Related papers: Spectrum as the Support of Functional Calculus
We revise a monogenic calculus for several non-commuting operators, which is defined through group representations. Instead of an algebraic homomorphism we use group covariance. The related notion of joint spectrum and spectral mapping…
A consistent functional calculus approach to the spectral theorem for strongly commuting normal operators on Hilbert spaces is presented. In contrast to the common approaches using projection-valued measures or multiplication operators,…
We present an approach to the spectrum and analytic functional calculus for quaternionic linear operators, following the corresponding results concerning the real linear operators. In fact, the construction of the analytic functional…
The main purpose of this work is the construction of an analytic functional calculus for Clifford operators, which are operators acting on certain modules over Clifford algebras. Unlike in some preceding works by other authors, we use a…
Classical spectral theory provides powerful tools for analyzing linear operators, but does not extend naturally to nonlinear or compositional settings. In particular, there is no general way to transport spectral invariants in a functorial…
This dissertation focuses on developing a new construction of a functional calculus using Henstock-Kurzweil integration methods. The assignment of a functional calculus will be applied to self-adjoint operators. We will address both the…
We give a direct non-abstract proof of the Spectral Mapping Theorem for the Helffer-Sj\"ostrand functional calculus for linear operators on Banach spaces with real spectra and consequently give a new non-abstract direct proof for the…
The spectral theory on the S-spectrum was introduced to give an appropriate mathematical setting to quaternionic quantum mechanics, but it was soon realized that there were different applications of this theory, for example, to fractional…
The aim of this work is to define a continuous functional calculus in quaternionic Hilbert spaces, starting from basic issues regarding the notion of spherical spectrum of a normal operator. As properties of the spherical spectrum suggest,…
Classical spectral theory gives a complete description of a single normal operator, but it fails for noncommuting operators, where no canonical joint spectrum or simultaneous diagonalization exists. Existing approaches provide only partial…
We define a smooth functional calculus for a non-commuting tuple of (unbounded) operators $A_j$ on a Banach space with real spectra and resolvents with temperate growth, by means of an iterated Cauchy formula. The construction is also…
We describe a closed operator functional calculus in Banach modules over the group algebra $L^1(\mathbb R)$ and illustrate its usefulness with a few applications. In particular, we deduce a spectral mapping theorem for operators in the…
Spectral representations of the dilation and translation operators on $L^2({\mathbb R})$ are built through appropriate bases. Orthonormal wavelets and multiresolution analysis are then described in terms of rigid operator-valued functions…
Joint spectra of tuples of operators are subsets in complex projective space. The corresponding tuple of operators can be viewed as an infinite dimensional analog of a determinantal representation of the joint spectrum. We investigate the…
Unlike in complex linear operator theory, polynomials or, more generally, Laurent series in antilinear operators cannot be modelled with complex analysis. There exists a corresponding function space, though, surfacing in spectral mapping…
In this paper, we shall consider the notion of bicomplex inner product and define bicomplex Hilbert space. We shall define $L^{2}[a,b]$ where the functions take bicomplex values. We shall prove the Theorem for a bounded self adjoint…
The spectral theory on the $S$-spectrum was born out of the need to give quaternionic quantum mechanics (formulated by Birkhoff and von Neumann) a precise mathematical foundation. Then it turned out that this theory has important…
Over the decades, Functional Analysis has been enriched and inspired on account of demands from neighboring fields, within mathematics, harmonic analysis (wavelets and signal processing), numerical analysis (finite element methods,…
Spectral theory and functional calculus for unbounded self-adjoint operators on a Hilbert space are usually treated through von Neumann's Cayley transform. Based on ideas of Woronowicz, we redevelop this theory from the point of view of…
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…