notes in Math., vol. 1843, Springer-Verlag, Berlin, 2004] 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.

Subfields: Algebra and Number Theory, Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/c27c5ddff4526c5a85847463fe1d1c4aacca22e5

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In Algebra and Number Theory

In Applied Mathematics

Citations in the corpus

A New Resolvent Equation for the $$S$$S-Functional Calculus (2013, unknown): “, Tn) of non necessarily commuting operators and is based on the so-called S-spectrum, see [14, 17].”
Regular Functions of a Quaternionic Variable (2013, background): “Moreover, we want to mention that the theory or regular quaternionic functions presented in this thesis and its generalizations to other hypercomplex algebras have been applied to construct new theories of functional calculus in non-commutative settings in [6, 7, 9, 8, 10, 13].”
Pontryagin-de Branges-Rovnyak spaces of slice hyperholomorphic functions (2012, unknown): “[16, 18, 21]; while slice monogenic functions admit a functional calculus for n-tuples of operators, see [24, 20, 22].”
A Cauchy kernel for slice regular functions (2009, background): “In the quaternionic case, the functional calculus is associated to quaternionic operators (see for example [4,7,2]), while in the Clifford algebras case it deals with n-tuples of non-commuting operators (see [1,5]).”
Extension results for slice regular functions of a quaternionic variable (2009, methodology): “The use of the Cauchy kernel which appears in [1] to prove the validity of a Cauchy formula on axially symmetric s-domains was suggested by the functional calculus introduced in [2,3,5] and [7].”
Non Commutative Functional Calculus: Bounded Operators (2007, background): “For the complex case, the reader is referred to [10] (for the commuting case) and to [3,7,11] for the case in which the operators do not commute.”
Slice monogenic functions (2007, methodology): “on with the study of the noncommutative Cauchy kernel series X n≥0 p~n~s−1−n, (10) for p~~s 6= ~sp~, which is a fundamental tool to deﬁne a functional calculus for several noncommuting operators, see [3]. To motivate the results of this section we make the following observations. We consider the Cau”
Regular Moebius transformations of the space of quaternions (2008, background): “This study is also the basis for a new functional calculus in a non commutative setting (see [2, 3, 5]).”
On Some Properties of the Quaternionic Functional Calculus (2009, unknown): “Such a theory is a natural tool to define a functional calculus for n-tuples of noncommuting operators, see [8] and [2, 3].”
An extension theorem for slice monogenic functions and some of its consequences (2010, background): “Note that the expression ( x 2 − 2Re[s] x + |s|2)−1( x − s ) cannot be simplified (see [7]), unless s ∈ R, and in this case it coincides with ( x − s )−1, i.”
A Structure Formula for Slice Monogenic Functions and Some of its Consequences (2008, methodology): “We recall some properties of the spectrum which are originally proven in [6] and can be found also in [4].”
The Fueter mapping theorem in integral form and the ℱ‐functional calculus (2010, background): “In [9], for T ∈B n (Vn), the S-spectrum for the n-tuple of noncommuting operators was defined as: S(T) :={s∈Rn+1 : T2 −2 Re[s]T +|s|2I is not invertible}.”
Entire slice regular functions (2015, background): “The S-functional calculus, see [9, 45, 48, 49, 59, 60, 61, 71], is the analogue of the Riesz-Dunford functional calculus in the quaternionic setting.”