The Geometry of Convex Transitive Banach Spaces

Throughout this paper, X will denote a Banach space, S=S(X)and B=B(X) will be the unit sphere and the closed unit ballof X, respectively, and G=G(X) will stand for the group of allsurjective linear isometries on X. Unless explicitly statedotherwise, all Banach spaces will be assumed to be real. Nevertheless,by passing to real structures, the results remain true for complexspaces. 1991 Mathematics Subject Classification 46B04, 46B10,46B22.     

Inertias and ranks of some Hermitian matrix functions with applications

Let S be a given set consisting of some Hermitian matrices with the same size. We say that a matrix A ∈ S is maximal if A − W is positive semidefinite for every matrix W ∈ S. In this paper, we consider the maximal and minimal inertias and ranks of the Hermitian matrix function f(X,Y) = P − QXQ* − TYT*, where * means the conjugate and transpose of a matrix, P = P*, Q, T are known matrices and for X and Y Hermitian solutions to the consistent matrix equations AX =B and YC = D respectively. As applications, we derive the necessary and sufficient conditions for the existence of maximal matrices of

Numerical Radius Norms on Operator Spaces

We introduce a numerical radius operator space (X, W_n). Theconditions to be a numerical radius operator space are weakerthan Ruan's axiom for an operator space (X, O_n). Let w(·)be the numerical radius on B(H). It is shown that, if X admitsa norm W_n(·) on the matrix space M_n(X) which satisfiesthe conditions, then there is a complete isometry, in the senseof the norms W_n(·) and w_n(·), from (X, W_n) into(B(H), w_n). We study the relationship between the operator space(X, O_n) and the numerical radius operator space (X, W_n). Thecategory of operator spaces can be regarded as a subcategoryof numerical radius operator spaces.     

Generalisation of the Bogomolov-Miyaoka-Yau Inequality to Singular Surfaces

The paper considers pairs (X, B) where X is a normal projectivesurface over C, and B is a Q-divisor whose coefficients are1 or 1–1/m for some natural number m. A log canonicalsingularity on such a pair is a quotient by a finite or infinitegroup, so if (X, B) has log canonical singularities, the orbifoldEuler number e_orb(X, B) can be defined. The main result is aBogomolov-Miyaoka-Yau-type inequality which implies that if(X, B) has log canonical singularities and (X, K_X + B) 0 then(K_X+B)² 3e_orb(X, B). The actual inequality proved is somewhatstronger and it also implies all the previously published versionsof the Bogomolov-Miyaoka-Yau inequality. The proof involvesthe Log Minimal Model Program, Q-sheaves when K_X+B is nef, anda study of the changes in the two sides of the inequality undera contraction. The paper also contains a further generalisationwhere the coefficients of B can be arbitrary rational numbersin [0, 1], a different condition is imposed on the singularitiesand K_X+B is required to be nef. Some applications of the inequalitiesare also given, for example, estimating the number of singularitiesor certain kinds of configurations of curves on surfaces. 1991Mathematics Subject Classification: 14J17, 14J60, 14C17.     

POLYNOMIAL NUMERICAL INDEX FOR SOME COMPLEX VECTOR-VALUED FUNCTION SPACES

We study the relation between the polynomial numerical indicesof a complex vector-valued function space and the ones of itsrange space. It is proved that the spaces C(K, X) and L(µ,X) have the same polynomial numerical index as the complex Banachspace X for every compact Hausdorff space K and every -finitemeasure µ, which does not hold any more in the real case.We give an example of a complex Banach space X such that, forevery k 2, the polynomial numerical index of order k of X isthe greatest possible, namely 1, while the one of X** is theleast possible, namely k^k/(1–k). We also give new examplesof Banach spaces with the polynomial Daugavet property, namelyL(µ, X) when µ is atomless, and C_w(K, X), C_w*(K,X*) when K is perfect.     

An interlacing theorem for matrices whose graph is a given tree

Let A and B be (n×n)-matrices. For an index set S ⊂ {1, …, n}, denote by A(S) the principal submatrix that lies in the rows and columns indexed by S. Denote by S′ the complement of S and define η(A, B) = det A(S) det B(S′), where the summation is over all subsets of {1, …, n} and, by convention, det A(∅) = det B(∅) = 1. C. R. Johnson conjectured that if A and B are Hermitian and A is positive semidefinite, then the polynomial η(λA,-B) has only real roots. G. Rublein and R. B. Bapat proved that this is true for n ⩽ 3. Bapat also proved this result for any n with the condition that both A and B are tridiagonal. In this paper, we generalize some little-known results concerning the characteristic polynomials and adjacency matrices of trees to matrices whose graph is a given tree and prove the conjecture for any n under the additional assumption that both A and B are matrices whose graph is a tree. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 10, No. 3, pp. 245–254, 2004.     

