Let $A:D(A)\subseteq\H\to\H$ be an injective self-adjoint operator and let $τ:D(A)\to\X$, X a Banach space, be a surjective linear map such that $\|τϕ\|_\X\le c \|Aϕ\|_\H$. Supposing that \text{\rm Range}$ (τ')\cap\H' =\{0\}$, we define a family $A^τ_Θ$ of self-adjoint operators which are extensions of the symmetric operator $A_{|\{τ=0\}.}$. Any $ϕ$ in the operator domain $D(A^τ_Θ)$ is characterized by a sort of boundary conditions on its univocally defined regular component $\phireg$, which belongs to the completion of D(A) w.r.t. the norm $\|Aϕ\|_\H$. These boundary conditions are written in terms of the map $τ$, playing the role of a trace (restriction) operator, as $τ\phireg=ΘQ_ϕ$, the extension parameter $Θ$ being a self-adjoint operator from X' to X. The self-adjoint extension is then simply defined by $A^τ_Θϕ:=A \phireg$. The case in which $Aϕ=T*ϕ$ is a convolution operator on LD, T a distribution with compact support, is studied in detail.
Given a self-adjoint operator $A:D(A)\subseteq\calH\to\calH$ and a continuous linear operator $τ:D(A)\to\X$ with Range$ τ'\cap\calH' ={0}$, $\X$ a Banach space, we explicitly construct a family $A^τ_Θ$ of self-adjoint operators such that any $A^τ_Θ$ coincides with the original $A$ on the kernel of $τ$. Such a family is obtained by giving a Kre\uın-like formula where the role of the deficiency spaces is played by the dual pair $(\X,\X')$; the parameter $Θ$ belongs to the space of symmetric operators from $\X'$ to $\X$. When $\X=\C$ one recovers the ``$\calH_{-2}$ -construction'' of Kiselev and …
This paper studies stability of essential spectra of self-adjoint subspaces (i.e., self-adjoint linear relations) under finite rank and compact perturbations in Hilbert spaces. Relationships between compact perturbation of closed subspaces and relatively compact perturbation of their operator parts are first established. This gives a characterization of compact perturbation in terms of difference between the operator parts of perturbed and unperturbed subspaces. It is shown that a self-adjoint subspace is still self-adjoint under either relatively bounded perturbation with relative bound less …
Let $A_\N$ be the symmetric operator given by the restriction of $A$ to $\N$, where $A$ is a self-adjoint operator on the Hilbert space $\H$ and $\N$ is a linear dense set which is closed with respect to the graph norm on $D(A)$, the operator domain of $A$. We show that any self-adjoint extension $A_Θ$ of $A_\N$ such that $D(A_Θ)\cap D(A)=\N$ can be additively decomposed by the sum $A_Θ=\A+T_Θ$, where both the operators $\A$ and $T_Θ$ take values in the strong dual of $D(A)$. The operator $\A$ is the closed extension of $A$ to the whole $\H$ whereas $T_Θ$ is explicitly written in terms of a (a…
Given the symmetric operator $A_N$ obtained by restricting the self-adjoint operator $A$ to $N$, a linear dense set, closed with respect to the graph norm, we determine a convenient boundary triple for the adjoint $A_N^*$ and the corresponding Weyl function. These objects provide us with the self-adjoint extensions of $A_N$ and their resolvents.
Assume that $T$ is a self-adjoint operator on a Hilbert space $\mathcal{H}$ and that the spectrum of $T$ is confined in the union $\bigcup_{j\in J}Δ_j$, $J\subseteq\mathbb{Z}$, of segments $Δ_j=[α_j, β_j]\subset\mathbb{R}$ such that $α_{j+1}>β_j$ and $$ \inf_{j} \left(α_{j+1}-β_j\right) = d > 0. $$ If $B$ is a bounded (in general non-self-adjoint) perturbation of $T$ with $\|B\|=:b<d/2$ then the spectrum of the perturbed operator $A=T+B$ lies in the union $\bigcup_{j\in J} U_{b}(Δ_j)$ of the mutually disjoint closed $b$-neighborhoods $U_{b}(Δ_j)$ of the segments $Δ_j$ in $\mathbb{C}$. Let $Q_j…
We extend Krupnik's criterion of self-adjointness of the Cauchy singular integral operator to the case of finitely connected domains. The main aim of the paper is to present a new approach for proof of the criterion.
