Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userPoly slice monogenic functions, Cauchy formulas and the PS-functional calculus
94
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Since 2006 the theory of slice hyperholomorphic functions and the related spectral theory on the S-spectrum have had a very fast development. This new spectral theory based on the S-spectrum has applications for example in the formulation of quaternionic quantum mechanics, in Schur analysis and in fractional diffusion problems. The notion of poly slice analytic function has been recently introduced for the quaternionic setting. In this paper we study the theory of poly slice monogenic functions and the associated functional calculus, called PS-functional calculus, which is the polyanalytic version of the S-functional calculus. Also for this poly monogenic functional calculus we use the notion of S-spectrum.
rate research
Read More
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 results 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.
In this article, we first study, in the framework of operator theory, Pusz and Woronowiczs functional calculus for pairs of bounded positive operators on Hilbert spaces associated with a homogeneous two-variable function on $[0,infty)^2$. Our construction has special features that functions on $[0,infty)^2$ are assumed only locally bounded from below and that the functional calculus is allowed to take extended semibounded self-adjoint operators. To analyze convexity properties of the functional calculus, we extend the notion of operator convexity for real functions to that for functions with values in $(-infty,infty]$. Based on the first part, we generalize the concept of operator convex perspectives to pairs of (not necessarily invertible) bounded positive operators associated with any operator convex function on $(0,infty)$. We then develop theory of such operator convex perspectives, regarded as an operator convex counterpart of Kubo and Andos theory of operator means. Among other results, integral expressions and axiomatization are discussed for our operator perspectives.
We present an extension of some results of higher order calculus of variations and optimal control to generalized functions. The framework is the category of generalized smooth functions, which includes Schwartz distributions, while sharing many nonlinear properties with ordinary smooth functions. We prove the higher order Euler-Lagrange equations, the DAlembert principle in differential form, the du Bois-Reymond optimality condition and the Noethers theorem. We start the theory of optimal control proving a weak form of the Pontryagin maximum principle and the Noethers theorem for optimal control. We close with a study of a singularly variable length pendulum, oscillations damped by two media and the Pais-Uhlenbeck oscillator with singular frequencies.
In this paper, two related types of dualities are investigated. The first is the duality between left-invertible operators and the second is the duality between Banach spaces of vector-valued analytic functions. We will examine a pair ($mathcal{B},Psi)$ consisting of a reflexive Banach spaces $mathcal{B}$ of vector-valued analytic functions on which a left-invertible multiplication operator acts and an operator-valued holomorphic function $Psi$. We prove that there exist a dual pair ($mathcal{B}^prime,Psi^prime)$ such that the space $mathcal{B}^prime$ is unitarily equivalent to the space $mathcal{B}^*$ and the following intertwining relations hold begin{equation*} mathscr{L} mathcal{U} = mathcal{U}mathscr{M}_z^* quadtext{and}quad mathscr{M}_zmathcal{U} = mathcal{U} mathscr{L}^*, end{equation*} where $mathcal{U}$ is the unitary operator between $mathcal{B}^prime$ and $mathcal{B}^*$. In addition we show that $Psi$ and $Psi^prime$ are connected through the relationbegin{equation*} langle(Psi^prime( bar{z}) e_1) (lambda),e_2rangle= langle e_1,(Psi( bar{ lambda}) e_2)(z)rangle end{equation*} for every $e_1,e_2in E$, $zin varOmega$, $lambdain varOmega^prime$. If a left-invertible operator $T$ satisfies certain conditions, then both $T$ and the Cauchy dual operator $T^prime$ can be modelled as a multiplication operator on reproducing kernel Hilbert spaces of vector-valued analytic functions $mathscr{H}$ and $mathscr{H}^prime$, respectively. We prove that Hilbert space of the dual pair of $(mathscr{H},Psi)$ coincide with $mathscr{H}^prime$, where $Psi$ is a certain operator-valued holomorphic function. Moreover, we characterize when the duality between spaces $mathscr{H}$ and $mathscr{H}^prime$ obtained by identifying them with $mathcal{H}$ is the same as the duality obtained from the Cauchy pairing.
We present an extension of the classical theory of calculus of variations to generalized functions. The framework is the category of generalized smooth functions, which includes Schwartz distributions while sharing many nonlinear properties with ordinary smooth functions. We prove full connections between extremals and Euler-Lagrange equations, classical necessary and sufficient conditions to have a minimizer, the necessary Legendre condition, Jacobis theorem on conjugate points and Noethers theorem. We close with an application to low regularity Riemannian geometry.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
