# Yu. M. Arlinskii

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### Compressed resolvents of selfadjoint contractive exit space extensions and holomorphic operator-valued functions associated with them

Methods Funct. Anal. Topology **21** (2015), no. 3, 199-224

Contractive selfadjoint extensions of a Hermitian contraction $B$ in a Hilbert space $\mathfrak H$ with an exit in some larger Hilbert space $\mathfrak H\oplus\mathcal H$ are investigated. This leads to a new geometric approach for characterizing analytic properties of holomorphic operator-valued functions of Krein-Ovcharenko type, a class of functions whose study has been recently initiated by the authors. Compressed resolvents of such exit space extensions are also investigated leading to some new connections to transfer functions of passive discrete-time systems and related classes of holomorphic operator-valued functions.

### Factorizations of nonnegative symmetric operators

Methods Funct. Anal. Topology **19** (2013), no. 3, 211-226

We prove that each closed denselydefined and nonnegative symmetric operator $\dot A$ having disjointnonnegative self-adjoint extensions admits infinitely manyfactorizations of the form $\dot A=\mathcal L\mathcal L_0$, where $\mathcal L_0$ is aclosed nonnegative symmetric operator and $\mathcal L$ its nonnegativeself-adjoint extension. The same factorizations are also establishedfor a non-densely defined nonnegative closed symmetric operator withinfinite deficiency indices while for operator with finitedeficiency indices we prove impossibility of such a kindfactorization. A construction of pairs $\mathcal L_0\subset\mathcal L$ ($\mathcal L_0$ isclosed and densely defined, $\mathcal L=\mathcal L^*\ge 0$) having the property${\rm dom\,}(\mathcal L\mathcal L_0)=\{0\}$ (and, in particular, ${\rm dom\,}(\mathcal L^2_0)=\{0\}$) is given.

### Operator-norm approximations of holomorphic one-parameter semigroups of contractions in Hilbert spaces

Methods Funct. Anal. Topology **18** (2012), no. 2, 101-110

We establish the operator-norm convergence of the Iosida and Dunford-Segal approximation formulas for one-parameter semigroups of the class $C_0$, gene ated by maximal sectorial generators in separable Hilbert spaces. Our approach is essentially based on the Crouzeix-Delyon theorem [8] related to the generalization of the von Neumann inequality.

### Conservative discrete time-invariant systems and block operator CMV matrices

Methods Funct. Anal. Topology **15** (2009), no. 3, 201-236

It is well known that an operator-valued function $\Theta$ from the Schur class ${\bf S}(\mathfrak M,\mathfrak N)$, where $\mathfrak M$ and $\mathfrak N$ are separable Hilbert spaces, can be realized as a transfer function of a simple conservative discrete time-invariant linear system. The known realizations involve the function $\Theta$ itself, the Hardy spaces or the reproducing kernel Hilbert spaces. On the other hand, as in the classical scalar case, the Schur class operator-valued function is uniquely determined by its so-called "Schur para-me ers". In this paper we construct simple conservative realizations using the Schur parameters only. It turns out that the unitary operators corresponding to the systems take the form of five diagonal block operator matrices, which are analogs of Cantero--Moral--Vel\'azquez (CMV) matrices appeared recently in the theory of scalar orthogonal polynomials on the unit circle. We obtain new models given by truncated block operator CMV matrices for an arbitrary completely non-unitary contraction. It is shown that the minimal unitary dilations of a contraction in a Hilbert space and the minimal Naimark dilations of a semi-spectral operator measure on the unit circle can also be expressed by means of block operator CMV matrices.

### Some remarks on singular perturbations of self-adjoint operators

Yu. M. Arlinskiĭ, E. R. Tsekanovskiĭ

Methods Funct. Anal. Topology **9** (2003), no. 4, 287-308

### On $M$-accretive extensions and restrictions

Methods Funct. Anal. Topology **4** (1998), no. 3, 1-26