Further results on the reverse-order law

An explicit expression is obtained for a pair of generalized inverses (B^?,A^?) such that B^?A^?=(AB)⁺_MN, and a class of pairs (B^?,A^? of this property is shown. A necessary and sufficient condition for (AB)^? to have the expression B^?A^? is also given.     

Simultaneous triangularization of a pair of matrices whose commutator has rank two

Let A,B be n×n matrices with entries in an algebraically closed field F of characteristic zero, and let C=AB?BA. It is shown that if C has rank two and AⁱB^jC^k is nilpotent for 0?i, j?n?1, 1?k?2, then A, B are simultaneously triangularizable over F. An example is given to show that this result is in some sense best possible.     

On a vector space analogue of Kneser’s theorem

In a previous paper [X. Hou, K.H. Leung, Q. Xiang, A generalization of an addition theorem of Kneser, J. Number Theory 97 (2002) 1-9], the following result was established: let E⊂K be fields such that the algebraic closure of E in K is separable over E. Let A,B be E-subspaces of K such that 0<dim_EA<∞ and 0<dim_EB<∞. Then dim_EAB?dim_EA+dim_EB-dim_EH(AB), where AB is the E-space generated by {ab:a∈A,b∈B} and H(AB)={x∈K:xAB⊂AB}. The separability assumption was essential in the proof of this result. However, even without the separability assumption, no counterexample is known. The present paper shows that no counterexample can be found if dim_EA?5.     

Closed range and relative regularity for products

Let B be a closed linear transformation of the Banach space X into the Banach space Y and let A be a bounded linear transformation of Y into the Banach space Z. A simple condition is shown to be necessary and sufficient for AB to have closed range. Provided B is relatively regular there is a simple necessary and sufficient condition for AB to be relatively regular. Provided B⁺ and A⁺ are pseudoinverses for B and A, respectively, the condition that B⁺A⁺ is a pseudoinverse for AB is completely characterized.     

A necessary and sufficient condition for a product relation to be total

Let A, B, and C be sets, let ? be a relation on A × B, and let σ be a relation on B × C. A necessary and sufficient condition for ? ° σ to be total is provided in terms of a DeMorgan algebra defined on B.     

On linear versions of some addition theorems

Let K ? L be a field extension. Given K-subspaces A, B of L, we study the subspace ?AB? spanned by the product set AB = {ab∣ a ∈ A, b ∈ B}. We obtain some lower bounds on dim _K ?AB? and dim _K ?B ⁿ ? in terms of dim _K A, dim _K B and n. This is achieved by establishing linear versions of constructions and results in additive number theory mainly due to Kemperman and Olson.     

Jordan forms for mutually annihilating nilpotent pairs

In this paper we completely characterize all possible pairs of Jordan canonical forms for mutually annihilating nilpotent pairs, i.e. pairs (A,B) of nilpotent matrices such that AB=BA=0.     

Preservers of eigenvalue inclusion sets of matrix products

For a square matrix A, let S(A) be an eigenvalue inclusion set such as the Gershgorin region, the union of Cassini ovals, and the Ostrowski’s set. Characterization is obtained for maps Φ on n×n matrices satisfying S(Φ(A)Φ(B))=S(AB) for all matrices A and B.     

Eigenvalue bounds for AB+BA, with A,B positive definite matrices

Bounds are derived for the eigenvalues of the Hermitian matrix C given by C=AB+BA, where A and B are positive definite, Hermitian, complex matrices. A sufficient condition is given for C to be positive definite.     

A normality criterion for an algebra of matrices

It is shown that the real algebra generated by a pair A,B of n × n (complex) matrices consists entirely of normal matrices if and only if A,B,AB,A + B and A + AB are normal.     

Generalized intersection patterns and two-symbol balanced arrays

Let m be a fixed integer, m = {0,1,?,m ? 1}; let C be a family of nonvoid subsets of m, and let R be a hereditary subfamily of C. Given finite sets A_m,…,A_m?1 such that ∩ _i?BA_i = Ø for all B ? m, B ? C, the vector of |∩_i?RA_i| (R_j?R) is called a C-supported R-intersection pattern. The characterization of the Y_RC of such patterns is a difficult combinatorial problem even for m=5 and simple families R and C. We study the algebraic structure of the convex cone Y_RC and its dual, and an integer linear-programming aspect of the problem; in particular we introduce the notion of content and pseudocontent. A relaxation leads to quadratic and higher forms over certain subsets of reals. As an application we study the natural link between highly symmetric patterns and two-symbol balanced arrays.     

Annihilator-preserving maps, multipliers, and derivations

For a commutative subspace lattice L in a von Neumann algebra N and a bounded linear map f:N∩algL→B(H), we show that if Af(B)C=0 for all A,B,C∈N∩algL satisfying AB=BC=0, then f is a generalized derivation. For a unital C^∗-algebra A, a unital Banach A-bimodule M, and a bounded linear map f:A→M, we prove that if f(A)B=0 for all A,B∈A with AB=0, then f is a left multiplier; as a consequence, every bounded local derivation from a C^∗-algebra to a Banach A-bimodule is a derivation. We also show that every local derivation on a semisimple free semigroupoid algebra is a derivation and every local multiplier on a free semigroupoid algebra is a multiplier.     

