Ordered Representations of Normal and Super-Differential Operators in Quaternion and Octonion Hilbert Spaces

2009 ◽  
Vol 20 (2) ◽  
pp. 321-342 ◽  
Author(s):  
S. V. Ludkowski ◽  
W. Sprössig
Keyword(s):  
Hilbert Spaces ◽  
Differential Operators

Existence Theorems for Some Weak Abstract Variable Domain Hyperbolic Problems

1971 ◽  
Vol 23 (4) ◽  
pp. 611-626 ◽  
Author(s):  
Robert Carroll ◽  
Emile State
Keyword(s):  
Hilbert Spaces ◽  
Differential Operators ◽  
Separable Hilbert Space ◽  
Hyperbolic Equations ◽  
Existence Theorems ◽  
Hyperbolic Problems ◽  
Linear Maps ◽  
Variable Domains ◽  
Elliptic Differential Operators ◽  
Image Position

In this paper we prove some existence theorems for some weak problems with variable domains arising from hyperbolic equations of the type1.1where A = {A(t)} is, for example, a family of elliptic differential operators in space variables x = (x1, …, xn). Thus let H be a separable Hilbert space and let V(t) ⊂ H be a family of Hilbert spaces dense in H with continuous injections i(t): V(t) → H (0 ≦ t ≦ T < ∞). Let V’ (t) be the antidual of V(t) (i.e. the space of continuous conjugate linear maps V(t) → C) and using standard identifications one writes V(t) ⊂ H ⊂ V‘(t).

scholarly journals A Local-in-Time Theory for Singular SDEs with Applications to Fluid Models with Transport Noise

2021 ◽  
Vol 31 (6) ◽  
Author(s):  
Diego Alonso-Orán ◽  
Christian Rohde ◽  
Hao Tang
Keyword(s):  
Hilbert Spaces ◽  
Differential Operators ◽  
Approximation Property ◽  
Blow Up ◽  
Nonlinear Models ◽  
Local Theory ◽  
Cancellation Property ◽  
Singular Drift ◽  
Two Component ◽  
And Diffusion

AbstractWe establish a local theory, i.e., existence, uniqueness and blow-up criterion, for a general family of singular SDEs in Hilbert spaces. The key requirement relies on an approximation property that allows us to embed the singular drift and diffusion mappings into a hierarchy of regular mappings that are invariant with respect to the Hilbert space and enjoy a cancellation property. Various nonlinear models in fluid dynamics with transport noise belong to this type of singular SDEs. By establishing a cancellation estimate for certain differential operators of order one with suitable coefficients, we give the detailed constructions of such regular approximations for certain examples. In particular, we show novel local-in-time results for the stochastic two-component Camassa–Holm system and for the stochastic Córdoba–Córdoba–Fontelos model.

scholarly journals Global solutions of semilinear heat equations in Hilbert spaces

1996 ◽  
Vol 1 (3) ◽  
pp. 263-276 ◽  
Author(s):  
G. Mihai Iancu ◽  
M. W. Wong
Keyword(s):  
Asymptotic Behavior ◽  
Hilbert Spaces ◽  
Differential Operators ◽  
Global Solutions ◽  
Linear Operators ◽  
Heat Equations ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Strongly Continuous Semigroups ◽  
Pseudo Differential Operators

The existence, uniqueness, regularity and asymptotic behavior of global solutions of semilinear heat equations in Hilbert spaces are studied by developing new results in the theory of one-parameter strongly continuous semigroups of bounded linear operators. Applications to special semilinear heat equations inL 2(ℝn)governed by pseudo-differential operators are given.

On the instability of non-semi-Fredholm closed operators under compact perturbations with applications to ordinary differential operators

1988 ◽  
Vol 109 (1-2) ◽  
pp. 97-108 ◽  
Author(s):  
L. E. Labuschagne
Keyword(s):  
Banach Spaces ◽  
Hilbert Spaces ◽  
Differential Operators ◽  
Compact Operators ◽  
Fredholm Operators ◽  
Closed Operators ◽  
Compact Perturbations ◽  
Ordinary Differential Operators ◽  
The Stability

SynopsisThe stability of several natural subsets of the bounded non-semi-Fredholm operators undercompact perturbations were studied by R. Bouldin [2] in separable Hilbert spaces and by M. Gonzales and V. M. Onieva [6] in Banach spaces. The aim of this paper is to study this problem for closed operators in operator ranges. The main results are a characterisation of the non-semi-Fredholm operators with respect to α-closed and α-compact operators as well as a generalisation of a result of M. Goldman [5]. We also give some applications of the theory developed to ordinary differential operators.

Spectra of a class of non-symmetric operators in Hilbert spaces with applications to singular differential operators

2019 ◽  
Vol 150 (4) ◽  
pp. 1769-1790
Author(s):  
Huaqing Sun ◽  
Bing Xie
Keyword(s):  
Essential Spectrum ◽  
Hilbert Spaces ◽  
Symmetric Operator ◽  
Differential Operators ◽  
Homogeneous Equation ◽  
Fundamental Theory ◽  
Linearly Independent ◽  
Sturm Liouville ◽  
Symmetric Operators ◽  
Complex Valued

AbstractThis paper is concerned with a class of non-symmetric operators, that is, 𝒥-symmetric operators, in Hilbert spaces. A sufficient condition for λ ∈ C being an element of the essential spectrum of a 𝒥-symmetric operator is given in terms of the number of linearly independent solutions of a certain homogeneous equation, and a characterization for points of the essential spectrum plus the set of all eigenvalues of a 𝒥-symmetric operator is obtained in terms of the numbers of linearly independent solutions of certain inhomogeneous equations. As direct applications, the corresponding results are obtained for singular 𝒥-symmetric Hamiltonian systems and their special forms of singular Sturm-Liouville equations with complex-valued coefficients, which enable us to study the spectra of singular 𝒥-symmetric differential expressions using numerous tools available in the fundamental theory of differential equations.

scholarly journals ON THE HERMITICITY OF q-DIFFERENTIAL OPERATORS AND FORMS ON THE QUANTUM EUCLIDEAN SPACES $\mathbb{R}_q^N$

2006 ◽  
Vol 18 (01) ◽  
pp. 79-117 ◽  
Author(s):  
GAETANO FIORE
Keyword(s):  
Hilbert Spaces ◽  
Differential Operators ◽  
Pseudodifferential Operators ◽  
Special Functions ◽  
Radial Coordinate ◽  
Complicated Structure ◽  
Similarity Transformations ◽  
Special Lattice ◽  
Free Parameters ◽  
Scalar Products

We show that the complicated ⋆-structure characterizing for positive q the Uqso(N)-covariant differential calculus on the noncommutative manifold [Formula: see text] boils down to similarity transformations involving the ribbon element of a central extension of Uqso(N) and its formal square root ṽ. Subspaces of the spaces of functions and of p-forms on [Formula: see text] are made into Hilbert spaces by introducing non-conventional "weights" in the integrals defining the corresponding scalar products, namely suitable positive-definite q-pseudodifferential operators ṽ′±1 realizing the action of ṽ±1; this serves to make the partial q-derivatives anti-hermitean and the exterior coderivative equal to the hermitean conjugate of the exterior derivative, as usual. There is a residual freedom in the choice of the weight m(r) along the "radial coordinate" r. Unless we choose a constant m, then the square-integrables functions/forms must fulfill an additional condition, namely, their analytic continuations to the complex r plane can have poles only on the sites of some special lattice. Among the functions naturally selected by this condition there are q-special functions with "quantized" free parameters.

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