Abstract:
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.

Abstract:
Using an operator-theoretic framework in a Hilbert-space setting, we perform a detailed spectral analysis of the one-dimensional Laplacian in a bounded interval, subject to specific non-self-adjoint connected boundary conditions modelling a random jump from the boundary to a point inside the interval. In accordance with previous works, we find that all the eigenvalues are real. As the new results, we derive and analyse the adjoint operator, determine the geometric and algebraic multiplicities of the eigenvalues, write down formulae for the eigenfunctions together with the generalised eigenfunctions and study their basis properties. It turns out that the latter heavily depend on Diophantine properties of the interior point. Finally, we find a closed formula for the metric operator that provides a similarity transform of the problem to a self-adjoint operator.

Abstract:
Let A and C be self-adjoint operators such that the spectrum of A lies in a gap of the spectrum of C and let d>0 be the distance between the spectra of A and C. We prove that under these assumptions the sharp value of the constant c in the condition ||B||

Abstract:
We develop Weyl-Titchmarsh theory for self-adjoint Schr\"odinger operators $H_{\alpha}$ in $L^2((a,b);dx;\cH)$ associated with the operator-valued differential expression $\tau =-(d^2/dx^2)+V(\cdot)$, with $V:(a,b)\to\cB(\cH)$, and $\cH$ a complex, separable Hilbert space. We assume regularity of the left endpoint $a$ and the limit point case at the right endpoint $b$. In addition, the bounded self-adjoint operator $\alpha= \alpha^* \in \cB(\cH)$ is used to parametrize the self-adjoint boundary condition at the left endpoint $a$ of the type $$ \sin(\alpha)u'(a)+\cos(\alpha)u(a)=0, $$ with $u$ lying in the domain of the underlying maximal operator $H_{\max}$ in $L^2((a,b);dx;\cH)$ associated with $\tau$. More precisely, we establish the existence of the Weyl-Titchmarsh solution of $H_{\alpha}$, the corresponding Weyl-Titchmarsh $m$-function $m_{\alpha}$ and its Herglotz property, and determine the structure of the Green's function of $H_{\alpha}$. Developing Weyl-Titchmarsh theory requires control over certain (operator-valued) solutions of appropriate initial value problems. Thus, we consider existence and uniqueness of solutions of 2nd-order differential equations with the operator coefficient $V$, -y" + (V - z) y = f \, \text{on} \, (a,b), y(x_0) = h_0, \; y'(x_0) = h_1, under the following general assumptions: $(a,b)\subseteq\bbR$ is a finite or infinite interval, $x_0\in(a,b)$, $z\in\bbC$, $V:(a,b)\to\cB(\cH)$ is a weakly measurable operator-valued function with $\|V(\cdot)\|_{\cB(\cH)}\in L^1_\loc((a,b);dx)$, and $f\in L^1_{\loc}((a,b);dx;\cH)$, with $\cH$ a complex, separable Hilbert space. We also study the analog of this initial value problem with $y$ and $f$ replaced by operator-valued functions $Y, F \in \cB(\cH)$. Our hypotheses on the local behavior of $V$ appear to be the most general ones to date.

Abstract:
A recent problem [B. Gardas, J. Math. Phys. 52, 042104 (2011)] concerning an antilinear solution of the Riccati equation is solved. We also exemplify that a simplification of the Riccati equation, even under reasonable assumptions, can lead to a not equivalent equation.

Abstract:
We consider linear systems on a separable Hilbert space $H$, which are null controllable at some time $T_0>0$ under the action of a point or boundary control. Parabolic and hyperbolic control systems usually studied in applications are special cases. To every initial state $ y_0 \in H$ we associate the minimal "energy" needed to transfer $ y_0 $ to $ 0 $ in a time $ T \ge T_0$ ("energy" of a control being the square of its $ L^2 $ norm). We give both necessary and sufficient conditions under which the minimal energy converges to $ 0 $ for $ T\to+\infty $. This extends to boundary control systems the concept of null controllability with vanishing energy introduced by Priola and Zabczyk (Siam J. Control Optim. 42 (2003)) for distributed systems. The proofs in Priola-Zabczyk paper depend on properties of the associated Riccati equation, which are not available in the present, general setting. Here we base our results on new properties of the quadratic regulator problem with stability and the Linear Operator Inequality.

Abstract:
The general spectral boundary value problem framework is utilized to restate boundary value problems of Poincare, Hilbert, and Riemann for harmonic and analytic functions in abstract operator-theoretic terms.

Abstract:
The problem of block diagonalization for diagonally dominant symmetric block operator matrices with self-adjoint diagonal entries is considered. We show that a reasonable block diagonalization with respect to a reducing graph subspace requires a related skew-symmetric operator to be a strong solution to the associated Riccati equation. Under mild additional regularity conditions, we also establish that this skew-symmetric operator is a strong solution to the Riccati equation if and only if the graph subspace is reducing for the given operator matrix. These regularity conditions are shown to be automatically fulfilled whenever the corresponding relative bound of the off-diagonal part is sufficiently small. This extends the results by Albeverio, Makarov, and Motovilov in [Canad. J. Math. Vol. \textbf{55}, 2003, 449--503], where the off-diagonal part is required to be bounded.

Abstract:
We introduce a new concept of unbounded solutions to the operator Riccati equation $A_1 X - X A_0 - X V X + V^\ast = 0$ and give a complete description of its solutions associated with the spectral graph subspaces of the block operator matrix $\mathbf{B} = \begin{pmatrix} A_0 & V V^\ast & A_1 \end{pmatrix}$. We also provide a new characterization of the set of all contractive solutions under the assumption that the Riccati equation has a contractive solution associated with a spectral subspace of the operator $\mathbf{B}$. In this case we establish a criterion for the uniqueness of contractive solutions.