Vol. 21 (2015), no. 2

We study the connection between the classes of (i) Livsic functions $s(z),$ i.e., the characteristic functions of densely defined symmetric operators $\dot A$ with deficiency indices $(1, 1)$; (ii) the characteristic functions $S(z)$ of a maximal dissipative extension $T$ of $\dot A,$ i.e., the Mobius transform of $s(z)$ determined by the von Neumann parameter $\kappa$ of the extension relative to an appropriate basis in the deficiency subspaces; and (iii) the transfer functions $W_\Theta(z)$ of a conservative L-system $\Theta$ with the main operator $T$. It is shown that under a natural hypothesis {the functions $S(z)$ and $W_\Theta(z)$ are reciprocal to each other. In particular, $W_\Theta(z)=\frac{1}{S(z)}=-\frac{1}{s(z)}$ whenever $\kappa=0$. It is established that the impedance function of a conservative L-system with the main operator $T$ belongs to the Donoghue class if and only if the von Neumann parameter vanishes ($\kappa=0$). Moreover, we introduce the generalized Donoghue class and obtain the criteria for an impedance function to belong to this class. We also obtain the representation of a function from this class via the Weyl-Titchmarsh function. All results are illustrated by a number of examples.

We study existence, uniqueness, and a limiting behavior of solutions to an abstract linear evolution equation in a scale of Banach spaces. The generator of the equation is a perturbation of the operator which satisfies the classical assumptions of Ovsyannikov's method by a generator of a $C_0$-semigroup acting in each of the spaces of the scale. The results are (slightly modified) abstract version of those considered in [10] for a particular equation. An application to a birth-and-death stochastic dynamics in the continuum is considered.

Using the Davies-Helffer-Sjostrand functional calculus based on almost analytic extensions, we address the following problem: Given a self-adjoint operator $S$ in $\mathcal H$, and functions $f$ in an appropriate class, for instance, $f \in C_0^{\infty}(\mathbb R)$, how to control the norm $\|f(S)\|_{\mathcal B(\mathcal H)}$ in terms of the norm of the resolvent of $S$, $\|(S - z_0 I_{\mathcal H})^{-1}\|_{\mathcal B(\mathcal H)}$, for some $z_0 \in \mathbb C\backslash\mathbb R$. We are particularly interested in the case where $\mathcal B(\mathcal H)$ is replaced by a trace ideal, $\mathcal B_p(\mathcal H)$, $p \in [1,\infty)$.

We consider differential equations of the form $\left(\frac{d^{2}}{dt^{2}} - B\right)^{m}y(t) = f(t)$, $m \in \mathbb{N}, \ t \in (-\infty, \infty)$, where $B$ is a positive operator in a Banach space $\mathfrak{B}, \ f(t)$ is a bounded continuous vector-valued function on $(-\infty, \infty)$ with values in $\mathfrak{B}$, and describe all their solutions. In the case, where $f(t) \equiv 0$, we prove that every solution of such an equation can be extended to an entire $\mathfrak{B}$-valued function for which the Phragmen-Lindel\"{o}f principle is fulfilled. It is also shown that there always exists a unique bounded on $\mathbb{R}^{1}$ solution, and if $f(t)$ is periodic or almost periodic, then this solution is the same as $f(t)$.

We consider the evolution of correlation functions in a non-Markov version of the contact model in the continuum. The memory effects are introduced by assuming the fractional evolution equation for the statistical dynamics. This leads to a behavior of time-dependent correlation functions, essentially different from the one known for the standard contact model.

We review several applications of Berezansky's projection spectral theorem to Jacobi fields in a symmetric Fock space, which lead to L\'evy white noise measures.