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.     

Generalized matrix version of reverse Hölder inequality

A generalized matrix version of reverse Cauchy-Schwarz/Hölder inequality is proved. This includes the recent results proved by Bourin, Fujii, Lee, Niezgoda and Seo.     

Differential invariants of surfaces

The algebra of differential invariants of a suitably generic surface S⊂R³, under either the usual Euclidean or equi-affine group actions, is shown to be generated, through invariant differentiation, by a single differential invariant. For Euclidean surfaces, the generating invariant is the mean curvature, and, as a consequence, the Gauss curvature can be expressed as an explicit rational function of the invariant derivatives, with respect to the Frenet frame, of the mean curvature. For equi-affine surfaces, the generating invariant is the third order Pick invariant. The proofs are based on the new, equivariant approach to the method of moving frames.     

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.     

On commutativity of projectors

The purpose of this paper is to revisit two problems discussed previously in the literature, both related to the commutativity property P₁P₂ = P₂P₁, where P₁ and P₂ denote projectors (i.e., idempotent matrices). The first problem was considered by Baksalary et al. [J.K. Baksalary, O.M. Baksalary, T. Szulc, A property of orthogonal projectors, Linear Algebra Appl. 354 (2002) 35-39], who have shown that if P₁ and P₂ are orthogonal projectors (i.e., Hermitian idempotent matrices), then in all nontrivial cases a product of any length having P₁ and P₂ as its factors occurring alternately is equal to another such product if and only if P₁ and P₂ commute. In the present paper a generalization of this result is proposed and validity of the equivalence between commutativity property and any equality involving two linear combinations of two any length products having orthogonal projectors P₁ and P₂ as their factors occurring alternately is investigated. The second problem discussed in this paper concerns specific generalized inverses of the sum P₁ + P₂ and the difference P₁ − P₂ of (not necessary orthogonal) commuting projectors P₁ and P₂. The results obtained supplement those provided in Section 4 of Baksalary and Baksalary [J.K. Baksalary, O.M. Baksalary, Commutativity of projectors, Linear Algebra Appl. 341 (2002) 129-142].     

Preservers of pairs of bivectors with bounded distance

We extend Liu’s fundamental theorem of the geometry of alternate matrices to the second exterior power of an infinite dimensional vector space and also use her theorem to characterize surjective mappings T from the vector space V of all n×n alternate matrices over a field with at least three elements onto itself such that for any pair A, B in V, rank(A-B)?2k if and only if rank(T(A)-T(B))?2k, where k is a fixed positive integer such that n?2k+2 and k?2.     

Generalizations of a property of orthogonal projectors

Generalizing the result in Lemma of Baksalary and Baksalary [J.K. Baksalary, O.M. Baksalary, Commutativity of projectors, Linear Algebra Appl. 341 (2002) 129-142], Baksalary et al. [J.K. Baksalary, O.M. Baksalary, T. Szulc, Linear Algebra Appl. 354 (2002) 35-39] have shown that if P₁ and P₂ are orthogonal projectors, then, in all nontrivial situations, a product of any length having P₁ and P₂ as its factors occurring alternately is equal to another such product if and only if P₁ and P₂ commute, in which case all products involving P₁ and P₂ reduce to the orthogonal projector P₁P₂ (= P₂P₁). In the present paper, further generalizations of this property are established. They consist in replacing a product of the type specified above, appearing on the left-hand side (say) of the equality under considerations, by an affine combination of two or three such products. Comments on the problem when the number of components in a combination exceeds three are also given.     

The cross-section body, plane sections of convex bodies and approximation of convex bodies, I

For a convex body K ^d we investigate three associated bodies, its intersection body IK (for 0int K), cross-section body CK, and projection body IIK, which satisfy IKCKIIK. Conversely we prove CKconst₁(d)I(K–x) for some xint K, and IIKconst₂ (d)CK, for certain constants, the first constant being sharp. We estimate the maximal k-volume of sections of 1/2(K+(-K)) with k-planes parallel to a fixed k-plane by the analogous quantity for K; our inequality is, if only k is fixed, sharp. For L ^d a convex body, we take n random segments in L, and consider their Minkowski average D. We prove that, for V(L) fixed, the supremum of V(D) (with also nN arbitrary) is minimal for L an ellipsoid. This result implies the Petty projection inequality about max V((IIM)*), for M ^d a convex body, with V(M) fixed. We compare the volumes of projections of convex bodies and the volumes of the projections of their sections, and, dually, the volumes of sections of convex bodies and the volumes of sections of their circumscribed cylinders. For fixed n, the pth moments of V(D) (1p<) also are minimized, for V(L) fixed, by the ellipsoids. For k=2, the supremum (nN arbitrary) and the pth moment (n fixed) of V(D) are maximized for example by triangles, and, for L centrally symmetric, for example by parallelograms. Last we discuss some examples for cross-section bodies.Research (partially) supported by Hungarian National Foundation for Scientific Research, Grant No. 41.     

