Generalized Specker lattice-ordered groups and two types of distributivity

Let A be a lattice-ordered group, B a generalized Boolean algebra. The Boolean extension A _B of A has been investigated in the literature; we will refer to A _B as a generalized Specker lattice-ordered group (namely, if A is the linearly ordered group of all integers, then A _B is a Specker lattice-ordered group). The paper establishes that some distributivity laws extend from A _B to both A and B, and (under certain circumstances) also conversely.     

Shorted operators of partitioned matrices and application

In this paper, our main objective is to study the effect of appending/deleting a column/row on the shorted operators. It turns out that for matrices A and B for which the shorted operator S(A|B) exists, S(A₁|B₁) of the matrix A₁=[A:a] with respect to the matrix B₁=[B:b], when it exists, is obtained by appending a suitable column to S(A|B). Moreover, if S(A₁|B₁) exists, then S(A|B) exists and is obtained from S(A₁|B₁) by dropping its last column. In the process, we study the effect of appending/deleting a column/row on the space pre-order and the parallel sum of parallel summable matrices. Finally, we specialize to the case of and matrices and study the effect of bordering (by an additional column and a row) on the shorted operator. We conclude the paper with an application to Linear Models with singular dispersion structure.     

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 spectral radius and the spectral norm of Hadamard products of nonnegative matrices

We prove the spectral radius inequality ρ(A₁°A₂°?°A_k)?ρ(A₁A₂?A_k) for nonnegative matrices using the ideas of Horn and Zhang. We obtain the inequality ‖A°B‖?ρ(A^TB) for nonnegative matrices, which improves Schur’s classical inequality ‖A°B‖?‖A‖‖B‖, where ‖·‖ denotes the spectral norm. We also give counterexamples to two conjectures about the Hadamard product.     

Sieve-equivalence and explicit bijections

Suppose A₁,…, A_n are subsets of a finite set A, and B₁,…, B_n are subsets of a finite set B. For each subset S of N = {1, 2,…, n}, let A_s = ∩_i?SA_i and B_S = ∩_i?SB_i. It is shown that if explicit bijections f_S:A_S → B_S for each S ? N are given, an explicit bijection h:A-∪_i=1A_i→B-∪_i=1B_i can be constructed. The map h is independent of any ordering of the elements of A and B, and of the order in which the subsets A_i and B_i are listed.     

Simultaneous quasidiagonalization of complex matrices

Let

be the complex algebra generated by a pair of n × n Hermitian matrices A, B. A recent result of Watters states that A, B are simultaneously unitarily quasidiagonalizable [i.e., A and B are simultaneously unitarily similar to direct sums C₁⊕…⊕C_t,D₁⊕…⊕D_t for some t, where C_i, D_i are k_i × k_i and k_i?2(1?i?t)] if and only if [p(A, B), A]² and [p(A, B), B]² belong to the center of

for all polynomials p(x, y) in the noncommuting variables x, y. In this paper, we obtain a finite set of conditions which works. In particular we show that if A, B are positive semidefinite, then A, B are simultaneously quasidiagonalizable if (and only if) [A, B]², [A², B]² and [A, B²]² commute with A, B.     

A Uniqueness Theorem in a Finitely Accessible Additive Category

It is proved that if A ₁,A ₂,..., A _m and B ₁,B ₂,..., B _n are objects in a finitely accessible additive category \(\mathcal{A}\) such that their pure injective envelopes are indecomposable, and there are pure monomorphisms μ:A ₁?⊕?A ₂?⊕?...?⊕?A _m →B ₁?⊕?B ₂?⊕?...?⊕?B _n and ν:B ₁?⊕?B ₂?⊕?...?⊕?B _n →A ₁?⊕?A ₂?⊕?...?⊕?A _m , then m?=?n and there are a permutation σ and pure monomorphisms A _i →B _σ(i) and B _σ(i)→A _i for every i?=?1, 2, ..., n.     

A criterion of diagonalizability of a pair of matrices over the ring of principal ideals by common row and separate column transformations

We establish that a pair A, B, of nonsingular matrices over a commutative domain R of principal ideals can be reduced to their canonical diagonal forms D ^A and D ^B by the common transformation of rows and separate transformations of columns. This means that there exist invertible matrices U, V _A, and V _B over R such that UAV _a=D^A and UAV _B=D^B if and only if the matrices B _*A and D _* ^B D^A where B _* 0 is the matrix adjoint to B, are equivalent.     

Continuity properties of the Drazin pseudoinverse

Let B^D denote that Drazin inverse of the n×n complex matrix B. Define the core-rank of B as rank (B^i(B)) where i(B) is the index of B. Let j = 1,2,…, and A_j and A be square matrices such that A_i converges to A with respect to some norm. The main result of this paper is that A_j^D converges to A^D if and only if there exist a j₀ such that core-rank A_j=core-rankA for j ? j₀.     

