Toka Diagana
This paper is concerned with the square root problem of Kato for the "sum" of linear operators in a Hilbert space ${\mathbb H}$. Under suitable assumptions, we show that if $A$ and $B$ are respectively m-scetroial linear operators satisfying the square root problem of Kato. Then the same conclusion still holds for their "sum". As application, we consider perturbed Schrödinger operators.
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We prove that the algebraic sum of unbounded normal operators satisfies the square root problem of Kato under appropriate hypotheses. As application, we consider perturbed Schrodinger operators.
In the infinite-dimensional separable complex Hilbert space we construct new abstract examples of unbounded maximal accretive and maximal sectorial operators $B$ for which ${\rm dom\,}B^{\frac{1}{2}}\ne{\rm dom\,}B^{*{\frac{1}{2}}}$. New criterions for the equality are established.
We consider the negative Laplacian subject to mixed boundary conditions on a bounded domain. We prove under very general geometric assumptions that slightly above the critical exponent $\frac{1}{2}$ its fractional power domains still coincide with suitable Sobolev spaces of optimal regularity. In combination with a reduction theorem recently obtained by the authors, this solves the Kato Square Root Problem for elliptic second order operators and systems in divergence form under the same geometric assumptions.
On a domain $Ω\subseteq \mathbb{R}^d$ we consider second order elliptic systems in divergence form with bounded complex coefficients, realized via a sesquilinear form with domain $V \subseteq H^1(Ω)$. Under very mild assumptions on $Ω$ and $V$ we show that the Kato Square Root Problem for such systems can be reduced to a regularity result for the fractional powers of the negative Laplacian in the same geometric setting. This extends an earlier result of McIntosh to non-smooth coefficients.
We obtain the Kato square root estimate for second order elliptic operators in divergence form with mixed boundary conditions on an open and possibly unbounded set in $\mathbb{R}^d$ under two simple geometric conditions: The Dirichlet boundary part is Ahlfors--David regular and a quantitative connectivity property in the spirit of locally uniform domains holds near the Neumann boundary part. This improves upon all existing results even in the case of pure Dirichlet or Neumann boundary conditions. We also treat elliptic systems with lower order terms. As a side product we establish new regulari…
The Kato square root problem for divergence form elliptic operators with potential $V : \mathbb{R}^{n} \rightarrow \mathbb{C}$ is the equivalence statement $\left\Vert (L + V)^{\frac{1}{2}} u\right\Vert_{2} \simeq \left\Vert \nabla u \right\Vert_{2} + \left\Vert V^{\frac{1}{2}} u \right\Vert_{2}$, where $L + V := - \mathrm{div} A \nabla + V$ and the perturbation $A$ is an $L^{\infty}$ complex matrix-valued function satisfying an accretivity condition. This relation is proved for any potential with range contained in some positive sector and satisfying $\left\Vert |V|^{\fracα{2}} u\right\Vert_{…
We solve the Kato square root problem for general elliptic operators and systems with measurable and complex coefficients on any domain of the Euclidean space. The operators are subject to Dirichlet boundary conditions. We also allow perturbations by general complex potentials.
The Kato square root estimate with Robin boundary conditions
本文研究了带有Robin边界条件的Kato平方根估计，针对二阶散度形式椭圆算子在局部均匀域上的估计问题，提出了新的分析方法。
Let $L$ be an elliptic differential operator with bounded measurable coefficients, acting in Bochner spaces $L^{p}(R^{n};X)$ of $X$-valued functions on $R^n$. We characterize Kato's square root estimates $\|\sqrt{L}u\|_{p} \eqsim \|\nabla u\|_{p}$ and the $H^{\infty}$-functional calculus of $L$ in terms of R-boundedness properties of the resolvent of $L$, when $X$ is a Banach function lattice with the UMD property, or a noncommutative $L^{p}$ space. To do so, we develop various vector-valued analogues of classical objects in Harmonic Analysis, including a maximal function for Bochner spaces. I…
Let $w$ be a Muckenhoupt $A_2(\mathbb{R}^n)$ weight and $L_w:=-w^{-1}\mathop\mathrm{div}(A\nabla)$ the degenerate elliptic operator on the Euclidean space $\mathbb{R}^n$, $n\geq 2$. In this article, the authors establish some weighted $L^p$ estimates of Kato square roots associated to the degenerate elliptic operators $L_w$. More precisely, the authors prove that, for $w\in A_{p}(\mathbb{R}^n)$, $p\in(\frac{2n}{n+1},\,2]$ and any $f\in C^\infty_c(\mathbb{R}^n)$, $\|L_w^{1/2}(f)\|_{L^p(w,\,\mathbb{R}^n)}\sim \|\nabla f\|_{L^p(w,\,\mathbb{R}^n)}$, where $C_c^\infty(\mathbb{R}^n)$ denotes the set…
On Kato's Square Root Property for the Generalized Stokes Operator
本文证明 $\mathbb{R}^d$ 上带有界可测椭圆系数的广义 Stokes 算子具有 Kato 平方根性质，刻画其平方根定义域并得到梯度等价估计，还推出平方根对系数的全纯依赖性。
A second order approach to the Kato square root problem on open sets
本文针对开集上带混合边界条件的耦合二阶椭圆系统，利用 Ahlfors–David 正则性与定量连通性条件，以二阶方法证明 Kato 平方根性质，从而获得更短、技术更简单的证明。
