Spectral conditions for admissibility of evolution equations in Hilbert space

We study properties of solutions of the evolution equation , where B is a closable operator on the space AP(R,H) of almost periodic functions with values in a Hilbert space H such that B commutes with translations. The operator B generates a family of closed operators on H such that (whenever eiλtx∈D(B)). For a closed subset Λ⊂R, we prove that the following properties (i) and (ii) are equivalent: (i) for every function f∈AP(R,H) such that σ(f)⊆Λ, there exists a unique mild solution u∈AP(R,H) of Eq. (∗) such that σ(u)⊆Λ; (ii) is invertible for all λ∈Λ and .     

Simple compact quantum groups I

The notion of simple compact quantum group is introduced. As non-trivial (noncommutative and noncocommutative) examples, the following families of compact quantum groups are shown to be simple: (a) The universal quantum groups Bu(Q) for Q∈GL(n,C) satisfying , n?2; (b) The quantum automorphism groups Aaut(B,τ) of finite-dimensional C^∗-algebras B endowed with the canonical trace τ when dim(B)?4, including the quantum permutation groups Aaut(Xn) on n points (n?4); (c) The standard deformations Kq of simple compact Lie groups K and their twists , as well as Rieffel's deformation KJ.     

Least and most colored bases

Consider a matroid M=(E,B), where B denotes the family of bases of M, and assign a color c(e) to every element e∈E (the same color can go to more than one element). The palette of a subset F of E, denoted by c(F), is the image of F under c. Assume also that colors have prices (in the form of a function π(?), where ? is the label of a color), and define the chromatic price as: π(F)=∑_?∈c(F)π(?). We consider the following problem: find a base B∈B such that π(B) is minimum. We show that the greedy algorithm delivers a lnr(M)-approximation of the unknown optimal value, where r(M) is the rank of matroid M. By means of a reduction from SETCOVER, we prove that the lnr(M) ratio cannot be further improved, even in the special case of partition matroids, unless . The results apply to the special case where M is a graphic matroid and where the prices π(?) are restricted to be all equal. This special case was previously known as the minimum label spanning tree (MLST) problem. For the MLST, our results improve over the ln(n-1)+1 ratio achieved by Wan, Chen and Xu in 2002. Inspired by the generality of our results, we study the approximability of coloring problems with different objective function π(F), where F is a common independent set on matroids M₁,…,M_k and, more generally, to independent systems characterized by the k-for-1 property.     

Essential approximate point spectra and Weyl's theorem for operator matrices

When A∈B(H) and B∈B(K) are given, we denote by MC the operator acting on the infinite dimensional separable Hilbert space H⊕K of the form . In this paper, it is shown that a 2×2 operator matrix MC is upper semi-Fredholm and ind(MC)?0 for some C∈B(K,H) if and only if A is upper semi-Fredholm and

     

Rank-1 perturbations of cosine functions and semigroups

Let A be the generator of a cosine function on a Banach space X. In many cases, for example if X is a UMD-space, A+B generates a cosine function for each B∈L(D((ω−A)^1/2),X). If A is unbounded and , then we show that there exists a rank-1 operator B∈L(D(γ(ω−A)),X) such that A+B does not generate a cosine function. The proof depends on a modification of a Baire argument due to Desch and Schappacher. It also allows us to prove the following. If A+B generates a distribution semigroup for each operator B∈L(D(A),X) of rank-1, then A generates a holomorphic C₀-semigroup. If A+B generates a C₀-semigroup for each operator B∈L(D(γ(ω−A)),X) of rank-1 where 0<γ<1, then the semigroup T generated by A is differentiable and ‖T^′(t)‖=O(t−α) as t↓0 for any α>1/γ. This is an approximate converse of a perturbation theorem for this class of semigroups.     

Boundary trace embedding theorems for variable exponent Sobolev spaces

We study boundary trace embedding theorems for variable exponent Sobolev space W1,p(⋅)(Ω). Let Ω be an open (bounded or unbounded) domain in RN satisfying strong local Lipschitz condition. Under the hypotheses that p∈L^∞(Ω), 1?infp(x)?supp(x)<N, |∇p|∈Lγ(⋅)(Ω), where γ∈L^∞(Ω) and infγ(x)>N, we prove that there is a continuous boundary trace embedding W1,p(⋅)(Ω)→Lq(⋅)(∂Ω) provided q(⋅), a measurable function on ∂Ω, satisfies condition for x∈∂Ω.     

Eventual differentiability of functional differential equations in Banach spaces

This paper concerns the regularity of a functional differential equation in the form: , t>0, where A is the generator of an analytic semigroup on a Banach space X, and B₁,B₂ are α(γ−A)-bounded linear operator for 0<α<1. By spectral analysis, it is shown that the associated solution semigroup of this equation is eventually differentiable.     

Strongly quasibounded maximal monotone perturbations for the Berkovits-Mustonen topological degree theory

Let X be a real reflexive Banach space with dual X^∗. Let L:X⊃D(L)→X^∗ be densely defined, linear and maximal monotone. Let T:X⊃D(T)→X^∗2, with 0∈D(T) and 0∈T(0), be strongly quasibounded and maximal monotone, and C:X⊃D(C)→X^∗ bounded, demicontinuous and of type (S₊) w.r.t. D(L). A new topological degree theory has been developed for the sum L+T+C. This degree theory is an extension of the Berkovits-Mustonen theory (for T=0) and an improvement of the work of Addou and Mermri (for T:X→X^∗2 bounded). Unbounded maximal monotone operators with are strongly quasibounded and may be used with the new degree theory.     

