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.     

Particular formulae for the Moore-Penrose inverse of a columnwise partitioned matrix

An essential part of Cegielski’s [Obtuse cones and Gram matrices with non-negative inverse, Linear Algebra Appl. 335 (2001) 167-181] considerations of some properties of Gram matrices with nonnegative inverses, which are pointed out to be crucial in constructing obtuse cones, consists in developing some particular formulae for the Moore-Penrose inverse of a columnwise partitioned matrix A = (A₁ : A₂) under the assumption that it is of full column rank. In the present paper, these results are generalized and extended. The generalization consists in weakening the assumption mentioned above to the requirement that the ranges of A₁ and A₂ are disjoint, while the extension consists in introducing the conditions referring to the class of all generalized inverses of A.     

The sum of unitary similarity orbits containing only special operators

Let B(H) be the algebra of bounded linear operator acting on a Hilbert space H (over the complex or real field). Characterization is given to A₁,…,A_k∈B(H) such that for any unitary operators is always in a special class S of operators such as normal operators, self-adjoint operators, unitary operators. As corollaries, characterizations are given to A∈B(H) such that complex, real or nonnegative linear combinations of operators in its unitary orbit U(A)={U^∗AU:Uunitary} always lie in S.     

On linear preservers of normal maps

We study group induced cone (GIC) orderings generating normal maps. Examples of normal maps cover, among others, the eigenvalue map on the space of n × n Hermitian matrices as well as the singular value map on n × n complex matrices. In this paper, given two linear spaces equipped with GIC orderings induced by groups of orthogonal operators, we investigate linear operators preserving normal maps of the orderings. A characterization of the preservers is obtained in terms of the groups. The result is applied to show that the normal structure of the spaces is preserved under the action of the operators. In addition, examples are given.     

Criteria of unitary equivalence of Hermitian operators with a degenerate spectrum

Nonimprovable, in general, estimates of the number of necessary and sufficient conditions for two Hermitian operators to be unitarily equaivalent in a unitary space are obtained when the multiplicities of eigenvalues of operators can be more than 1. The explicit form of these conditions is given. In the Appendix the concept of conditionally functionally independent functions is given and the corresponding necessary and sufficient conditions are presented.     

An extension of a result of Hwang and Pyo

In this paper, we extend a majorization result of Hwang and Pyo [LAA 332-334 (2001) pp. 15-21] from the ordinary majorization ordering to the class of group induced cone orderings induced by non-effective groups. The case of effective groups is also investigated. A particular attention is paid to positive operators.     

The characterization of operators preserving primitivity for matrix k-tuples

We obtain a complete characterization of surjective additive operators acting on the Cartesian product of several matrix spaces over an antinegative semiring without zero divisors, which map primitive matrix k-tuples to primitive matrix k-tuples.     

Transitive Spaces of Operators

We investigate algebraic and topological transitivity and, more generally, k-transitivity for linear spaces of operators. In finite dimensions, we determine minimal dimensions of k-transitive spaces for every k, and find relations between the degree of transitivity of a product or tensor product on the one hand and those of the factors on the other. We present counterexamples to some natural conjectures. Some infinite dimensional analogues are discussed. A simple proof is given of Arveson’s result on the weak-operator density of transitive spaces that are masa bimodules. Authors partially supported by NSERC grants.     

Perturbations on the subdiagonals of Toeplitz matrices

Let L be an Hermitian linear functional defined on the linear space of Laurent polynomials. It is very well known that the Gram matrix of the associated bilinear functional in the linear space of polynomials is a Toeplitz matrix. In this contribution we analyze some linear spectral transforms of L such that the corresponding Toeplitz matrix is the result of the addition of a constant in a subdiagonal of the initial Toeplitz matrix. We focus our attention in the analysis of the quasi-definite character of the perturbed linear functional as well as in the explicit expressions of the new monic orthogonal polynomial sequence in terms of the first one.     

On simultaneous triangularization of collections of compact operators

In this paper we consider collections of compact (resp. C_p class) operators on arbitrary Banach (resp. Hilbert) spaces. For a subring R of reals, it is proved that an R-algebra of compact operators with spectra in R on an arbitrary Banach space is triangularizable if and only if every member of the algebra is triangularizable. It is proved that every triangularizability result on certain collections, e.g., semigroups, of compact operators on a complex Banach (resp. Hilbert) space gives rise to its counterpart on a real Banach (resp. Hilbert) space. We use our main results to present new proofs as well as extensions of certain classical theorems (e.g., those due to Kolchin, McCoy, and others) on arbitrary Banach (resp. Hilbert) spaces.     

