Vol. 23 (2017), no. 3
Valentina Ivanivna Gorbachuk (to 80th birthday anniversary)
MFAT 23 (2017), no. 3, 207-208
207-208
Eduard R. Tsekanovskiĭ (to 80th birthday anniversary)
MFAT 23 (2017), no. 3, 208-210
208-210
Transformations of Nevanlinna operator-functions and their fixed points
MFAT 23 (2017), no. 3, 212-230
212-230
We give a new characterization of the class N0M[−1,1] of the operator-valued in the Hilbert space M Nevanlinna functions that admit representations as compressed resolvents (m-functions) of selfadjoint contractions. We consider the auto\-morphism Γ: M(λ)↦MΓ(λ):=((λ2−1)M(λ))−1 of the class N0M[−1,1] and construct a realization of MΓ(λ) as a compressed resolvent. The unique fixed point of Γ is the m-function of the block-operator Jacobi matrix related to the Chebyshev polynomials of the first kind. We study a transformation ˆΓ: M(λ)↦MˆΓ(λ):=−(M(λ)+λIM)−1 that maps the set of all Nevanlinna operator-valued functions into its subset. The unique fixed point M0 of ˆΓ admits a realization as the compressed resolvent of the "free" discrete Schrödinger operator ˆJ0 in the Hilbert space H0=ℓ2(N0)⨂M. We prove that M0 is the uniform limit on compact sets of the open upper/lower half-plane in the operator norm topology of the iterations {Mn+1(λ)=−(Mn(λ)+λIM)−1} of ˆΓ. We show that the pair {H0,ˆJ0} is the inductive limit of the sequence of realizations {ˆHn,ˆAn} of {Mn}. In the scalar case (M=C), applying the algorithm of I.S. Kac, a realization of iterates {Mn(λ)} as m-functions of canonical (Hamiltonian) systems is constructed.
A-regular–A-singular factorizations of generalized J-inner matrix functions
Volodymyr Derkach, Olena Sukhorukova
MFAT 23 (2017), no. 3, 231-251
231-251
Let J be an m×m signature matrix, i.e., J=J∗=J−1. An m×m mvf (matrix valued function) W(λ) that is meromorphic in the unit disk D is called J-inner if W(λ)JW(λ)∗≤J for every λ from h+W, the domain of holomorphy of W, in D, and W(μ)JW(μ)∗=J for a.e. μ∈T=∂D. A J-inner mvf W(λ) is called A-singular if it is outer and it is called right A-regular if it has no non-constant A-singular right divisors. As was shown by D. Arov [18] every J-inner mvf admits an essentially unique A-regular--A-singular factorization W=W(1)W(2). In the present paper this factorization result is extended to the class Urκ(J) of right generalized J-inner mvf's introduced in~\cite{DD09}. The notion and criterion of A-regularity for right generalized J-inner mvf's are presented. The main result of the paper is that we find a criterion for existence of an A-regular--A-singular factorization for a rational generalized J-inner mvf.
Overdamped modes and optimization of resonances in layered cavities
Illya M. Karabash, Olga M. Logachova, Ievgen V. Verbytskyi
MFAT 23 (2017), no. 3, 252-260
252-260
We study the problem of optimizing the imaginary parts Imω of quasi-normal-eigenvalues ω associated with the equation y″=−ω2By. It is assumed that the coefficient B(x), which describes the structure of an optical or mechanical resonator, is constrained by the inequalities 0≤b1≤B(x)≤b2. Extremal quasi-normal-eigenvalues belonging to the imaginary line iR are studied in detail. As an application, we provide examples of ω with locally minimal |Imω| (without additional restrictions on Reω) and show that a structure generating an optimal quasi-normal-eigenvalue on iR is not necessarily unique.
On a function system making a basis in a weight space
V. A. Zolotarev, V. N. Levchuk
MFAT 23 (2017), no. 3, 261-269
261-269
We find necessary and sufficient conditions for systems of functions generated by a second order differential equation to form a basis. The results are applied to show that Mathieu functions make a basis.
On Barcilon’s formula for Krein’s string
MFAT 23 (2017), no. 3, 270-276
270-276
We find conditions on two sequences of positive numbers that are sufficient for the sequences to be the Neumann and the Dirichlet spectra of a Krein string such that Barcilon’s formula holds true.
Linear maps preserving the index of operators
MFAT 23 (2017), no. 3, 277-284
277-284
Let H be an infinite-dimensional separable complex Hilbert space and B(H) the algebra of all bounded linear operators on H. In this paper, we prove that if a surjective linear map ϕ:B(H)⟶B(H) preserves the index of operators, then ϕ preserves compact operators in both directions and the induced map φ:C(H)⟶C(H), determined by φ(π(T))=π(ϕ(T)) for all T∈B(H), is a continuous automorphism multiplied by an invertible element in C(H).
On the graph K1,n related configurations of subspaces of a Hilbert space
MFAT 23 (2017), no. 3, 285-300
285-300
We study systems of subspaces H1,…,HN of a complex Hilbert space H that satisfy the following conditions: for every index k>1, the set {θk,1,…,θk,mk} of angles θk,i∈(0,π/2) between H1 and Hk is fixed; all other pairs Hk, Hj are orthogonal. The main tool in the study is a construction of a system of subspaces of a Hilbert space on the basis of its Gram operator (the G-construction).
Polarization inequality in complex normed spaces
MFAT 23 (2017), no. 3, 301-308
301-308
We introduce a product in all complex normed vector spaces, which generalizes the inner product of complex inner product spaces. Naturally the question occurs whether the polarization inequality of line (3.1) is fulfilled. We show that the polarization inequality holds for the product from Definition 1.1. This also yields a new proof of the Cauchy-Schwarz inequality in complex inner product spaces, which does not rely on the linearity of the inner product. The proof depends only on the norm in the vector space.