2019
Том 71
№ 5

# Samoilenko Yu. S.

Articles: 35
Article (Ukrainian)

### Structure of the Systems of Orthogonal Projections Connected with Countable Coxeter Trees

Ukr. Mat. Zh. - 2014. - 66, № 9. - pp. 1185–1192

The paper is devoted to the investigation of representations of Temperley–Lieb-type algebras generated by orthogonal projections connected with countable Coxeter trees. The theorem on the structure of these systems of orthogonal projections is proved. Some examples are presented.

Anniversaries (Ukrainian)

### Myroslav L’vovych Horbachuk (on his 75 th birthday)

Ukr. Mat. Zh. - 2013. - 65, № 3. - pp. 451-454

Anniversaries (Ukrainian)

### Anatolii Mykhailovych Samoilenko (on his 75th birthday)

Ukr. Mat. Zh. - 2013. - 65, № 1. - pp. 3 - 6

Article (Russian)

### Scalar operators equal to the product of unitary roots of the identity operator

Ukr. Mat. Zh. - 2012. - 64, № 6. - pp. 819-825

We study the set of all $\gamma \in \mathbb{C}$ for which there exist unitary operators $U_i$ such that $U_1U_2 ... U_n = \gamma I$ and $U_i^{m_i} = I$, where $m_i \in \mathbb{N}$ are given.

Article (Russian)

### On simple $n$-tuples of subspaces of a Hilbert space

Ukr. Mat. Zh. - 2009. - 61, № 12. - pp. 1668-1703

This survey is devoted to the structure of “simple” systems $S = (H;H_1,…,H_n)$ of subspaces $H_i,\; i = 1,…, n,$ of a Hilbert space $H$, i.e., $n$-tuples of subspaces such that, for each pair of subspaces $H_i$ and $H_j$, the angle $0 < θ_{ij} ≤ π/2$ between them is fixed. We give a description of “simple” systems of subspaces in the case where the labeled graphs naturally associated with these systems are trees or unicyclic graphs and also in the case where all subspaces are one-dimensional. If the cyclic range of a graph is greater than or equal to two, then the problem of description of all systems of this type up to unitary equivalence is *-wild.

Brief Communications (Russian)

### Growth of generalized Temperley–Lieb algebras connected with simple graphs

Ukr. Mat. Zh. - 2009. - 61, № 11. - pp. 1579-1585

We prove that the generalized Temperley–Lieb algebras associated with simple graphs Γ have linear growth if and only if the graph Γ coincides with one of the extended Dynkin graphs ${\tilde A_n}$, ${\tilde D_n}$, ${\tilde E_6}$, or ${\tilde E_7}$. An algebra $T{L_{\Gamma, \tau }}$ has exponential growth if and only if the graph Γ coincides with none of the graphs ${A_n}$, ${D_n}$, ${E_n}$, ${\tilde A_n}$, ${\tilde D_n}$, ${\tilde E_6}$, and ${\tilde E_7}$.

Article (Ukrainian)

### Myroslav L’vovych Horbachuk (on his 70th birthday)

Ukr. Mat. Zh. - 2008. - 60, № 4. - pp. 439–442

Article (Ukrainian)

### On the *-representation of one class of algebras associated with Coxeter graphs

Ukr. Mat. Zh. - 2008. - 60, № 4. - pp. 545–556

We investigate *-representations of a class of algebras that are quotient algebras of the Hecke algebras associated with Coxeter graphs. A description of all unitarily nonequivalent irreducible *-representations of finite-dimensional algebras is given. We prove that only trees that have at most one edge of type s > 3 define algebras of finite Hilbert type for all values of parameters.

Anniversaries (Ukrainian)

### Fifty years devoted to science (on the 70th birthday of Anatolii Mykhailovych Samoilenko)

Ukr. Mat. Zh. - 2008. - 60, № 1. - pp. 3–7

Article (Ukrainian)

### On the growth of deformations of algebras associated with Coxeter graphs

Ukr. Mat. Zh. - 2007. - 59, № 6. - pp. 826–837

We investigate a class of algebras that are deformations of quotient algebras of group algebras of Coxeter groups. For algebras from this class, a linear basis is found by using the “diamond lemma.” A description of all finite-dimensional algebras of this class is given, and the growth of infinite-dimensional algebras is determined.

Article (Ukrainian)

### On spectral theorems for families of linearly connected self-adjoint operators with given spectra associated with extended Dynkin graphs

Ukr. Mat. Zh. - 2006. - 58, № 11. - pp. 1556–1570

We prove spectral theorems for families of linearly connected self-adjoint operators with given special spectra associated with extended Dynkin graphs. We establish that all irreducible families of linearly connected operators with arbitrary spectra associated with extended Dynkin graphs are finite-dimensional.

Obituaries (Ukrainian)

### Andrei Reuter (1937-2006)

Ukr. Mat. Zh. - 2006. - 58, № 11. - pp. 1584-1585

Anniversaries (Ukrainian)

### Leonid Pavlovych Nyzhnyk (on his 70-th birthday)

Ukr. Mat. Zh. - 2005. - 57, № 8. - pp. 1120-1122

Anniversaries (Ukrainian)

### Yurij Makarovich Berezansky (the 80th anniversary of his birth)

Ukr. Mat. Zh. - 2005. - 57, № 5. - pp. 3-11

Article (Ukrainian)

### On the Group $C^{*}$-Algebras of a Semidirect Product of Commutative and Finite Groups

Ukr. Mat. Zh. - 2005. - 57, № 5. - pp. 697–705

By using representations of general position and their properties, we give the description of group $C^{*}$-algebras for semidirect products $\mathbb{Z}^d \times G_f$, where $G_f$ is a finite group, in terms of algebras of continuous matrix-functions defined on some compact set with boundary conditions. We present examples of the $C^{*}$-algebras of affine Coxeter groups.

