Functional calculus
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The spectral theorem for quaternionic unbounded normal operators based on the <i>S</i>-spectrum Open
In this paper we prove the spectral theorem for quaternionic unbounded normal operators using the notion of S-spectrum. The proof technique consists of first establishing a spectral theorem for quaternionic bounded normal operators and the…
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Fine scales of decay of operator semigroups Open
Motivated by potential applications to partial differential equations, we develop a theory of fine scales of decay rates for operator semigroups. The theory contains, unifies, and extends several notable results in the literature on decay …
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Functional Itô versus Banach space stochastic calculus and strict solutions of semilinear path-dependent equations Open
Functional Itô calculus was introduced in order to expand a functional [Formula: see text] depending on time [Formula: see text], past and present values of the process [Formula: see text]. Another possibility to expand [Formula: see text]…
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General Fractional Calculus in Multi-Dimensional Space: Riesz Form Open
An extension of the general fractional calculus (GFC) is proposed as a generalization of the Riesz fractional calculus, which was suggested by Marsel Riesz in 1949. The proposed Riesz form of GFC can be considered as an extension GFC from …
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Paley-Littlewood decomposition for sectorial operators and interpolation spaces Open
We prove Paley-Littlewood decompositions for the scales of fractional powers\nof $0$-sectorial operators $A$ on a Banach space which correspond to\nTriebel-Lizorkin spaces and the scale of Besov spaces if $A$ is the classical\nLaplace oper…
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On multidimensional Bochner-Phillips functional calculus Open
The functional calculus of semigroup generators, based on the class of Bernstein functions in several variables is developed, the condition for holomorphy of semigroups, generated by operators which arisen in the calculus is given, and in …
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Operator-Lipschitz estimates for the singular value functional calculus Open
We consider a functional calculus for compact operators, acting on the\nsingular values rather than the spectrum, which appears frequently in applied\nmathematics. Necessary and sufficient conditions for this singular value\nfunctional cal…
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A Besov algebra calculus for generators of operator semigroups and related norm-estimates Open
We construct a new bounded functional calculus for the generators of bounded semigroups on Hilbert spaces and generators of bounded holomorphic semigroups on Banach spaces. The calculus is a natural (and strict) extension of the classical …
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Functions of the infinitesimal generator of a strongly continuous quaternionic group Open
The quaternionic analogue of the Riesz–Dunford functional calculus and the theory of semigroups and groups of linear quaternionic operators have recently been introduced and studied. In this paper, we suppose that [Formula: see text] is th…
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Connecting quantum calculus and harmonic starlike functions Open
Quantum calculus or q-calculus plays an important role in hypergeometric series, quantum physics, operator theory, approximation theory, sobolev spaces, geometric functions theory and others. But role of q-calculus in the theory of harmoni…
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Applications of Symmetric Quantum Calculus to the Class of Harmonic Functions Open
In the past few years, many scholars gave much attention to the use of q-calculus in geometric functions theory, and they defined new subclasses of analytic and harmonic functions. While using the symmetric q-calculus in geometric function…
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Universality property of the S-functional calculus, noncommuting matrix variables and Clifford operators Open
Spectral theory on the S-spectrum was born out of the need to give quaternionic quantum mechanics a precise mathematical foundation (Birkhoff and von Neumann [8] showed that a general set theoretic formulation of quantum mechanics can be r…
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The Fine Structure of the Spectral Theory on the S-Spectrum in Dimension Five Open
Holomorphic functions play a crucial role in operator theory and the Cauchy formula is a very important tool to define the functions of operators. The Fueter–Sce–Qian extension theorem is a two-step procedure to extend holomorphic function…
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Harmonic univalent functions defined by q-calculus operators Open
The fractional q-calculus is the q-extension of the ordinary fractional calculus and dates back to early 20-th century. The theory of q-calculus operators are used in various areas of science such as ordinary fractional calculus, optimal c…
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Harmonic univalent functions defined by post quantum calculus operators Open
