Aɛ-bounded, finite rank perturbations of s.c. group generators A: Counterexamples to generation and to another condition for well-posedness

1984 ◽  
pp. 227-253 ◽  
Author(s):  
R. Triggiani
Keyword(s):  
Finite Rank ◽  
Well Posedness

Jordan Canonical Form for Finite Rank Elements in Jordan Algebras

1997 ◽  
Vol 260 (1-3) ◽  
pp. 151-167
Author(s):  
A López
Keyword(s):  
Canonical Form ◽  
Finite Rank ◽  
Jordan Algebras ◽  
Jordan Canonical Form ◽  
Finite Rank Elements

Cauchy Problem for Equations with Fractional Differentiation Bessel Operator in the Space of Temperate Distributions

2003 ◽  
Vol 8 (1) ◽  
pp. 61-75
Author(s):  
V. Litovchenko
Keyword(s):  
Functional Analysis ◽  
Cauchy Problem ◽  
Fundamental Solution ◽  
Initial Function ◽  
Well Posedness ◽  
Fractional Differentiation ◽  
Basic Tool ◽  
Conjugate Space ◽  
Analysis Technique ◽  
The Cauchy Problem

The well-posedness of the Cauchy problem, mentioned in title, is studied. The main result means that the solution of this problem is usual C∞ - function on the space argument, if the initial function is a real functional on the conjugate space to the space, containing the fundamental solution of the corresponding problem. The basic tool for the proof is the functional analysis technique.

WELL-POSEDNESS AND NUMERICAL SOLUTION OF A NONLINEAR VOLTERRA PARTIAL INTEGRO-DIFFERENTIAL EQUATION MODELING A SWELLING POROUS MATERIAL

2014 ◽  
Vol 17 (9) ◽  
pp. 763-784 ◽  
Author(s):  
Keith J. Wojciechowski ◽  
Jinhai Chen ◽  
Lynn Schreyer-Bennethum ◽  
Kristian Sandberg
Keyword(s):  
Differential Equation ◽  
Porous Material ◽  
Numerical Solution ◽  
Equation Modeling ◽  
Well Posedness ◽  
Integro Differential Equation ◽  
Differential Equation Modeling

scholarly journals The BV solution of the parabolic equation with degeneracy on the boundary

2016 ◽  
Vol 14 (1) ◽  
pp. 272-282
Author(s):  
Huashui Zhan ◽  
Shuping Chen
Keyword(s):  
Boundary Condition ◽  
Parabolic Equation ◽  
Well Posedness ◽  
Test Functions ◽  
The Stability

AbstractConsider a parabolic equation which is degenerate on the boundary. By the degeneracy, to assure the well-posedness of the solutions, only a partial boundary condition is generally necessary. When 1 ≤ α < p – 1, the existence of the local BV solution is proved. By choosing some kinds of test functions, the stability of the solutions based on a partial boundary condition is established.

scholarly journals Rigorous derivation and well-posedness of a quasi-homogeneous ideal MHD system

2021 ◽  
Vol 60 ◽  
pp. 103284
Author(s):  
Dimitri Cobb ◽  
Francesco Fanelli
Keyword(s):  
Well Posedness ◽  
Homogeneous Ideal ◽  
Rigorous Derivation ◽  
Ideal Mhd ◽  
Mhd System

scholarly journals Dimension of the Exceptional Set in the Aronszajn–Donoghue Theorem for Finite Rank Perturbations

2020 ◽  
Author(s):  
Constanze Liaw ◽  
Sergei Treil ◽  
Alexander Volberg
Keyword(s):  
Finite Rank ◽  
Measure Zero ◽  
Adjoint Operator ◽  
Cyclic Vector ◽  
Spectral Measures ◽  
Hermitian Matrices ◽  
Exceptional Set ◽  
Direct Sum ◽  
Rank One Perturbation ◽  
Rank One

Abstract The classical Aronszajn–Donoghue theorem states that for a rank-one perturbation of a self-adjoint operator (by a cyclic vector) the singular parts of the spectral measures of the original and perturbed operators are mutually singular. As simple direct sum type examples show, this result does not hold for finite rank perturbations. However, the set of exceptional perturbations is pretty small. Namely, for a family of rank $d$ perturbations $A_{\boldsymbol{\alpha }}:= A + {\textbf{B}} {\boldsymbol{\alpha }} {\textbf{B}}^*$, ${\textbf{B}}:{\mathbb C}^d\to{{\mathcal{H}}}$, with ${\operatorname{Ran}}{\textbf{B}}$ being cyclic for $A$, parametrized by $d\times d$ Hermitian matrices ${\boldsymbol{\alpha }}$, the singular parts of the spectral measures of $A$ and $A_{\boldsymbol{\alpha }}$ are mutually singular for all ${\boldsymbol{\alpha }}$ except for a small exceptional set $E$. It was shown earlier by the 1st two authors, see [4], that $E$ is a subset of measure zero of the space $\textbf{H}(d)$ of $d\times d$ Hermitian matrices. In this paper, we show that the set $E$ has small Hausdorff dimension, $\dim E \le \dim \textbf{H}(d)-1 = d^2-1$.

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