On a Topological Property of certain Calkin Algebras

Let X = 1^p, 1 p < , or X = c₀, B(X) be the algebra of allbounded linear operators on X, H(X) be the ideal of compactoperators in B(X), and C(X) = B(X)/H(X) be the Calkin algebraon X. For TB(X), let ||T||_c = dist(T, H(X)) be the essentialnorm of T that is the norm of T+H(X) in C(X). It is shown thatfor any operator TB(X) and any number 0 < t < 1, thereexists a closed infinite dimensional subspace Z Z X such that ||Tx|| t||T||_c, for all x Z. As a consequence, it is shown that every (not necessarily complete)submultiplicative norm on the Calkin algebra C(X) is equivalentto the quotient norm || ||_c on C(X).     

Norm Estimates for Commutators of Operators

Given two self-adjoint Hilbert space operators A, B, and a continuousfunction f, we prove several inequalities of the form ||f(A)X–Xf(B)||C(f)||AX–XB|| involving the L_p-norm of the derivative f' and the Besov B^r_,1-normof f.     

Kazhdan-Lusztig Cells, q-Schur Algebras and James' Conjecture

We consider the Dipper–James q-Schur algebra S_q(n, r)_k,defined over a field k and with parameter q 0. An understandingof the representation theory of this algebra is of considerableinterest in the representation theory of finite groups of Lietype and quantum groups; see, for example, [6] and [11]. Itis known that S_q(n, r)_k is a semisimple algebra if q is nota root of unity. Much more interesting is the case when S_q(n,r)_k is not semisimple. Then we have a corresponding decompositionmatrix which records the multiplicities of the simple modulesin certain ‘standard modules’ (or ‘Weyl modules’).A major unsolved problem is the explicit determination of thesedecomposition matrices.     

On the Number of Solutions of an Equation Over a Finite Field

The Stöhr–Voloch approach is used to obtain a newbound for the number of solutions in (F_q)² of an equation f(X,Y) = 0, where f(X, Y) is an absolutely irreducible polynomialwith coefficients in a finite field F_q.     

Rationally convex sets on the unit sphere in ℂ2

Let X be a rationally convex compact subset of the unit sphere S in ?², of three-dimensional measure zero. Denote by R(X) the uniform closure on X of the space of functions P/Q, where P and Q are polynomials and Q≠0 on X. When does R(X)=C(X)? Our work makes use of the kernel function for the $\bar{\delta}_{b}Let X be a rationally convex compact subset of the unit sphere S in ℂ², of three-dimensional measure zero. Denote by R(X) the uniform closure on X of the space of functions P/Q, where P and Q are polynomials and Q≠0 on X. When does R(X)=C(X)? Our work makes use of the kernel function for the operator on S, introduced by Henkin in [5] and builds on results obtained in Anderson–Izzo–Wermer [3]. We define a real-valued function ε _X on the open unit ball intB, with ε _X (z,w) tending to 0 as (z,w) tends to X. We give a growth condition on ε _X (z,w) as (z,w) approaches X, and show that this condition is sufficient for R(X)=C(X) (Theorem 1.1). In Section 4, we consider a class of sets X which are limits of a family of Levi-flat hypersurfaces in intB. For each compact set Y in ℂ², we denote the rationally convex hull of Y by . A general reference is Rudin [8] or Aleksandrov [1].     

Finiteness of integrals of functions of Levy processes

We prove necessary and sufficient conditions for the almostsure convergence of the integrals

and thus of ,where M_t = sup{|X_s|: s t} is the two-sided maximum processcorresponding to a Lévy process (X_t)_{t 0}, a(·)is a non-decreasing function on [0, ) with a(0) = 0, g(·)is a positive non-increasing function on (0, ), possibly withg(0 + ) = , and f(·) is a positive non-decreasing functionon [0, ) with f(0) = 0. The conditions are expressed in termsof the canonical measure, (·), of the process X_t. Thespecial case when a(x) = 0, f(x) = x and g(·) is equivalentto the tail of (at zero or infinity) leads to an interestingcomparison of M_t with the largest jump of X_t in (0, t]. Some results concerning the convergence at zero and infinityof integrals like t g(a(t) + |X_t|) dt, t g(S_t) dt,and t g(R_t) dt, where S_t is the supremum process and R_t= S_t – X_t is the process reflected in its supremum, arealso given. We also consider the convergence of integrals suchas , etc.     

The Milnor number of a function on a space curve germ

Given a finite function germ f:(X, 0) (, 0) on a reduced spacecurve singularity (X, 0), we show that µ(f) = µ(X,0) + deg(f) – 1. Here, µ(f) and µ(X, 0) denotethe Milnor numbers of the function and the curve, respectively,and deg(f) is the degree of f. We use this formula to obtainseveral consequences related to the topological triviality andWhitney equisingularity of families of curves and families offunctions on curves.     

