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 4I + 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 P1,...,Pn 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 $n \geqslant \sqrt {k - 1}$.