Given, on the Hilbert space $\H_0$, the self-adjoint operator $B$ and the skew-adjoint operators $C_1$ and $C_2$, we consider, on the Hilbert space $\H\simeq D(B)\oplus\H_0$, the skew-adjoint operator $$W=[\begin{matrix} C_2&\uno -B^2&C_1\end{matrix}]$$ corresponding to the abstract wave equation $\ddotϕ-(C_1+C_2)\dotϕ=-(B^2+C_1C_2)ϕ$. Given then an auxiliary Hilbert space $\fh$ and a linear map $τ:D(B^2)\to\fh$ with a kernel $\K$ dense in $\H_0$, we explicitly construct skew-adjoint operators $W_Θ$ on a Hilbert space $\H_Θ\simeq D(B)\oplus\H_0\oplus \fh$ which coincide with $W$ on $\N\simeq\K…
We consider a perturbation problem for embedded eigenvalues of a self-adjoint differential operator in $L^2(\mathbb R;\mathbb R^n)$. In particular, we study the set of all small perturbations in an appropriate Banach space for which the embedded eigenvalue remains embedded in the continuous spectrum. We show that this set of small perturbations forms a smooth manifold and we specify its co-dimension. Our methods involve the use of exponential dichotomies, their roughness property and Lyapunov-Schmidt reduction.
We study the trace class perturbations of the half-line, discrete Laplacian and obtain a new bound for the perturbation determinant of the corresponding non-self-adjoint Jacobi operator. Based on this bound, we obtain the Lieb--Thirring inequalities for such operators. The spectral enclosure for the discrete spectrum and embedded eigenvalues are also discussed.
We study one-dimensional Schroedinger operators S with real-valued distributional potentials q in W^{-1}_{2,loc}(R) and prove an extension of the Povzner-Wienholtz theorem on self-adjointness of bounded below S thus providing additional information on its domain. The results are further specified for q in W^{-1}_{2,unif}(R).
We survey the relationships of rank one self-adjoint and unitary perturbations as well as finite rank unitary perturbations with various branches of analysis and mathematical physics. We include the case of non-inner characteristic operator functions. For rank one perturbations and non-inner characteristic functions, we prove a representation formula for the adjoint of the Clark operator. Throughout we mention many open problems at varying levels of difficulty.
In this paper we study properties of functions of triples of not necessarily commuting self-adjoint operators. The main result of the paper shows that unlike in the case of functions of pairs of self-adjoint operators there is no Lipschitz type estimates in the trace norm for arbitrary functions in the Besov class $B_{\infty,1}^1({\Bbb R}^3)$.
We establish Lieb-Thirring type inequalities for non self-adjoint relatively compact perturbations of certain operators of mathematical physics. We apply our results to quantum Hamiltonians of Schr{ö}dinger and Pauli with constant magnetic field of strength $b\textgreater{}0$. In particular, we use these bounds to obtain some information on the distribution of the eigenvalues of the perturbed operators in the neighborhood of their essential spectrum.
We use recent results on the boundary behavior of Cauchy integrals to study the Krein spectral shift of a rank one perturbation problem for self-adjoint operators. As an application, we prove that all self-adjoint rank one perturbations of a self-adjoint operator are pure point if and only if the spectrum of the operator is countable. We also study pairs of pure point operators unitarily equivalent up to a rank one perturbation and give various examples of rank one perturbations of singular spectra.
In this note self-adjoint realizations of second order elliptic differential expressions with non-local Robin boundary conditions on a domain $Ω\subset\dR^n$ with smooth compact boundary are studied. A Schatten--von Neumann type estimate for the singular values of the difference of the $m$th powers of the resolvents of two Robin realizations is obtained, and for $m>\tfrac{n}{2}-1$ it is shown that the resolvent power difference is a trace class operator. The estimates are slightly stronger than the classical singular value estimates by M. Sh. Birman where one of the Robin realizations is repla…
In this paper I prove existence of an irreducible pair of operators $H$ and $H+V,$ where $H$ is a self-adjoint operator and $V$ is a self-adjoint trace-class operator, such that the singular spectral shift function of the pair is non-zero on the absolutely continuous spectrum of the operator $H.$
Let $H:D(H)\subseteq{\mathscr F}\to{\mathscr F}$ be self-adjoint and let $A:D(H)\to{\mathscr F}$ (playing the role of the annihilator operator) be $H$-bounded. Assuming some additional hypotheses on $A$ (so that the creation operator $A^{*}$ is a singular perturbation of $H$), by a twofold application of a resolvent Krein-type formula, we build self-adjoint realizations $\hat H$ of the formal Hamiltonian $H+A^{*}+A$ with $D(H)\cap D(\hat H)=\{0\}$. We give an explicit characterization of $D(\hat H)$ and provide a formula for the resolvent difference $(-\hat H+z)^{-1}-(-H+z)^{-1}$. Moreover, we…
We study the trace class perturbations of the whole-line, discrete Laplacian and obtain a new bound for the perturbation determinant of the corresponding non-self-adjoint Jacobi operator. Based on this bound, we refine the Lieb--Thirring inequality due to Hansmann--Katriel. The spectral enclosure for such operators is also discussed.