Norms on Free Topological Groups

A norm on a group G is a function N mapping G into the set ofnon-negative real numbers such that for each x and y in G, N(xy^–1) N(x)+N(y) and N(e) = 0, where e is the identity element ofG. It is shown here that if F(X) is the free topological groupon any completely regular Hausdorff space X and H is a subgroupof F(X) generated by a finite subset of X, then any norm onH can be extended to a continuous norm on F(X).     

Sets of nonnegative matrices with positive inhomogeneous products

Let X be a set of k×k matrices in which each element is nonnegative. For a positive integer n, let P(n) be an arbitrary product of n matrices from X, with any ordering and with repetitions permitted. Define X to be a primitive set if there is a positive integer n such that every P(n) is positive [i.e., every element of every P(n) is positive]. For any primitive set X of matrices, define the index g(X) to be the least positive n such that every P(n) is positive. We show that if X is a primitive set, then g(X)?2^k?2. Moreover, there exists a primitive set Y such that g(Y) = 2^k?2.     

Mappings Preserving Submodules of Hilbert C*-Modules

A Hilbert module over a C*-algebra B is a right B-module X,equipped with an inner product ·, · which is linearover B in the second factor, such that X is a Banach space withthe norm ||x||:=||x, x||^1/2. (We refer to [8] for the basictheory of Hilbert modules; the basic example for us will beX=B with the inner product x, y=x*y.) We denote by B(X) thealgebra of all bounded linear operators on X, and we denoteby L(X) the C*-algebra of all adjointable operators. (In thebasic example X=B, L(X) is just the multiplier algebra of B.)Let A be a C*-subalgebra of L(X), so that X is an A-B-bimodule.We always assume that A is nondegenerate in the sense that [AX]=X,where [AX] denotes the closed linear span of AX. Denote by A_X the algebra of all mappings on X of the form (1.1) where m is an integer and a_iA, b_iB for all i. Mappings of form(1.1) will be called elementary, and this paper is concernedwith the question of which mappings on X can be approximatedby elementary mappings in the point norm topology.     

Realizing irreducible semigroups and real algebras of compact operators

Let B(X) be the algebra of bounded operators on a complex Banach space X. Viewing B(X) as an algebra over R, we study the structure of those irreducible subalgebras which contain nonzero compact operators. In particular, irreducible algebras of trace-class operators with real trace are characterized. This yields an extension of Brauer-type results on matrices to operators in infinite dimensions, answering the question: is an irreducible semigroup of compact operators with real spectra realizable, i.e., simultaneously similar to a semigroup whose matrices are real?     

A Numerical Method for the Solution of the Double Eigenvalue Problem

A generalization of the Rayleigh quotient iterative method,called the Minimum Residual Quotient Iteration (MRQI), is derivedfor the numerical solution of the 2-parameter eigenvalue problem;i.e. to find scalars µ and a corresponding vector x satisfyingthe following equations, Ax = B₁x + µB₂x, ||x|| = 1, f(x) = 0, where A and B are nxn real matrices, ||.|| denotes the l₂ normand f is a real functional. The method is applied to doubleeigenvalue problems for ordinary differential equations andcomputational results are presented.     

Lorentz Spaces Of Vector-Valued Measures

Given a non-atomic, finite and complete measure space (,,µ)and a Banach space X, the modulus of continuity for a vectormeasure F is defined as the function _F(t) = sup_µ(E)t |F|(E)and the space V^p,q(X) of vector measures such that t^–1/p'_F(t) L^q((0,µ()],dt/t) is introduced. It is shown thatV^p,q(X) contains isometrically L^p,q(X) and that L^p,q(X) = V^p,q(X)if and only if X has the Radon–Nikodym property. It isalso proved that V^p,q(X) coincides with the space of cone absolutelysumming operators from L^p',q^' into X and the duality V^p,q(X^*)=(Lp^',q^'(X))^*where 1/p+1/p^'= 1/q+1/q^' = 1. Finally, V^p,q(X) is identifiedwith the interpolation space obtained by the real method (V¹(X),V(X))_1/p^',q. Spaces where the variation of F is replaced bythe semivariation are also considered.     

On the Hermitian positive defnite solution of the nonlinear matrix equation X + A*X −1 A + B*X −1 B = I

In this paper, we study the matrix equation X + A*X ⁻¹ A + B*X ⁻¹ B = I, where A, B are square matrices, and obtain some conditions for the existence of the positive definite solution of this equation. Two iterative algorithms to find the positive definite solution are given. Some numerical results are reported to illustrate the effectiveness of the algorithms. This research supported by the National Natural Science Foundation of China 10571047 and Doctorate Foundation of the Ministry of Education of China 20060532014.