# K. A. Makarov

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Articles: 3

### Conservative L-systems and the Livšic function

Methods Funct. Anal. Topology 21 (2015), no. 2, 104-133

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.

### On $\mu$-scale invariant operators

Methods Funct. Anal. Topology 13 (2007), no. 2, 181-186

We introduce the concept of a $\mu$-scale invariant operator with respect to a unitary transformation in a separable complex Hilbert space. We show that if a nonnegative densely defined symmetric operator is $\mu$-scale invariant for some $\mu>0$, then both the Friedrichs and the Krein-von Neumann extensions of this operator are also $\mu$-scale invariant.

### Generalized eigenfunctions under singular perturbation

Methods Funct. Anal. Topology 5 (1999), no. 1, 13-28