We give a criterion of asymptotic completeness and provide a representation of the scattering matrix for the scattering couple $(A_{0},A)$, where $A_{0}$ and $A$ are semi-bounded self-adjoint operators in $L^{2}(M,{\mathscr B},m)$ such that the set $\{u\in D(A_{0})\cap D(A):A_{0}u=Au\}$ is dense. No sort of trace-class condition on resolvent differences is required. Applications to the case in which $A_{0}$ corresponds to the free Laplacian in $L^{2}({\mathbb R}^{n})$ and $A$ describes the Laplacian with self-adjoint boundary conditions on rough compact hypersurfaces are given.
Given a self-adjoint involution J on a Hilbert space H, we consider a J-self-adjoint operator L=A+V on H where A is a possibly unbounded self-adjoint operator commuting with J and V a bounded J-self-adjoint operator anti-commuting with J. We establish optimal estimates on the position of the spectrum of L with respect to the spectrum of A and we obtain norm bounds on the operator angles between maximal uniformly definite reducing subspaces of the unperturbed operator A and the perturbed operator L. All the bounds are given in terms of the norm of V and the distances between pairs of disjoint s…
We give a sufficient condition for the essential self-adjointness of a perturbation of the square of the magnetic Laplacian on an infinite weighted graph. The main result is applicable to graphs whose degree function is not necessarily bounded. The result allows perturbations that are not necessarily bounded from below by a constant.
We define functions of noncommuting self-adjoint operators with the help of double operator integrals. We are studying the problem to find conditions on a function $f$ on ${\Bbb R}^2$, for which the map $(A,B)\mapsto f(A,B)$ is Lipschitz in the operator norm and in Schatten--von Neumann norms $\boldsymbol{S}_p$. It turns out that for functions $f$ in the Besov class $B_{\infty,1}^1({\Bbb R}^2)$, the above map is Lipschitz in the $\boldsymbol{S}_p$ norm for $p\in[1,2]$. However, it is not Lipschitz in the operator norm, nor in the $\boldsymbol{S}_p$ norm for $p>2$. The main tool is triple opera…
For a pair $(A,B)$ of not necessarily bounded and not necessarily commuting self-adjoint operators and for a function $f$ on the Euclidean space ${\Bbb R}^2$ that belongs to the inhomogeneous Besov class $B_{\infty,1}^1({\Bbb R}^2)$, we define the function $f(A,B)$ of these operators as a densely defined operator. We consider the problem of estimating the functions $f(A,B)$ under perturbations of the pair $(A,B)$. It is established that if $1\le p\le2$, and $(A_1,B_1)$ and $(A_2,B_2)$ are pairs of not necessarily bounded and not necessarily commuting self-adjoint operators such that the operat…
In this paper we study properties of functions of triples of not necessarily commuting self-adjoint operators. The main result of the paper shows that unlike in the case of functions of pairs of self-adjoint operators there is no Lipschitz type estimates in any Schatten--von Neumann norm $\boldsymbol S_p$, $1\le p\le\infty$, for arbitrary functions in the Besov class $B_{\infty,1}^1({\Bbb R}^3)$. In other words, we prove that for $p\in[1,\infty]$, there is no constant $K>0$ such that the inequality \begin{align*} \|f(A_1,B_1,C_1)&-f(A_2,B_2,C_2)\|_{\boldsymbol S_p}\\[.1cm] &\le K\|f\|_{B_{\inf…
The work is devoted to the study of the similarity of a correct restriction to some self-adjoint operator in the case when the minimal operator is symmetric. The resulting theorem was applied to the Sturm-Liouville operator and the Laplace operator. It is shown that the spectrum of a non self-adjoint singularly perturbed operator is real and the corresponding system of eigenvectors forms a Riesz basis.
