# Hatamleh R.

### Jacobi operators and orthonormal matrix-valued polynomials. II

Ukr. Mat. Zh. - 2017. - 69, № 6. - pp. 836-847

We use a system of operator-valued orthogonal polynomials to construct analogs of L. de Branges spaces and establish their relationship with the theory of nonself-adjoint operators.

### Jacobi operators and orthonormal matrix-valued polynomials. I

Ukr. Mat. Zh. - 2017. - 69, № 2. - pp. 228-239

It is shown that every self-adjoint operator in a separable Hilbert space is unitarily equivalent to a block Jacobi operator. A system of orthogonal operator-valued polynomials is constructed.

### Triangular models of commutative systems of linear operators close to unitary operators

Ukr. Mat. Zh. - 2016. - 68, № 5. - pp. 694-711

Triangular models are constructed for commutative systems of linear bounded operators close to unitary operators. The construction of these models is based on the continuation of basic relations for the characteristic function along the general chain of invariant subspaces.

### On Two-Dimensional Model Representations of One Class of Commuting Operators

Ukr. Mat. Zh. - 2014. - 66, № 1. - pp. 108–127

In the work by V. A. Zolotarev, *Dokl. Akad. Nauk Arm. SSR*, **63**, No. 3, 136–140 (1976), a triangular model is constructed for a system of twice-commuting linear bounded completely nonself-adjoint operators {*A* _{1}, *A* _{2}} ([*A* _{1}, *A* _{2}] = 0, [*A* _{1} ^{∗} , *A* _{2}] = 0) such that rank (*A* _{1})_{ I }(*A* _{2})_{ I } = 1 (2*i*(*A* _{ k })_{ I } = *A* _{ k } − *A* _{ k } ^{∗} , *k* = 1, 2) and the spectrum of each operator *A* _{ k }, *k* = 1, 2*,* is concentrated at zero. The indicated triangular model has the form of a system of operators of integration over the independent variable in *L* _{ Ω } ^{2} where the domain *Ω* = [0, *a*] × [0, *b*] is a compact set in ℝ^{2} bounded by the lines *x* = *a* and *y* = *b* and a decreasing smooth curve *L* connecting the points (0*, b*) and (*a,* 0)*.*