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Let ϕ be a subadditive weight on a C* -algebra A, and let Mϕ+ be the set of all elements x in A+ with ϕ(x) < +00. A seminorm ‖ • ‖ is introduced on the lineal Mϕsa = linRMϕ+, and a sufficient condition for the seminorm to be a norm is given. Let I be the unit of the algebra A, and let ϕ(I) = 1. Then, for every element x of Asa, the limit ρϕ(x) = limt→0+(ϕ(I + tx) - 1)/t exists and is finite. Properties of ρϕ are investigated, and examples of subadditive weights on C* -algebras are considered. On the basis of Lozinskii’s 1958 results, specific subadditive weights on Mn(C) are considered. An estimate for the difference of Cayley transforms of Hermitian elements of a von Neumann algebra is obtained.

Given α ∈ (0, 1) and c = h + iβ, h, β ∈ R, the function fα,c: R → C defined as follows is considered: (1) fα,c is Hermitian, i.e., $${f_{\alpha ,c}}\left( { - x} \right)\overline {{f_{\alpha ,c}}\left( x \right)} ,x \in \mathbb{R};$$ , x ∈ R; (2) fα,c(x) = 0 for x > 1; moreover, on each of the closed intervals [0, α] and [α, 1], the function fα,c is linear and satisfies the conditions fα,c(0) = 1, fα,c(α) = c, and fα,c(1) = 0. It is proved that the complex piecewise linear function fα,c is positive definite on R if and only if m(α) ≤ h ≤ 1 − α and |β| ≤ γ(α, h), $$where m\left( \alpha \right) = \left\{ \begin{gathered} 0if1/\alpha \notin \mathbb{N}, \hfill \\ - \alpha if1/\alpha \in \mathbb{N}. \hfill \\ \end{gathered} \right.$$ If m(α) ≤ h ≤ 1 − α and α ∈ Q, then γ(α, h) > 0; otherwise, γ(α, h) = 0. This result is used to obtain a criterion for the complete monotonicity of functions of a special form and prove a sharp inequality for trigonometric polynomials.

Consider the sesquilinearmatrix equation X*DX + AX + X*B + C = 0, where all the matrices are square and have the same order n. With this equation, we associate a block matrix M of double order 2n. The solvability of the above equation turns out to be related to the existence of n-dimensional neutral subspaces for the matrix M. We indicate sufficiently general conditions ensuring the existence of such subspaces.

Let R+:= [0, +∞), and let the matrix functions P, Q, and R of order n, n ∈ N, defined on the semiaxis R+ be such that P(x) is a nondegenerate matrix, P(x) and Q(x) are Hermitian matrices for x ∈ R+ and the elements of the matrix functions P−1, Q, and R are measurable on R+ and summable on each of its closed finite subintervals. We study the operators generated in the space Ln2(R+) by formal expressions of the form l[f] = −(P(f' − Rf))' − R*P(f' − Rf) + Qf and, as a particular case, operators generated by expressions of the form l[f] = −(P0f')' + i((Q0f)' + Q0f') + P'1f, where everywhere the derivatives are understood in the sense of distributions and P0, Q0, and P1 are Hermitianmatrix functions of order n with Lebesgue measurable elements such that P0−1 exists and ǁP0ǁ, ǁP0−1ǁ, ǁP0−1ǁǁP1ǁ2, ǁP0−1ǁǁQ0ǁ2 ∈ Lloc1(R+). Themain goal in this paper is to study of the deficiency index of the minimal operator L0 generated by expression l[f] in Ln2(R+) in terms of the matrix functions P, Q, and R (P0, Q0, and P1). The obtained results are applied to differential operators generated by expressions of the form \(l[f] = - f'' + \sum\limits_{k = 1}^{ + \infty } {{H_k}} \delta \left( {x - {x_k}} \right)f\), where xk, k = 1, 2,..., is an increasing sequence of positive numbers, with limk→+∞xk = +∞, Hk is a number Hermitian matrix of order n, and δ(x) is the Dirac δ-function.

Probability measures on the space of Hermitian matrices which are ergodic for the conjugation action of an infinite-dimensional unitary group are considered. It is established that the eigenvalues of random matrices distributed with respect to these measures satisfy the law of large numbers. The relationship between such models of random matrices and objects in free probability, freely infinitely divisible measures, is also established.

The volume of the parallelepiped spanned by functions of a square complex matrix defined on its spectrum is considered. The sharp lower bound of such volumes is found.

The present paper is devoted to the Hermitian spectral pseudoinversion and its applications to analysis, the solution and reduction of Hermitian differential-algebraic systems. New explicit formulas for the solutions of such systems and the solutions of related generalized Lyapunov equations are proposed. Attainable upper bounds for the norms of the solutions are obtained. A realization of the balanced truncation method not requiring computations involving projections onto deflating subspaces is proposed.