Maps preserving the nilpotency of products of operators

Let B(X) be the algebra of all bounded linear operators on the Banach space X, and let N(X) be the set of nilpotent operators in B(X). Suppose ?:B(X)→B(X) is a surjective map such that A,B∈B(X) satisfy AB∈N(X) if and only if ?(A)?(B)∈N(X). If X is infinite dimensional, then there exists a map f:B(X)→C?{0} such that one of the following holds:

(a)

There is a bijective bounded linear or conjugate-linear operator S:X→X such that ? has the form A?S[f(A)A]S^-1.

(b)

The space X is reflexive, and there exists a bijective bounded linear or conjugate-linear operator S : X′ → X such that ? has the form A ? S[f(A)A′]S⁻¹.

If X has dimension n with 3 ? n < ∞, and B(X) is identified with the algebra M_n of n × n complex matrices, then there exist a map f:M_n→C?{0}, a field automorphism ξ:C→C, and an invertible S ∈ M_n such that ? has one of the following forms:

     

A note on generating <Emphasis Type="Italic">P</Emphasis>-matrices

We prove that for any \({A,B\in\mathbb{R}^{n\times n}}\) such that each matrix S satisfying min(A, B) ≤ S ≤ max(A, B) is nonsingular, all four matrices A ^?1 B, AB ^?1, B ^?1 A and BA ^?1 are P-matrices. A practical method for generating P-matrices is drawn from this result.     

The characteristic polynomial of a principal subpencil of a Hermitian matrix pencil

Given n-square Hermitian matrices A,B, let A_i,B_i denote the principal (n?1)- square submatrices of A,B, respectively, obtained by deleting row i and column i. Let μ, λ be independent indeterminates. The first main result of this paper is the characterization (for fixed i) of the polynomials representable as det(μA_i?λB_i) in terms of the polynomial det(μA?λB) and the elementary divisors, minimal indices, and inertial signatures of the pencil μA?λB. This result contains, as a special case, the classical interlacing relationship governing the eigenvalues of a principal sub- matrix of a Hermitian matrix. The second main result is the determination of the number of different values of i to which the characterization just described can be simultaneously applied.     

On the a-Browder and a-Weyl spectra of tensor products

Given Banach space operators A ∈ B( ) and B ∈ B( ), let A?B ∈ B( ? ) denote the tensor product of A and B. Let σ _a , σ _aw and σ _ab denote the approximate point spectrum, the Weyl approximate point spectrum and the Browder approximate point spectrum, respectively. Then σ _aw (A?B) ? σ _a (A)σ _aw (B) ? σ _aw (A)σ _a (B) ? σ _a (A)σ _ab (B) ? σ _ab (A)σ _a (B) = σ _ab (A?B), and a sufficient condition for the (a-Weyl spectrum) identity σ _aw (A?B) = σ _a (A)σ _aw (B) ? σ _aw (A)σ _a (B) to hold is that σ _aw (A?B) = σ _ab (A?B). Equivalent conditions are proved in Theorem 1, and the problem of the transference of a-Weyl’s theorem for a-isoloid operators A and B to their tensor product A?B is considered in Theorem 2. Necessary and sufficient conditions for the (plain) Weyl spectrum identity are revisited in Theorem 3.     

Spin Leonard pairs

Let ${\mathbb K}$ denote a field, and let V denote a vector space over ${\mathbb K}$ of finite positive dimension. A pair A, A* of linear operators on V is said to be a Leonard pair on V whenever for each B∈{A, A*}, there exists a basis of V with respect to which the matrix representing B is diagonal and the matrix representing the other member of the pair is irreducible tridiagonal. A Leonard pair A, A* on V is said to be a spin Leonard pair whenever there exist invertible linear operators U, U* on V such that UA = A U, U*A* = A*U*, and UA* U ^?1 = U*^?1 AU*. In this case, we refer to U, U* as a Boltzmann pair for A, A*. We characterize the spin Leonard pairs. This characterization involves explicit formulas for the entries of the matrices that represent A and A* with respect to a particular basis. The formulas are expressed in terms of four algebraically independent parameters. We describe all Boltzmann pairs for a spin Leonard pair in terms of these parameters. We then describe all spin Leonard pairs associated with a given Boltzmann pair. We also describe the relationship between spin Leonard pairs and modular Leonard triples. We note a modular group action on each isomorphism class of spin Leonard pairs.     

Some isomorphisms between pairs of latin squares

Let A, B, C, D be latin squares with A orthogonal to B and C orthogonal to D. The pair A, B is isomorphic with the pair C, D if the graph of A, B is graph-isomorphic with the graph of C, D. A characterization is given for determining when a pair A, B of latin squares is isomorphic with a self-orthogonal square C and its transpose. Self-orthogonal squares are important because they are both abundant and easy to store. An algorithm either displays a self-orthogonal square C and an isomorphism from A, B to C, C^T or, if none exists, gives a small set of blocks to the existence of such a square isomorphism.     

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.