We study a family of harmonic univalent functions in the open unit disc defined by using post quantum calculus operators. We first obtained a coefficient characterization of these functions. Using this, coefficients estimates, distortion a…
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Axially Harmonic Functions and the Harmonic Functional Calculus on the S-spectrum Open
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…
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Calderón reproducing formulas and applications to Hardy spaces Open
We establish new Calderón holomorphic functional calculus whilst the synthesising function interacts with D through functional calculus based on the Fourier transform. We apply these to prove the embedding H^p_D(\wedge T^*M) \subseteq L^p(…
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An Introduction to Hyperholomorphic Spectral Theories and Fractional Powers of Vector Operators Open
The aim of this paper is to give an overview of the spectral theories associated with the notions of holomorphicity in dimension greater than one. A first natural extension is the theory of several complex variables whose Cauchy formula is…
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Joint Spectral Multipliers for Mixed Systems of Operators Open
We obtain a general Marcinkiewicz-type multiplier theorem for mixed systems of strongly commuting operators $$L=(L_1,\ldots ,L_d);$$ where some of the operators in L have only a holomorphic functional calculus, while others have additional…
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Multivariate trace inequalities, p-fidelity, and universal recovery beyond tracial settings Open
Trace inequalities are general techniques with many applications in quantum information theory, often replacing the classical functional calculus in noncommutative settings. The physics of quantum field theory and holography, however, moti…
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FUNCTIONAL CALCULI FOR SECTORIAL OPERATORS AND RELATED FUNCTION THEORY Open
We construct two bounded functional calculi for sectorial operators on Banach spaces, which enhance the functional calculus for analytic Besov functions, by extending the class of functions, generalising and sharpening estimates and adapti…
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Oversmoothing Tikhonov regularization in Banach spaces <sup>*</sup> Open
This paper develops a Tikhonov regularization theory for nonlinear ill-posed operator equations in Banach spaces. As the main challenge, we consider the so-called oversmoothing state in the sense that the Tikhonov penalization is not able …
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Holomorphic functional calculus and vector-valued Littlewood–Paley–Stein theory for semigroups Open
We study vector-valued Littlewood–Paley–Stein theory for semigroups \{T_{t}\}_{t>0} of regular contractions on L_{p}(\Omega) for a fixed 10} is the Poisson semigroup subordinated to \{T_{t}\}_{t>0} . Let \mathsf{L}^{P}_{\mathsf{c}, q, p}(X…
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Fractional Calculus Operator and Certain Applications in Geometric Function Theory Open
Using a operator involving fractional calculus introduced by Owa and Srivastava [8], two novel families: $${\mathcal V}_{\delta}^{\alpha, \beta}(\nu;\gamma)\;\; \mbox{and} \;\;{\mathcal W}_{\delta}^{\alpha, \beta}(\mu;\gamma)$$ $$(\delta\n…
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Divided differences in noncommutative geometry: Rearrangement Lemma, functional calculus and expansional formula Open
We state a generalization of the Connes–Tretkoff–Moscovici Rearrangement Lemma and give a surprisingly simple (almost trivial) proof of it. Secondly, we put on a firm ground the multivariable functional calculus used implicitly in the Rear…
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H^∞-functional calculus for commuting families of Ritt operators and sectorial operators Open
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On functional calculus properties of Ritt operators Open
We compare various functional calculus properties of Ritt operators. We show the existence of a Ritt operator T: X → X on some Banach space X with the following property: T has a bounded H ∞ -functional calculus with respect to the unit di…
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Smooth perturbations of the functional calculus and applications to Riemannian geometry on spaces of metrics Open
We show that the functional calculus, which maps operators A to functionals f(A), is holomorphic for a certain class of operators A and holomorphic functions f. Using this result we are able to prove that fractional Laplacians depend real …
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Calculus in Locally Convex Spaces Open
It is well known that multidimensional calculus, aka Fréchet calculus, carries over to the realm of Banach spaces and Banach manifolds. Banach spaces are often not sufficient for our purposes. To generalise derivatives, we will, as a minim…
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Spectrum and Analytic Functional Calculus for Clifford Operators via Stem Functions Open
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 sp…