A regular symmetric operator on a Hilbert module is self-adjoint whenever there exists a suitable approximate identity. We say an operator is 'locally bounded' if the composition of the operator with each element in the approximate identity is bounded. We prove that the perturbation of a regular self-adjoint operator by a locally bounded symmetric operator is again regular and self-adjoint. We use this result to show that the Kasparov class represented by an unbounded Kasparov module is stable under locally bounded perturbations. As an application, we show that we obtain a converse to the 'dou…
We consider an off-diagonal self-adjoint finite rank perturbation of a self-adjoint operator in a complex separable Hilbert space $\mathfrak{H}_0 \oplus \mathfrak{H}_1$, where $\mathfrak{H}_1$ is finite dimensional. We describe the singular spectrum of the perturbed operator and establish a connection with solutions to the operator Riccati equation. In particular, we prove existence results for solutions in the case where the whole Hilbert space is finite dimensional.
If a Banach-space operator has a complemented range, then its normed-space adjoint has a complemented kernel and the converse holds on a reflexive Banach space. It is also shown when complemented kernel for an operator is equivalent to complemented range for its normed-space adjoint. This is applied to compact operators and to compact perturbations. In particular, compact perturbations of semi-Fredholm operators have complemented range and kernel for both the perturbed operator and its normed-space adjoint
Symplectic self-adjointness of Hamiltonian operator matrices is studied, which arises in symplectic elasticity and optimal control. For the cases of diagonal domain and off-diagonal domain, necessary and sufficient conditions are shown. The proofs use Frobenius-Schur fractorizations of unbounded operator matrices. Under additional assumptions, sufficient conditions based on perturbation method are obtained. The theory is applied to a problem in symplectic elasticity.
We describe large classes of compact self-adjoint Hankel operators whose eigenvalues have power asymptotics and obtain explicit expressions for the coefficient in front of the leading term. The results are stated both in the discrete and continuous representations for Hankel operators. We also elucidate two key principles underpinning the proof of such asymptotic relations. We call them {\it the localization principle} and {\it the symmetry principle}. The localization principle says that disjoint components of the singular support of the symbol of a Hankel operator make independent contributi…
We construct self-adjoint Laplacians and symmetric Markov semigroups on partially hyperbolic attractors and on hyperbolic attractors with singularities, endowed with Gibbs u-measures. If the measure has full support, we can also guarantee the existence of an associated symmetric Hunt diffusion process. In the special case of partially hyperbolic diffeomorphisms induced by geodesic flows on manifolds of negative sectional curvature the Laplacians we consider are self-adjoint extensions of well-known classical leafwise Laplacians.
Some preliminaries and basic facts regarding unbounded Wiener-Hopf operators (WH) are provided. WH with rational symbols are studied in detail showing that they are densely defined closed and have finite dimensional kernels and deficiency spaces. The latter spaces as well as the domains and ranges are explicitly determined. A further topic concerns semibounded WH. Expressing a semibounded WH by a product of a closable operator and its adjoint this representation allows for a natural self-adjoint extension. It is shown that it coincides with the Friedrichs extension. Polar decomposition gives r…
The work is devoted to the study of Laplace operator when the potential is a singular generalized function and plays the role of a singular perturbation of a Laplace operator. Abstract theorem obtained earlier by the authors B.N. Biyarov and G.K. Abdrasheva applies to this. The main purpose of the study is the spectral issue. Singular perturbations for differential operators have been studied by many authors for the mathematical substantiation of solvable models of quantum mechanics, atomic physics, and solid state physics. In all these cases, the problems were self-adjoint. In this paper, we …
We give an elementary proof of a formula recently obtained by Hammond, Moorhouse, and Robbins for the adjoint of a rationally induced composition operator on the Hardy space H^2. We discuss some variants and implications of this formula, and use it to provide a sufficient condition for a rationally induced composition operator adjoint to be a compact perturbation of a weighted composition operator.
We present an abstract result on removing regularization for singular potentials which are not semibounded from below. The relation between ``right'' regularizations and ``right'' self-adjoint extensions of the perturbed Schrödinger operators is examined.
The quantum Ito formula has so far been proved for regular (bounded) quantum semimartingales We give three different extensions to classes of essentially self-adjoint (unbounded) quantum semimartingales. The first extension is to quantum semimartingales whose integrands have chaos matrices controlled by certain scalar matrices The second extension is to perturbations of classical, possibly unbounded, quantum semimartingales by bounded quantum semimartingales The third extension is obtained by perturbing an essentially self-adjoint quantum semimartingale satisfying the quantum Duhamel formula b…