Article (Russian)

### On C*-Algebras Generated by Deformations of CCR

Ukr. Mat. Zh. - 2004. - 56, № 11. - pp. 1527-1538

We consider C*-algebras generated by deformations of classical commutation relations (CCR), which are generalizations of commutation relations for generalized quons and twisted CCR. We show that the Fock representation is a universal bounded representation. We discuss the connection between the presented deformations and extensions of many-dimensional noncommutative tori.

Article (Russian)

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

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.

Article (Russian)

### On Homomorphisms of Algebras Generated by Projectors and Coxeter Functors

Ukr. Mat. Zh. - 2003. - 55, № 9. - pp. 1224-1237

We consider algebras generated by idempotents in Banach spaces and orthoprojectors in Hilbert spaces whose sum is a multiple of the identity. We construct several functors generated by homomorphisms of the algebras considered between categories of representations. We investigate properties of these functors and present their applications.

Article (Russian)

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

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.

Article (Russian)

### Scalar Operators Representable as a Sum of Projectors

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 $\left\{ {0,1,1 + \frac{1}{{n - 1}},\left[ {1 + \frac{1}{{n - 2}},n - 1 - \frac{1}{{n - 2}}} \right],n - 1 - \frac{1}{{n - 1}},n - 1,n} \right\} \supset$ $\Sigma _n \supset \left\{ {0,1,1 + \frac{k}{{k\left( {n - 3} \right) + 2}},k \in \mathbb{N},\left[ {1 + \frac{1}{{n - 3}},n - 1 - \frac{1}{{n - 3}}} \right],n - 1 - \frac{k}{{k\left( {n - 3} \right) + 2}},k \in \mathbb{N},n - 1,n} \right\}$ .

Article (Russian)

### Structure theorems for families of idempotents

Ukr. Mat. Zh. - 1998. - 50, № 4. - pp. 523–533

For *-algebras generated by idempotents and orthoprojectors, we study the complexity of the problem of description of *-representations to within unitary equivalence. In particular, we prove that the *-algebra generated by two orthogonal idempotents is *-wild as well as the *-algebra generated by three orthoprojectors, two of which are orthogonal.

Brief Communications (Russian)

### Periodic groups are not *-wild

Ukr. Mat. Zh. - 1997. - 49, № 5. - pp. 729–730

We consider certain properties of *-wild groups and prove that periodic groups are not *-wild.

Article (Russian)

### Wild problems in the theory of representations of *-algebras generated by generatrices and relations

Ukr. Mat. Zh. - 1995. - 47, № 1. - pp. 70–78

Two definitions of wild *-algebras are considered from the point of view of the theory of representations. We present methods for proving the fact that a *-algebra is wild. It is also shown that if a algebra isp-wild, then it isf-wild. The converse statement is not true.

Article (Ukrainian)

### Representations of relations of the form i [A, B] = f (A) + g (B)

Ukr. Mat. Zh. - 1991. - 43, № 1. - pp. 110-114

It is proved that all nontrivial representations of quadratic relation i[A, B]=f(A)+g(B) with self-adjoint operators A, B are unbounded if f and g are nonnegative; for any f and g this relation does not have nontrivial finite-dimensional representations and factor-representations of type II1, but can have infinite-dimensional irreducible representations with bounded operators.

Article (Ukrainian)

### Unbounded self-adjoint operators connected by algebraic relations

Ukr. Mat. Zh. - 1989. - 41, № 12. - pp. 1664–1668

Article (Ukrainian)

### An application of the projective spectral theorem to commuting families of operators

Ukr. Mat. Zh. - 1988. - 40, № 4. - pp. 469-481

Article (Ukrainian)

### Expansion in eigenfunctions of families of commuting operators and representations of commutation relations

Ukr. Mat. Zh. - 1988. - 40, № 1. - pp. 106-109

Article (Ukrainian)

### Garding domain and entire vectors for inductive limits of commutative locally compact groups

Ukr. Mat. Zh. - 1983. - 35, № 4. - pp. 427—434

Article (Ukrainian)

### Families of commuting self-adjoint operators

Ukr. Mat. Zh. - 1979. - 31, № 5. - pp. 555–558

Article (Ukrainian)

### A countable set of commuting self-adjoint operators and the algebra of local observables

Ukr. Mat. Zh. - 1979. - 31, № 4. - pp. 365–371

Article (Ukrainian)

### Irreducible representations of inductive limits of groups

Ukr. Mat. Zh. - 1977. - 29, № 4. - pp. 526–531

Article (Ukrainian)

### Decomposition of a representation of the group SL (2,C) of regular type into indecomposables

Ukr. Mat. Zh. - 1975. - 27, № 4. - pp. 464–470

Article (Ukrainian)

### Nuclear spaces of functions of infinitely many variables

Ukr. Mat. Zh. - 1973. - 25, № 6. - pp. 723—737

Article (Ukrainian)

### Integral representations of invariant Hermitian functionals

Ukr. Mat. Zh. - 1970. - 22, № 1. - pp.

Article (Russian)

### Integral representation of invariant positive-definite matrix kernels

Ukr. Mat. Zh. - 1969. - 21, № 4. - pp. 487–510