We study monotone operators in reflexive Banach spaces that are invariant with respect to a group of suitable isometric isomorphisms, and we show that they always admit a maximal extension which preserves the same invariance. A similar result applies to Lipschitz maps in Hilbert spaces, thus providing an invariant version of Kirszbraun–Valentine extension theorem. We then provide a relevant application to the case of monotone operators in $L^{p}$-spaces of random variables which are invariant with respect to measure-preserving isomorphisms, proving that they always admit maximal dissipative extensions which are still invariant by measure-preserving isomorphisms. We also show that such operators are law invariant, a much stronger property which is also inherited by their resolvents, the Moreau–Yosida approximations, and the associated semigroup of contractions. These results combine explicit representation formulae for the maximal extension of a monotone operator based on self-dual Lagrangians and a refined study of measure-preserving maps in standard Borel spaces endowed with a nonatomic measure, with applications to the approximation of arbitrary couplings between measures by sequences of maps.

Let $c_{kl} \in W^{1,\infty }(\Omega , \mathbb{C})$ for all $k,l \in \{1, \ldots , d\};$ and $\Omega \subset \mathbb{R}^{d}$ be open with uniformly $C^{2}$ boundary. We consider the divergence form operator $A_p = - \sum \nolimits _{k,l=1}^{d} \partial _l (c_{kl} \partial _k)$ in $L_p(\Omega )$ when the coefficient matrix satisfies $(C(x) \xi , \xi ) \in \Sigma _\theta$ for all $x \in \Omega$ and $\xi \in \mathbb{C}^{d}$, where $\Sigma _\theta$ be the sector with vertex 0 and semi-angle $\theta$ in the complex plane. We show that a sectorial estimate holds for $A_p$ for all $p$ in a suitable range. We then apply these estimates to prove that the closure of $-A_p$ generates a holomorphic semigroup under further assumptions on the coefficients. The contractivity and consistency properties of these holomorphic semigroups are also considered.

We consider Kolmorogov operator $-\Delta +b \cdot \nabla $ with drift b in the class of form-bounded vector fields (containing vector fields having critical-order singularities). We characterize quantitative dependence of the Sobolev and Hölder regularity of solutions to the corresponding elliptic equation on the value of the form-bound of b.

This paper is concerned with the modified Wigner (respectively, Wigner–Fokker–Planck) Poisson equation. The quantum mechanical model describes the transport of charged particles under the influence of the modified Poisson potential field without (respectively, with) the collision operator. Existence and uniqueness of a global mild solution to the initial boundary value problem in one dimension are established on a weighted $L^{2}$ -space. The main difficulties are to derive a priori estimates on the modified Poisson equation and prove the Lipschitz properties of the appropriate potential term.

We improve a previous result about the local energy decay for the damped wave equation on $\mathbb{R}^{d}$. The problem is governed by a Laplacian associated with a long-range perturbation of the flat metric and a short-range absorption index. Our purpose is to recover the decay ${\mathcal{O}}(t^{-d+\unicode[STIX]{x1D700}})$ in the weighted energy spaces. The proof is based on uniform resolvent estimates, given by an improved version of the dissipative Mourre theory. In particular, we have to prove the limiting absorption principle for the powers of the resolvent with inserted weights.

Assuming $T_{0}$ to be an m-accretive operator in the complex Hilbert space ${\mathcal{H}}$, we use a resolvent method due to Kato to appropriately define the additive perturbation $T=T_{0}+W$ and prove stability of square root domains, that is,

$$\begin{eqnarray}\text{dom}((T_{0}+W)^{1/2})=\text{dom}(T_{0}^{1/2}).\end{eqnarray}$$

$\text{dom}(T_{0}^{1/2})=\text{dom}((T_{0}^{\ast })^{1/2})$

$$\begin{eqnarray}\text{dom}((T_{0}+W)^{1/2})=\text{dom}(T_{0}^{1/2})=\text{dom}((T_{0}^{\ast })^{1/2})=\text{dom}(((T_{0}+W)^{\ast })^{1/2}),\end{eqnarray}$$

$$\begin{eqnarray}-\text{div}(a{\rm\nabla}\,\cdot )+(\vec{B}_{1}\cdot {\rm\nabla}\,\cdot )+\text{div}(\vec{B}_{2}\,\cdot )+V\end{eqnarray}$$

$L^{2}({\rm\Omega})$

${\rm\Omega}\subseteq \mathbb{R}^{n}$

$n\in \mathbb{N}$

$\partial {\rm\Omega}$

$\partial {\rm\Omega}$

We solve the Kato square root problem for second order elliptic systems in divergence form under mixed boundary conditions on Lipschitz domains. This answers a question posed by Lions in 1962. To do this we develop a general theory of quadratic estimates and functional calculi for complex perturbations of Dirac-type operators on Lipschitz domains.

We obtain a new representation for the solution to the operator Sylvester equation in the form of a Stieltjes operator integral. We also formulate new sufficient conditions for the strong solvability of the operator Riccati equation that ensures the existence of reducing graph subspaces for block operator matrices. Next, we extend the concept of the Lifshits-Krein spectral shift function associated with a pair of self-adjoint operators to the case of pairs of admissible operators that are similar to self-adjoint operators. Based on this new concept we express the spectral shift function arising in a perturbation problem for block operator matrices in terms of the angular operators associated with the corresponding perturbed and unperturbed eigenspaces.

In this paper we discuss the asymptotic behaviour, as t → ∞, of the integral solution u(t) of the non-linear evolution equation where {A(t)}t≥0 is a family of m-dissipative operators in a Hilbert space H, and g ∈ Lloc (0, ∞ H).We give some sufficient conditions and some sufficient and necessary conditions to ensure that are weakly convergent.

We show that the Kato conjecture is true for m-accretive operators with highly singular coefficients. For operators of the form A = *F, where formally corresponds to d/dx + zδ on L2 (R), we prove that Dom (A1/2) = Dom() = e-zHH1(R) where H is the Heavysied function. By adapting recent methods of Auscher and Tchamitchian, we characterize Dom (A) in terms of an unconditional wavelet basis for L2(R).