Estimates for norms of matrix-valued and operator-valued functions and some of their applications

A survey is presented of estimates for a norm of matrix-valued and operator-valued functions obtained by the author. These estimates improve the Gel'fand-Shilov estimate for regular functions of matrices and Carleman's estimates for resolvents of matrices and compact operators.From the estimates for resolvents, the well-known result for spectrum perturbations of self-adjoint operators is extended to quasi-Hermitian operators. In addition, the classical Schur and Brown's inequalities for eigenvalues of matrices are improved.From estimates for the exponential function (semigroups), bounds for solution norms of nonlinear differential equations are derived. These bounds give the stability criteria which make it possible to avoid the construction of Lyapunov functions in appropriate situations.     

The q-numerical radius of weighted shift operators with periodic weights

We deal with the q-numerical radius of weighted unilateral and bilateral shift operators. In particular, the q-numerical radius of weighted shift operators with periodic weights is discussed and computed.     

The normed finiteness property of compact contraction operators

We study the joint spectral radius given by a finite set of compact operators on a Hilbert space. It is shown that the normed finiteness property holds in this case, that is, if all the compact operators are contractions and the joint spectral radius is equal to 1 then there exists a finite product that has a spectral radius equal to 1. We prove an additional statement in that the requirement that the joint spectral radius be equal to 1 can be relaxed to the asking that the maximum norm of finite products of a length norm is equal to 1. The length of this product is related to the dimension of the subspace on which the set of operators is norm preserving.     

Unitary similarity of projectors

Summary We deal with linear operators acting in a finite dimensional complex Hilbert space. We show that there exists a simple canonical form for projectors (not necessarily orthogonal) under unitary similarity. As a consequence we obtain a simple test for unitary similarity of projectors. IfP is a projector we show thatP andP ^* are unitarily similar. We also determine the isomorphism type of the algebra generated by the projectorsP andP ^*.Dedicated to the memory of Alexander M. Ostrowski on the occasion of the 100th anniversary of his birth     

Reverse order laws in C-algebras

In this paper, we offer purely algebraic necessary and sufficient conditions for reverse order laws for generalized inverses in C^∗-algebras, extending rank conditions for matrices and range conditions for Hilbert space operators.     

Bijective linear rank preservers for spaces of matrices over antinegative semirings

We classify the bijective linear operators on spaces of matrices over antinegative commutative semirings with no zero divisors which preserve certain rank functions such as the symmetric rank, the factor rank and the tropical rank. We also classify the bijective linear operators on spaces of matrices over the max-plus semiring which preserve the Gondran-Minoux row rank or the Gondran-Minoux column rank.     

Polar decompositions and related classes of operators in spaces ∏κ

Polar decompositions with respect to an indefinite inner product are studied for bounded linear operators acting on a space. Criteria are given for existence of various forms of the polar decompositions, under the conditions that the range of a given operatorX is closed and that zero is not an irregular critical point of the selfadjoint operatorX ^[*]X. Both real and complex spaces are considered. Relevant classes of operators having a selfadjoint (in the sense of the indefinite inner product) square root, or a selfadjoint logarithm, are characterized.The work of this author was partially supported by INdAM-GNCS and MURSTThe work of this author was partially supported by NSF grant DMS-9988579.     

Invariant Subspaces of Collectively Compact Sets of Linear Operators

In this paper, we first give some invariant subspace results for collectively compact sets of operators in connection with the joint spectral radius of these sets. We then prove that any collectively compact set M in algΓ satisfies Berger-Wang formula, where Γ is a complete chain of subspaces of X.      

On existence of finite universal Korovkin sets in Segal algebras

We prove that there exists a finite universal Korovkin set w.r.t positive operators for the centre of a Segal algebra on a compact groupG if and only ifG is metrizable. As a consequence it follows that a Segal algebra on a compact abelian group admits a finite universal Korovkin set w.r.t. positive operators iff the group is metrizable.