Schröder matrix as inverse of Delannoy matrix

Using Riordan arrays, we introduce a generalized Delannoy matrix by weighted Delannoy numbers. It turns out that Delannoy matrix, Pascal matrix, and Fibonacci matrix are all special cases of the generalized Delannoy matrices, meanwhile Schröder matrix and Catalan matrix also arise in involving inverses of the generalized Delannoy matrices. These connections are the focus of our paper. The half of generalized Delannoy matrix is also considered. In addition, we obtain a combinatorial interpretation for the generalized Fibonacci numbers.     

Revisitation of the product of two orthogonal projectors

Several results involving a product of two orthogonal projectors (i.e., Hermitian idempotent matrices) are established by exploring a representation of the product as a partitioned matrix. These results concern, for instance, rank, trace, range, null space, generalized inverses, and spectral properties of the product and its various functions. Particular attention is paid to the conditions equivalent to the requirement that the product of two orthogonal projectors is an orthogonal projector itself, and these characterizations refer to such known classes of matrices as Hermitian, involutory, normal, star-dagger, unitary as well as partial isometries and semi-orthogonal projectors. Moreover, some results dealing with the notions of parallel sum and spectral norm are obtained. The variety of problems considered shows that the approach utilized in the paper provides a powerful tool of wide applicability.     

Characterization of an extremal sum of ridge functions

The approximation problem considered in the paper is to approximate a continuous multivariate function f(x)=f(_x1,…,_xd)$f(\mathbf{x})=f(x_1,\dots ,x_d)$ by sums of two ridge functions in the uniform norm. We give a necessary and sufficient condition for a sum of two ridge functions to be a best approximation to f(x)$f(\mathbf{x})$. This main result is next used in a special case to obtain an explicit formula for the approximation error and to construct one best approximation. The problem of well approximation by such sums is also considered.     

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.     

Isometries of a generalized numerical radius

For 0<q<1, the q-numerical range is defined on the algebra M_n of all n×n complex matrices by

W_q(A)={x^∗Ay:x,y∈Cⁿ,∥x∥=∥y∥=1,〈y,x〉=q}.     

A system of real quaternion matrix equations with applications

Let H be the real quaternion algebra and H^n×m denote the set of all n×m matrices over H. Let P∈H^n×n and Q∈H^m×m be involutions, i.e., P²=I,Q²=I. A matrix A∈H^n×m is said to be (P,Q)-symmetric if A=PAQ. This paper studies the system of linear real quaternion matrix equations

     

On the span of Hadamard products of vectors

Let A₁, … , A_k be positive semidefinite matrices and B₁, … , B_k arbitrary complex matrices of order n. We show that

span{(A₁x)°(A₂x)°?°(A_kx)|x∈Cⁿ}=range(A₁°A₂°?°A_k)     

Topological vector spaces, compacta, and unions of subspaces

In the first two sections, we study when a σ-compact space can be covered by a point-finite family of compacta. The main result in this direction concerns topological vector spaces. Theorem 2.4 implies that if such a space L admits a countable point-finite cover by compacta, then L has a countable network. It follows that if f is a continuous mapping of a σ-compact locally compact space X onto a topological vector space L, and fibers of f are compact, then L is a σ-compact space with a countable network (Theorem 2.10). Therefore, certain σ-compact topological vector spaces do not have a stronger σ-compact locally compact topology.In the last, third section, we establish a result going in the orthogonal direction: if a compact Hausdorff space X is the union of two subspaces which are homeomorphic to topological vector spaces, then X is metrizable (Corollary 3.2).     

Determinantal divisors of products of matrices over Dedekind domains

Given three lists of ideals of a Dedekind domain, the question is raised whether there exist two matrices A and B with entries in the given Dedekind domain, such that the given lists of ideals are the determinantal divisors of A, B, and AB, respectively. To answer this question, necessary and sufficient conditions are developed in this article.     

Positive definite solutions of some matrix equations

In this paper we investigate some existence questions of positive semi-definite solutions for certain classes of matrix equations known as the generalized Lyapunov equations. We present sufficient and necessary conditions for certain equations and only sufficient for others.     

Balanced minimum covers of a finite set

In this paper a method of enumeration for n-balanced, labelled, minimum covers of a finite set of m indistinguishable elements is developed. The method is then used to tabulate the number of such covers for small values of m and n by means of a generating function.