A short proof of a cross-intersection theorem of Hilton

Families A₁,…,A_k of sets are said to be cross-intersecting if for any A_i∈A_i and A_j∈A_j, i≠j. A nice result of Hilton that generalises the Erd?s-Ko-Rado (EKR) Theorem says that if r≤n/2 and A₁,…,A_k are cross-intersecting sub-families of , then

     

Mapping properties for oscillatory integrals in d-dimensions

For aj,bj?1, j=1,2,…,d, we prove that the operator maps into itself for , where , and k(x,y)=φ(x,y)eig(x,y), φ(x,y) satisfies (1.2) (e.g. φ(x,y)=|x−y|^iτ,τ real) and the phase g(x,y)=xa⋅yb. We study operators with more general phases and for these operators we require that aj,bj>1, j=1,2,…,d, or al=bl?1 for some l∈{1,2,…,d}.     

On the perturbation of the group generalized inverse for a class of bounded operators in Banach spaces

Given a bounded operator A on a Banach space X with Drazin inverse AD and index r, we study the class of group invertible bounded operators B such that I+AD(B−A) is invertible and R(B)∩N(Ar)={0}. We show that they can be written with respect to the decomposition X=R(Ar)⊕N(Ar) as a matrix operator, , where B₁ and are invertible. Several characterizations of the perturbed operators are established, extending matrix results. We analyze the perturbation of the Drazin inverse and we provide explicit upper bounds of ‖B^?−AD‖ and ‖BB^?−ADA‖. We obtain a result on the continuity of the group inverse for operators on Banach spaces.     

A numerical characterization of the S2-ification of a Rees algebra

Let A be a local ring with maximal ideal . For an arbitrary ideal I of A, we define the generalized Hilbert coefficients . When the ideal I is -primary, j_k(I)=(0,…,0,(−1)^ke_k(I)), where e_k(I) is the classical kth Hilbert coefficient of I. Using these coefficients we give a numerical characterization of the homogeneous components of the S₂-ification of S=A[It,t⁻¹], extending previous results obtained by the author to not necessarily -primary ideals.     

Schur-class multipliers on the Arveson space: De Branges-Rovnyak reproducing kernel spaces and commutative transfer-function realizations

An interesting and recently much studied generalization of the classical Schur class is the class of contractive operator-valued multipliers S(λ) for the reproducing kernel Hilbert space H(kd) on the unit ball Bd⊂Cd, where kd is the positive kernel kd(λ,ζ)=1/(1−〈λ,ζ〉) on Bd. The reproducing kernel space H(KS) associated with the positive kernel KS(λ,ζ)=(I−S(λ)S^∗(ζ))⋅kd(λ,ζ) is a natural multivariable generalization of the classical de Branges-Rovnyak canonical model space. A special feature appearing in the multivariable case is that the space H(KS) in general may not be invariant under the adjoints of the multiplication operators on H(kd). We show that invariance of H(KS) under for each j=1,…,d is equivalent to the existence of a realization for S(λ) of the form S(λ)=D+C⁻¹(I−λ₁A₁−?−λdAd)(λ₁B₁+?+λdBd) such that connecting operator has adjoint U^∗ which is isometric on a certain natural subspace (U is “weakly coisometric”) and has the additional property that the state operators A₁,…,Ad pairwise commute; in this case one can take the state space to be the functional-model space H(KS) and the state operators A₁,…,Ad to be given by (a de Branges-Rovnyak functional-model realization). We show that this special situation always occurs for the case of inner functions S (where the associated multiplication operator MS is a partial isometry), and that inner multipliers are characterized by the existence of such a realization such that the state operators A₁,…,Ad satisfy an additional stability property.     

Multilinear maps on products of operator algebras

Let A₁,A₂,…,A_r be C∗-algebras with second duals A₁″,A₂″,…,A_r″, and let X be an arbitrary Banach space. Let be a bounded r-linear map, and denote by the Johnson-Kadison-Ringrose extension (i.e., the separately to continuous r-linear extension) of Γ. The problem of characterising those Γ for which Γ″ takes its values in X was solved by Villanueva when the algebras are all commutative. Because the Dunford-Pettis property fails for noncommutative C∗-algebras, the ‘obvious’ extension of Villanueva's characterisation does not give the correct condition. In this paper we solve this problem for general C∗-algebras. This result is then applied to obtaining a multilinear generalisation of the normal-singular decomposition of a bounded linear operator on a von Neumann algebra.     

Transition matrices for well-conditioned Markov chains

Let T∈R^n×n be an irreducible stochastic matrix with stationary distribution vector π. Set A = I − T, and define the quantity , where A_j, j = 1, … , n, are the (n − 1) × (n − 1) principal submatrices of A obtained by deleting the jth row and column of A. Results of Cho and Meyer, and of Kirkland show that κ₃ provides a sensitive measure of the conditioning of π under perturbation of T. Moreover, it is known that .In this paper, we investigate the class of irreducible stochastic matrices T of order n such that , for such matrices correspond to Markov chains with desirable conditioning properties. We identify some restrictions on the zero-nonzero patterns of such matrices, and construct several infinite classes of matrices for which κ₃ is as small as possible.     