Necessary and sufficient conditions for the Stieltjes moment problem to have a unique solution and for the Hamburger moment problem with the same moments to have infinitely many solutions were obtained in Hamburger’s papers on the classical moment problem. In this paper, we obtain an analog of Hamburger’s criterion for the matrix moment problem.

We consider a class of almost geodesic mappings, namely, almost geodesic mappings of class π 2, and obtain conditions under which almost Hermitian manifolds admit almost geodesic mappings of class π 2. We prove that an almost Hermitian manifold admits a π 2-mapping with respect to a Riemannian connection if and only if it is an NK-manifold. We obtain a condition on the defining form ψ of any nontrivial π 2(e)-mapping under which a properNK-structure is taken to a proper NK-structure.

In this note, for a system of q matrix equations of the form $$ A_i XA_i^* = B_i B_i^* ,i = 1,2,...,q, $$ we consider the problem of the existence of Hermitian nonnegative-definite solutions. We offer an alternative with simplification and regularity to the result on necessary and sufficient conditions for the above matrix equations with q = 2 to have a Hermitian nonnegative-definite solution obtained by Zhang [1], who proposed a revised version of Young et al. [2]. Moreover, we give a necessary condition for the general case and then put forward a conjecture, with which at least some special situations are in agreement.

For matrices whose eigenvalues are real (such as Hermitian or real symmetric matrices), we derive simple explicit estimates for the maximal (λmax) and the minimal (λmin) eigenvalues in terms of determinants of order less than 3. For 3 × 3 matrices, we derive sharper estimates, which use det A but do not require to solve cubic equations.

Let A be a complex n × n matrix, and let A = B + iC, B = B*, C = C* be its Toeplitz decomposition. Then A is said to be (strictly) accretive if B > 0 and (strictly) dissipative if C > 0. We study the properties of matrices that satisfy both these conditions, in other words, of accretive-dissipative matrices. In many respects, these matrices behave as numbers in the first quadrant of the complex plane. Some other properties are natural extensions of the corresponding properties of Hermitian positive-definite matrices.

We construct a new almost-Hermitian structure of anti-invariant type on tangent bundles and deduce criteria for this structure to belong to all the Gray--Hervella classes. In particular, we prove that the tangent bundles over Kahlerian and semi-Kahlerian manifolds carry, respectively, a Kahlerian and a semi-Kahlerian structure.

Homogeneous reductive almost Hermitian spaces are considered. For such spaces satisfying a certain simple algebraic condition, criteria providing simple descriptions of Kahler, nearly Kahler, almost Kahler, quasi-Kahler, and G1 structures are obtained. It is found that, under this condition, Kahler structures can occur only on locally symmetric spaces and nearly Kahler structures, on naturally reductive spaces. Almost Kahler, quasi-Kahler, and G1 structures are described by simple conditions imposed on the Nomizu function α of the Riemannian connection of a homogeneous reductive almost Hermitian space.

It is proved that a generic-type 6-dimensional almost Hermitian submanifold of the algebra of octaves is minimal if and only if it belongs to the Gray--Hervella class G2. This is a maximal strengthening of the well-known result of Gray, who proved the minimality of the 6-dimensional Kahler submanifolds of the Cayley algebra.

We obtain necessary and sufficient conditions for the solvability of the augmentation and modification problems of order \(r\) for Hermitian matrices. The augmentation problem consists in the construction of a Hermitian \(((n + r) \times (n + r))\)-matrix with a given \((n \times n)\)-block \(A_{11} \) in block \((2 \times 2)\)-representation and with the prescribed eigenvalues. The modification problem consists in the construction of a Hermitian \((n \times n)\)-matrix \(B\) of rank not greater than \(r\) so that the obtained matrix, being added to a given Hermitian \((n \times n)\)-matrix \(A\), will have the required spectrum. We give an estimate for the minimal number of different eigenvalues of the solutions to these problems.

Solvability conditions are examined for the matrix equation \(X^{ - 1} X* = A\), which cannot be found in the well-known reference books on matrix theory. Methods for constructing solutions to this equation are indicated.

We suggest a most natural generalization of the notion of constant type for nearly Kahlerian manifolds introduced by A. Gray to arbitrary almost Hermitian manifolds. We prove that the class of almost Hermitian manifolds of zero constant type coincides with the class of Hermitian manifolds. We show that the class of G1-manifolds of zero constant type coincides with the class of 6-dimensional G1-manifolds with a non-integrable structure. Finally, we prove that the class of normal G2-manifolds of nonzero constant type coincides with the class of 4-dimensional G2-manifolds with a nonintegrable structure.

Automorphisms of nondegenerate (4, 3)-quadrics are studied. We classify quadrics of this kind with nonlinear automorphisms and find the dimensions of their groups of automorphisms.

Sufficient conditions for the “separation of the spectrum” of a quadratic matrix pencil with self-adjoint coefficients are found. A criterion for the representability of a positive operator in a finite-dimensional space in the form of a matrix with dominating main diagonal in each orthonormal basis is obtained.