We study the persistence of quadratic estimates related to the Kato square root problem across a change of metric on smooth manifolds by defining a class of Riemannian-like metrics that are permitted to be of low regularity and degenerate on sets of measure zero. We also demonstrate how to transmit quadratic estimates between manifolds which are homeomorphic and locally bi-Lipschitz. As a consequence, we demonstrate the invariance of the Kato square root problem under Lipschitz transformations of the space and obtain solutions to this problem on functions and forms on compact manifolds with a …
Let L(t) = --div (A(x, t)$\nabla$ x) for t $\in$ (0, $τ$) be a uniformly elliptic operator with boundary conditions on a domain $Ω$ of R d and $\partial$ = $\partial$ $\partial$t. Define the parabolic operator L = $\partial$ + L on L 2 (0, $τ$, L 2 ($Ω$)) by (Lu)(t) := $\partial$u(t) $\partial$t + L(t)u(t). We assume a very little of regularity for the boundary of $Ω$ and assume that the coefficients A(x, t) are measurable in x and piecewise C $α$ in t for some $α$ > 1 2. We prove the Kato square root property for $\sqrt$ L and the estimate $\sqrt$ L u L 2 (0,$τ$,L 2 ($Ω$)) $\approx$ $\nabla$ …
We provide a direct proof of a quadratic estimate that plays a central role in the determination of domains of square roots of elliptic operators and, as shown more recently, in some boundary value problems with $L^2$ boundary data. We develop the application to the Kato conjecture and to a Neumann problem. This quadratic estimate enjoys some equivalent forms in various settings. This gives new results in the functional calculus of Dirac type operators on forms.
We provide first order perturbation formulas for the matrix square root (in the positive semi-definite case) and the matrix modulus (in the general case). The results are new for singular matrices, and extend previously known Fréchet differentiability formulas provided by the Daleckii-Krein theorem.
In this paper we give complete descriptions of the set of square roots of certain classical operators, often providing specific formulas. The classical operators included in this discussion are the square of the unilateral shift, the Volterra operator, certain compressed shifts, the unilateral shift plus its adjoint, the Hilbert matrix, and the Cesàro operator.
We briefly review results on Colombeau type generalized solutions to the Cauchy problem for linear Schrödinger-type equations with non-smooth principal part and their compatibility with classical and distributional solutions. In the main part, we study convergence properties of regularized solutions to the standard Schrödinger equation with initial values corresponding to "square roots" of Dirac measures in various duals of classical subspaces of the space of continuous functions.
The primary purpose of this paper is to show the existence of normal square and nth roots of some classes of bounded operators on Hilbert spaces. Two interesting simple results hold. Namely, under simple conditions we show that if any operator $T$ is such that $T^2=0$, then this implies that $T$ is normal and so $T=0$. Also, we will see when the square root of an arbitrary bounded operator is normal.
In this article we show that the semigroup operation of a strictly linearly ordered semigroup on a real interval is automatically continuous if each element of the semigroup admits a square root. Hence, by a result of Aczél, such a semigroup is isomorphic to an additive subsemigroup of the real numbers.
In this paper, we are mainly concerned with studying arbitrary unbounded square roots of linear operators as well as some of their basic properties. The paper contains many examples and counterexamples. As an illustration, we give explicit everywhere defined unbounded non-closable $nth$ roots of the identity operator as well as the zero operator. We also show a non-closable unbounded operator without any non-closable square root. Among other consequences, we have a way of finding everywhere defined bijective operators, everywhere defined operators which are surjective without being injective a…
L.Accardi shows that the Banach space of singed measures is homeomorphic to the Hilbert space formed by so-called root measures. In this paper, we redefine root measures in view of the theory of measures on infinite dimensional spaces, and introduce a notion of differentiation, Fourier transform, and convolution product for root measures, and examine those relations. We also study about Lévy Laplacian on Wiener space as application. It is shown that the symbol of Lévy Laplacian is equal to the quadratic variation of paths.
Square Root Operators and the Well-Posedness of Pseudodifferential Parabolic Models of Wave Phenomena
本文研究了伪微分抛物型方程中平方根算子的严格定义及其适定性，通过强连续半群理论证明了该方程的适定性，并为波场模拟提供了理论支持。
On root mean square approximation by exponential functions
研究了用有限线性组合的指数函数对平方可积函数进行均方逼近问题，解决线性部分并探讨非线性情况的特定例子。
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