Quantum effects, sequential independence and majorization

A quantum effect is a positive Hilbert space contraction operator. If {E_i}, 1?i?n, are n quantum effects (defined on some Hilbert space H), then their sequential product is the operator . It is proved that the quantum effects {E_i}, 1?i?n, are sequentially independent if and only if for every permutation r₁r₂…r_n of the set S_n={1,2,…,n}. The sequential independence of the effects E_i, 1?i?n, implies E_noE_n-1o…oE_j+1oE_jo…oE₁=(E_noE_n-1o…E_j+1)oE_jo…oE₁ for every 1?j?n. It is proved that if there exists an effect E_j, 1?j?n, such that E_j?(E_noE_n-1o…E_j+1)oE_jo…oE₁, then the effects {E_i} are sequentially independent and satisfy .     

Davenport constant with weights and some related questions, II

Let G be a finite abelian group of order n and let A⊆Z be non-empty. Generalizing a well-known constant, we define the Davenport constant of G with weight A, denoted by DA(G), to be the least natural number k such that for any sequence (x₁,…,xk) with xi∈G, there exists a non-empty subsequence (xj₁,…,xjl) and a₁,…,al∈A such that . Similarly, for any such set A, EA(G) is defined to be the least t∈N such that for all sequences (x₁,…,xt) with xi∈G, there exist indices j₁,…,jn∈N,1?j₁<?<jn?t, and ?₁,…,?n∈A with . In the present paper, we establish a relation between the constants DA(G) and EA(G) under certain conditions. Our definitions are compatible with the previous generalizations for the particular group G=Z/nZ and the relation we establish had been conjectured in that particular case.     

Weakly Birkhoff recurrent switching signals, almost sure and partial stability of linear switched dynamical systems

Let {A₁,…,AK}⊂Cd×d be arbitrary K matrices, where K and d both ?2. For any 0<Δ<∞, we denote by the set of all switching sequences u=(λ.,t.):N→{1,…,K}×R₊ satisfying tj−tj−1?Δ and

     

Riesz transforms and conjugacy for Laguerre function expansions of Hermite type

Riesz transforms and conjugate Poisson integrals for multi-dimensional Laguerre function expansions of Hermite type with index α are defined and investigated. It is proved that for any multi-index α=(α₁,…,αd) such that αi?−1/2, αi∉(−1/2,1/2), the appropriately defined Riesz transforms , j=1,2,…,d, are Calderón-Zygmund operators, hence their mapping properties follow from a general theory. Similar mapping results are obtained in one dimension, without excluding α∈(−1/2,1/2), by means of a local Calderón-Zygmund theory and weighted Hardy's inequalities. The conjugate Poisson integrals are shown to satisfy a system of Cauchy-Riemann type equations and to recover the Riesz-Laguerre transforms on the boundary. The two specific values of α, (−1/2,…,−1/2) and (1/2,…,1/2), are distinguished since then a connection with Riesz transforms for multi-dimensional Hermite function expansions is established.     

On various restricted sumsets

For finite subsets A₁,…,A_n of a field, their sumset is given by . In this paper, we study various restricted sumsets of A₁,…,A_n with restrictions of the following forms:

     

Riesz transforms for multi-dimensional Laguerre function expansions

Riesz transforms and conjugate Poisson integrals for multi-dimensional Laguerre function expansions of type α are defined and investigated. It is proved that for any multi-index α=(α₁,…,αd) such that αi?−1/2, the appropriately defined Riesz-Laguerre transforms , j=1,2,…,d, are Calderón-Zygmund operators in the sense of the associated space of homogeneous type, hence their mapping properties follow from the general theory. Similar results are obtained for all higher order Riesz-Laguerre transforms. The conjugate Poisson integrals are shown to satisfy a system of equations of Cauchy-Riemann type and to recover the Riesz-Laguerre transforms on the boundary.     

Operator algebras associated with unitary commutation relations

We define nonselfadjoint operator algebras with generators Le₁,…,Len,Lf₁,…,Lfm subject to the unitary commutation relations of the form

     

A Robertson-type uncertainty principle and quantum Fisher information

Let A₁,…,A_N be complex self-adjoint matrices and let ρ be a density matrix. The Robertson uncertainty principle

     

The spectra of some trees and bounds for the largest eigenvalue of any tree

Let T be an unweighted tree of k levels such that in each level the vertices have equal degree. Let n_k−j+1 and d_k−j+1 be the number of vertices and the degree of them in the level j. We find the eigenvalues of the adjacency matrix and Laplacian matrix of T for the case of two vertices in level 1 (n_k = 2), including results concerning to their multiplicity. They are the eigenvalues of leading principal submatrices of nonnegative symmetric tridiagonal matrices of order k × k. The codiagonal entries for these matrices are , 2 ? j ? k, while the diagonal entries are 0, …, 0, ±1, in the case of the adjacency matrix, and d₁, d₂, …, d_k−1, d_k ± 1, in the case of the Laplacian matrix. Finally, we use these results to find improved upper bounds for the largest eigenvalue of the adjacency matrix and of the Laplacian matrix of any given tree.