The representation and perturbation of theW-weighted Drazin inverse

LetA andE bem x n matrices andW an n xm matrix, and letA _d,W denote the W-weighted Drazin inverse ofA. In this paper, a new representation of the W-weighted Drazin inverse ofA is given. Some new properties for the W-weighted Drazin inverseA _d,W and B_d,W are investigated, whereB =A+E. In addition, the Banach-type perturbation theorem for the W-weighted Drazin inverse ofA andB are established, and the perturbation bounds for ∥B_d,W∥ and ∥B_{d, W}, -A_d,W∥/∥A_d,W∥ are also presented. WhenA andB are square matrices andW is identity matrix, some known results in the literature related to the Drazin inverse and the group inverse are directly reduced by the results in this paper as special cases.     

Generalizations of the reverse order law with related results

This paper presents conditions which are necessary and sufficient for (AB>)⁺ = B⁺A^ω for all normalized generalized inverses A^ω of the complex matrix A. Corresponding conditions are stated which are equivalent to the situation where (AB)⁺ = B_ωA⁺ is satisfied by each weak generalized inverse B_ω of B. The results are applied to theorems by Baskett and Katz and by Schwerdtfeger.     

Identities in combinatorics,I: on sorting two ordered sets

A binomial coefficient identity equivalent to Saalschutz's summation of a ₃F₂ hypergeometric series is proved combinatorially. The proof depends on the enumeration of ordered pairs (A,B) of subsets of{1,2,3,…,ν} in which |A|=n, |B|=m, and B contains exactly r elements of the first n elements of A∪B.     

Explicit solutions to the reverse order law (AB)+ = B-mrA-lr

Shinozaki and Sibuya have shown that the Moore-Penrose inverse (AB)⁺ can always be expressed as B^-A^- for generalized inverses A^- and B^- of matrices A and B, respectively. In this paper, explicit solutions B^-_mr and A^-_lr to (AB)⁺ = B^-_mrA^-_lr are given. A class of solutions is obtained which is related to an equation of Greville, and expressions for the general solutions are presented.     

A localization inequality for set functions

We prove the following theorem, which is an analog for discrete set functions of a geometric result of Lovász and Simonovits. Given two real-valued set functions f₁,f₂ defined on the subsets of a finite set S, satisfying for i∈{1,2}, there exists a positive multiplicative set function μ over S and two subsets A,B⊆S such that for i∈{1,2}μ(A)f_i(A)+μ(B)f_i(B)+μ(A∪B)f_i(A∪B)+μ(A∩B)f_i(A∩B)?0. The Ahlswede-Daykin four function theorem can be deduced easily from this.     

Boundaries for algebras of holomorphic functions

Let A_u(B_G) be the Banach algebra of all complex valued functions defined on the closed unit ball B_G of a complex Banach space G which are uniformly continuous on B_G and holomorphic in the interior of B_G, endowed with the sup norm. A characterization of the boundaries for A_u(B_G) is given in case G belongs to a class of Banach spaces that includes the pre-dual of a Lorentz sequence space studied by Gowers in Israel J. Math. 69 (1990) 129-151. The non-existence of the Shilov boundary for A_u(B_G) is also proved.     

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.     

Going up along absolutely flat morphisms

Flat morphisms from A to B (commutative and unitary rings) such that the multiplication B?_AB→B is flat, have many of the properties of ind-etale morphisms. They don't raise weak dimension. As a consequence they preserve integral closure. In the local case they are the extensions of A that have the same strict henselian extensions as A.     

On rational invariants of a set of symmetric matrices

Some identities resulting from the Cayley-Hamilton theorem are derived. Some applications include: (a) for k = 1,2,…,n ? 1 a condition is found for a pair (A,B) of symmetric operators acting in Euclidean n-space to have common invariant k-subspace (provided that A does not have multiple eigenvalues); (b) it is shown that the field of rational invariants of (A,B) is isomorphic to a subfield of a rational function field with n(n+3)/2 generators consisting of elements symmetric with respect to the permutaion group P_n; (c) it is shown that any rational invariant of (g+2) symmetric operators A,B,C₁,C₂,…, C_g can be expressed as a rational function of invariants of one or two operators that are taken for pairs (A,B), (A,C₂),…, (A,C_g, (A,B+C₁), (A,B+C₂),…,(A,B+C_g).     

A general Trotter-Kato formula for a class of evolution operators

In this article we prove new results concerning the existence and various properties of an evolution system UA+B(t,s)_0?s?t?T generated by the sum −(A(t)+B(t)) of two linear, time-dependent and generally unbounded operators defined on time-dependent domains in a complex and separable Banach space B. In particular, writing L(B) for the algebra of all linear bounded operators on B, we can express UA+B(t,s)_0?s?t?T as the strong limit in L(B) of a product of the holomorphic contraction semigroups generated by −A(t) and −B(t), respectively, thereby proving a product formula of the Trotter-Kato type under very general conditions which allow the domain D(A(t)+B(t)) to evolve with time provided there exists a fixed set D⊂_?t∈[0,T]D(A(t)+B(t)) everywhere dense in B. We obtain a special case of our formula when B(t)=0, which, in effect, allows us to reconstruct UA(t,s)_0?s?t?T very simply in terms of the semigroup generated by −A(t). We then illustrate our results by considering various examples of nonautonomous parabolic initial-boundary value problems, including one related to the theory of time-dependent singular perturbations of self-adjoint operators. We finally mention what we think remains an open problem for the corresponding equations of Schrödinger type in quantum mechanics.