Topics on operator inequalities

This paper has two main goals. First, we are concerned with a description of all self-adjoint extensions of the Laplacian $$ - \Delta {|_{C_0^\infty (\Omega )}}$$
 in L
 2(Ω; d
 n
 x). Here, the domain Ω belongs to a subclass of bounded Lipschitz domains (which we term quasi-convex domains), that contains all convex domains as well as all domains of class C
 1,r
 , for r > 1/2. Second, we establish Kreĭn-type formulas for the resolvents of the various self-adjoint extensions of the Laplacian in quasiconvex domains and study the well-posedness of boundary value problems for the Laplacian as well as basic properties of the corresponding Weyl-Titchmarsh operators (or energy-dependent Dirichlet-to-Neumann maps). One significant innovation in this paper is an extension of the classical boundary trace theory for functions in spaces that lack Sobolev regularity in a traditional sense, but are suitably adapted to the Laplacian.

A description of all self-adjoint extensions of the Laplacian and Kreĭn-type resolvent formulas on non-smooth domains

We systematically develop Weyl-Titchmarsh theory for singular differential operators on arbitrary intervals $(a,b) \subseteq \mathbb{R}$ associated with rather general differential expressions of the type \[ 
\tau f = \frac{1}{r} (- \big(p[f' + s f]\big)' + s p[f' + s f] + qf),] where the coefficients $p$, $q$, $r$, $s$ are real-valued and Lebesgue measurable on $(a,b)$, with $p\neq 0$, $r>0$ a.e.\ on $(a,b)$, and $p^{-1}$, $q$, $r$, $s \in L^1_{\text{loc}}((a,b); dx)$, and $f$ is supposed to satisfy [f \in AC_{\text{loc}}((a,b)), \; p[f' + s f] \in AC_{\text{loc}}((a,b)).] In particular, this setup implies that $\tau$ permits a distributional potential coefficient, including potentials in $H^{-1}_{\text{loc}}((a,b))$. 
We study maximal and minimal Sturm-Liouville operators, all self-adjoint restrictions of the maximal operator $T_{\text{max}}$, or equivalently, all self-adjoint extensions of the minimal operator $T_{\text{min}}$, all self-adjoint boundary conditions (separated and coupled ones), and describe the resolvent of any self-adjoint extension of $T_{\text{min}}$. In addition, we characterize the principal object of this paper, the singular Weyl-Titchmarsh-Kodaira $m$-function corresponding to any self-adjoint extension with separated boundary conditions and derive the corresponding spectral transformation, including a characterization of spectral multiplicities and minimal supports of standard subsets of the spectrum. We also deal with principal solutions and characterize the Friedrichs extension of $T_{\text{min}}$. 
Finally, in the special case where $\tau$ is regular, we characterize the Krein-von Neumann extension of $T_{\text{min}}$ and also characterize all boundary conditions that lead to positivity preserving, equivalently, improving, resolvents (and hence semigroups).

Weyl-Titchmarsh Theory for Sturm-Liouville Operators with Distributional Potentials

Spectral theory for perturbed Krein Laplacians in nonsmooth domains

We systematically develop Weyl-Titchmarsh theory for singular differential operators on arbitrary intervals $(a,b) \subseteq \mathbb{R}$ associated with rather general differential expressions of the type \begin{equation*}\tau f = \frac{1}{\tau} (-(p[f'+sf])'+sp[f'+sf]+qf),\end{equation*} where the coefficients $p, q, r, s$ are real-valued and Lebesgue measurable on $(a,b)$, with $p \neq 0$, $r \gt 0$ a.e. on $(a,b)$, and $p^{-1}, q, r, s \in L_{loc}^1((a,b),dx)$, and $f$ is supposed to satisfy \begin{equation*} f \in AC_{loc}((a,b)), p[f'+sf] \in AC_{loc}((a,b)). \end{equation*}
In particular, this setup implies that $\tau$ permits a distributional potential coefficient, including potentials in $H_{loc}^{-1}((a,b))$. We study maximal and minimal Sturm-Liouville operators, all self-adjoint restrictions of the maximal operator $T_{max}$, or equivalently, all self-adjoint extensions of the minimal operator $T_{min}$, all self-adjoint boundary conditions (separated and coupled ones), and describe the resolvent of any self-adjoint extension of $T_{min}$. In addition, we characterize the principal object of this paper, the singular Weyl-Titchmarsh-Kodaira m-function corresponding to any self-adjoint extension with separated boundary conditions and derive the corresponding spectral transformation, including a characterization of spectral multiplicities and minimal supports of standard subsets of the spectrum. We also deal with principal solutions and characterize the Friedrichs extension of $T_{min}$. Finally, in the special case where $\tau$ is regular, we characterize the Krein-von Neumann extension of $T_{min}$ and also characterize all boundary conditions that lead to positivity preserving, equivalently, improving, 
resolvents (and hence semigroups).

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Weyl-Titchmarsh theory for Sturm-Liouville operators with distributional potentials

We are going to consider the M Kreĭn classical papers on the theory of semi-bounded operators and the theory of contractive self-adjoint extensions of Hermitian contractions, and discuss their impact and role in the solution of J von Neumann’s problem about parametrization in terms of his formulas of all nonnegative self-adjoint extensions of nonnegative symmetric operators, in the solution of the Phillips-Kato extension problems (in restricted sense) about existence and parametrization of all proper sectorial (accretive) extensions of nonnegative operators, in bi-extension theory of non-negative operators with the exit into triplets of Hilbert spaces, in the theory of singular perturbations of nonnegative self-adjoint operators, in general realization problems (in system theory) of Stieltjes matrix-valued functions, in Nevanlinna-Pick system interpolation in the class of sectorial Stieltjes functions, in conservative systems theory with accretive main Schrodinger operator, in the theory of semi-bounded symmetric and self-adjoint operators invariant with respect to some groups of transformations New developments and applications to the singular differential operators are discussed as well

E. Tsekanovskiĭ

Papers

M. Kreĭn’s Research on Semi-Bounded Operators, its Contemporary Developments, and Applications