# Rabanovych V. I.

### On Decompositions of a Scalar Operator into a Sum of Self-Adjoint Operators with Finite Spectrum

Ukr. Mat. Zh. - 2015. - 67, № 5. - pp. 701–716

We consider the problem of classification of nonequivalent representations of a scalar operator $λI$ in the form of a sum of $k$ self-adjoint operators with at most $n_1 , ...,n_k$ points in their spectra, respectively. It is shown that this problem is *-wild for some sets of spectra if $(n_1 , ...,n_k)$ coincides with one of the following $k$ -tuples: $(2, ..., 2)$ for $k ≥ 5,\; (2, 2, 2, 3),\; (2, 11, 11),\; (5, 5, 5)$, or $(4, 6, 6)$. It is demonstrated that, for the operators with points 0 and 1 in the spectra and $k ≥ 5$, the classification problems are *-wild for every rational $λ ϵ 2 [2, 3]$.

### On the Decomposition of a Diagonal Operator into a Linear Combination of Idempotents or Projectors

Ukr. Mat. Zh. - 2005. - 57, № 3. - pp. 388–393

For a bounded operator that is not a sum of scalar and compact operators and is similar to a diagonal operator, we prove that it is a linear combination of three idempotents. It is also proved that any self-adjoint diagonal operator is a linear combination of four orthoprojectors with real coefficients.

### On the identities in algebras generated by linearly connected idempotents

Rabanovych V. I., Samoilenko Yu. S., Strilets O. V.

Ukr. Mat. Zh. - 2004. - 56, № 6. - pp. 782–795

We investigate the problem of the existence of polynomial identities (PI) in algebras generated by idempotents whose linear combination is equal to identity. In the case where the number of idempotents is greater than or equal to five, we prove that these algebras are not *PI*-algebras. In the case of four idempotents, in order that an algebra be a *PI*-algebra, it is necessary and sufficient that the sum of the coefficients of the linear combination be equal to two. In this case, these algebras are *F* _{4}-algebras.

### On the Decomposition of an Operator into a Sum of Four Idempotents

Ukr. Mat. Zh. - 2004. - 56, № 3. - pp. 419-424

We prove that operators of the form (2 ± 2/*n*)*I* + *K* are decomposable into a sum of four idempotents for integer *n* > 1 if there exists the decomposition *K* = *K* _{1} ⊕ *K* _{2} ⊕ ... ⊕ *K* _{n}, \(\sum\nolimits_1^n {K_i = 0} \) , of a compact operator *K*. We show that the decomposition of the compact operator 4*I* + *K* or the operator *K* into a sum of four idempotents can exist if *K* is finite-dimensional. If *n* tr *K* is a sufficiently large (or sufficiently small) integer and *K* is finite-dimensional, then the operator (2 − 2/*n*)*I* + *K* [or (2 + 2/*n*)*I* + *K*] is a sum of four idempotents.

### On Identities in Algebras $Q_{n,λ}$ Generated by Idempotents

Rabanovych V. I., Samoilenko Yu. S., Strilets O. V.

Ukr. Mat. Zh. - 2001. - 53, № 10. - pp. 1380-1390

We investigate the presence of polynomial identities in the algebras $Q_{n,λ}$ generated by $n$ idempotents with the sum $λe$ ($λ ∈ C$ and $e$ is the identity of an algebra). We prove that $Q_{4,2}$ is an algebra with the standard polynomial identity $F_4$, whereas the algebras $Q_{4,2},\; λ ≠ 2$, and $Q_{n,λ},\; n ≥ 5$, do not have polynomial identities.

### Scalar Operators Representable as a Sum of Projectors

Rabanovych V. I., Samoilenko Yu. S.

Ukr. Mat. Zh. - 2001. - 53, № 7. - pp. 939-952

We study sets \(\Sigma _n = \{ \alpha \in \mathbb{R}^1 |\) there exist *n* projectors P_{1},...,P_{n} such that \(\sum\nolimits_{k = 1}^n {P_k = \alpha I} \}\) . We prove that if *n* ≥ 6, then \(\left\{ {0,1,1 + \frac{1}{{n - 1}},\left[ {1 + \frac{1}{{n - 2}},n - 1 - \frac{1}{{n - 2}}} \right],n - 1 - \frac{1}{{n - 1}},n - 1,n} \right\} \supset\) \(\Sigma _n \supset \left\{ {0,1,1 + \frac{k}{{k\left( {n - 3} \right) + 2}},k \in \mathbb{N},\left[ {1 + \frac{1}{{n - 3}},n - 1 - \frac{1}{{n - 3}}} \right],n - 1 - \frac{k}{{k\left( {n - 3} \right) + 2}},k \in \mathbb{N},n - 1,n} \right\}\) .

### Singly generatedC *-algebras

Ukr. Mat. Zh. - 1999. - 51, № 8. - pp. 1136-1141

We consider a $C*$-algebra $A$ generated by $k$ self-adjoint elements. We prove that, for $n \geqslant \sqrt {k - 1}$ , the algebra $M_n (A)$ is singly generated, i.e., generated by one non-self-adjoint element. We present an example of algebraA for which the property that $M_n (A)$ is singly generated implies the relation $n \geqslant \sqrt {k - 1}$.