When Is a Sum of Partial Reflections Equal to a Scalar Operator?

We describe the set of values of the parameter for which there exists a Hilbert space H and n partial reflections A ₁,...,A _n (self-adjoint operators such that A _k ³ =A_k or, which is the same, self-adjoint operators whose spectra belong to the set {-1,0,1}) whose sum is equal to the scalar operator I _H .     

Decomposition of a scalar operator into a product of unitary operators with two points in spectrum

We consider products of unitary operators with at most two points in their spectra, 1 and e^iα. We prove that the scalar operator e^iγI is a product of k such operators if α(1+1/(k-3))?γ?α(k-1-1/(k-3)) for k?5. Also we prove that for e^iα≠-1, only a countable number of scalar operators can be decomposed in a product of four operators from the mentioned class. As a corollary we show that every unitary operator on an infinite-dimensional space is a product of finitely many such operators.     

Singly generatedC *-algebras

We consider aC ^*-algebraA generated byk self-adjoint elements. We prove that, for , the algebraM _n (A) is singly generated, i.e., generated by one non-self-adjoint element. We present an example of algebraA for which the property thatM _n (A) is singly generated implies the relation . Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 8, pp. 1136–1141, August, 1999.     

On the identities in algebras generated by linearly connected idempotents

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 F₄-algebras.Translated from Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 6, pp. 782–795, June, 2004.     

Decomposition of a Hermitian Matrix into a Sum of Fixed Number of Orthogonal Projections

Ukrainian Mathematical Journal - We prove that any Hermitian matrix whose trace is integer and all eigenvalues lie in the segment [1 + 1/(k ? 3),k ? 1 ? 1/(k ? 3)] can be...     

On Identities in Algebras Q n,λ Generated by Idempotents

We investigate the presence of polynomial identities in the algebras Q _n, generated by n idempotents with the sum e ( 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, and Q _n,, n 5, do not have polynomial identities.     

Scalar Operators Representable as a Sum of Projectors

We study sets there exist n projectors P₁,...,P_n such that . We prove that if n 6, then .     

On Sums of Projections

In the paper, for all n, we describe the set _n of all real numbers admitting a collection of projections P ₁,...,P _n on a Hilbert space H such that _k=1 ⁿ P _k= I (I is the identity operator on H) and study the problem to find all collections of this kind for a given _n .