# Mozel’ V. A.

### On a Banach algebra generated by the Bergman operator, constant coefficients, and finitely generated groups of shifts

Ukr. Mat. Zh. - 2017. - 69, № 11. - pp. 1515-1523

We study a Banach algebra generated by the Bergman operator, constant coefficients, and shifts formed by finitely generated commutative groups of hyperbolic transformations of the unit disk acting in the Lebesgue space $L_p, p > 1$, and obtain an effective criterion for the operators from the considered Banach algebra to be Fredholm operators.

### On the $C^{*}$-Algebra Generated by the Bergman Operator, Carleman Second-Order Shift, and Piecewise Continuous Coefficients

Ukr. Mat. Zh. - 2015. - 67, № 9. - pp. 1244–1252

We study the $C^{*}$ -algebra generated by the Bergman operator with piecewise continuous coefficients in the Hilbert space $L_2$ and extended by the Carleman rotation by an angle $π$. As a result, we obtain an efficient criterion for the operators from the indicated $C^{*}$ -algebra to be Fredholm operators.

### Banach algebra generated by a finite number of bergman polykernel operators, continuous coefficients, and a finite group of shifts

Ukr. Mat. Zh. - 2010. - 62, № 9. - pp. 1247–1255

We study the Banach algebra generated by a finite number of Bergman polykernel operators with continuous coefficients that is extended by operators of weighted shift that form a finite group. By using an isometric transformation, we represent the operators of the algebra in the form of a matrix operator formed by a finite number of mutually complementary projectors whose coefficients are Toeplitz matrix functions of finite order. Using properties of Bergman polykernel operators, we obtain an efficient criterion for the operators of the algebra considered to be Fredholm operators.

### Algebra of Bergman Operators with Automorphic Coefficients and Parabolic Group of Shifts

Chernetskii V. A., Mozel’ V. A.

Ukr. Mat. Zh. - 2001. - 53, № 9. - pp. 1218-1223

UDC 517.983

We study the algebra of operators with the Bergman kernel extended by isometric weighted shift operators. The coefficients of the algebra are assumed to be automorphic with respect to a cyclic parabolic group of fractional-linear transformations of a unit disk and continuous on the Riemann surface of the group. By using an isometric transformation, we obtain a quasiautomorphic matrix operator on the Riemann surface with properties similar to the properties of the Bergman operator. This enables us to construct the algebra of symbols, devise an efficient criterion for the Fredholm property, and calculate the index of the operators of the algebra considered.