Upper triangular operator matrices with the single-valued extension property

If A=(Aij)_1?i,j?n∈B(X) is an upper triangular Banach space operator such that AiiAij=AijAjj for all 1?i?j?n, then A has SVEP or satisfies (Dunford's) condition (C) or (Bishop's) property (β) or (the decomposition) property (δ) if and only if Aii, 1?i?n, has the corresponding property.     

On the regularity and stability of Pritchard-Salamon systems

Let Σ(S(⋅),B,−) be a Pritchard-Salamon system for (W,V), where W and V are Hilbert spaces. Suppose U is a Hilbert space and F∈L(W,U) is an admissible output operator, SBF(⋅) is the corresponding admissible perturbation C₀-semigroup. We show that the C₀-semigroup SBF(⋅) persists norm continuity, compactness and analyticity of C₀-semigroup S(⋅) on W and V, respectively. We also characterize the compactness and norm continuity of ΔBF(t)=SBF(t)−S(t) for t>0. In particular, we unexpectedly find that ΔBF(t) is norm continuous for t>0 on W and V if the embedding from W into V is compact. Moreover, from this we give some relations between the spectral bounds and growth bounds of SBF(⋅) and S(⋅), so we obtain some new stability results.     

Biholomorphic mappings on bounded starlike circular domains

Let Ω⊂Cn be a bounded starlike circular domain with 0∈Ω. In this paper, we introduce a class of holomorphic mappings Mg on Ω. Let f(z) be a normalized locally biholomorphic mapping on Ω such that and z=0 is the zero of order k+1 of f(z)−z. We obtain a sharp growth theorem and sharp coefficient bounds for f(z). As applications, sharp distortion theorems for a subclass of starlike mappings are obtained. These results unify and generalize many known results.     

Decomposition rank and absorbing extensions of type I algebras

We prove the following: Let A and B be separable C^*-algebras. Suppose that B is a type I C^*-algebra such that

(i)

B has only infinite dimensional irreducible *-representations, and

(ii)

B has finite decomposition rank.

If

0→B→C→A→0     

Nonexistence of global solutions for a class of complex vector fields on two-torus

The goal of this paper is study the global solvability of a class of complex vector fields of the special form L=∂/∂t+(a+ib)(x)∂/∂x, a,b∈C^∞(S¹;R), defined on two-torus T²≅R²/2πZ². The kernel of transpose operator is described and the solvability near the characteristic set is also studied.     

Semigroup crossed products and the induced algebras of lattice-ordered groups

Let (G,G₊) be a quasi-lattice-ordered group with positive cone G₊. Laca and Raeburn have shown that the universal C^∗-algebra C^∗(G,G₊) introduced by Nica is a crossed product BG₊α_×G₊ by a semigroup of endomorphisms. The goal of this paper is to extend some results for totally ordered abelian groups to the case of discrete lattice-ordered abelian groups. In particular given a hereditary subsemigroup H₊ of G₊ we introduce a closed ideal IH₊ of the C^∗-algebra BG₊. We construct an approximate identity for this ideal and show that IH₊ is extendibly α-invariant. It follows that there is an isomorphism between C^∗-crossed products and B₊(G/H)β_×G₊. This leads to our main result that B₊(G/H)β_×G₊ is realized as an induced C^∗-algebra .     

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 perturbations of spectra of upper triangular operator matrices

In this paper we investigate perturbations of the regular spectrum of an upper triangular operator matrix such as acting on a Hilbert space H⊕K.     

Solutions of non-periodic super-quadratic Dirac equations

This paper is concerned with solutions to the Dirac equation: −i∑αkk_∂u+aβu+M(x)u=Ru(x,u). Here M(x) is a general potential and R(x,u) is a self-coupling which is super-quadratic in u at infinity. We use variational methods to study this problem. By virtue of some auxiliary system related to the “limit equation” of the Dirac equation, we construct linking levels of the variational functional ΦM such that the minimax value cM based on the linking structure of ΦM satisfies , where is the least energy of the “limit equation”. Thus we can show the c(C)-condition holds true for all and consequently obtain one least energy solution to the Dirac equation.     

Graphic sequences with a realization containing a complete multipartite subgraph

A nonincreasing sequence of nonnegative integers π=(d₁,d₂,…,d_n) is graphic if there is a (simple) graph G of order n having degree sequence π. In this case, G is said to realizeπ. For a given graph H, a graphic sequence π is potentiallyH-graphic if there is some realization of π containing H as a (weak) subgraph. Let σ(π) denote the sum of the terms of π. For a graph H and n∈Z⁺, σ(H,n) is defined as the smallest even integer m so that every n-term graphic sequence π with σ(π)≥m is potentially H-graphic. Let denote the complete t partite graph such that each partite set has exactly s vertices. We show that and obtain the exact value of σ(K_j+K_s,s,n) for n sufficiently large. Consequently, we obtain the exact value of for n sufficiently large.     

The quantitative difference between countable compactness and compactness

We establish here some inequalities between distances of pointwise bounded subsets H of RX to the space of real-valued continuous functions C(X) that allow us to examine the quantitative difference between (pointwise) countable compactness and compactness of H relative to C(X). We prove, amongst other things, that if X is a countably K-determined space the worst distance of the pointwise closure of H to C(X) is at most 5 times the worst distance of the sets of cluster points of sequences in H to C(X): here distance refers to the metric of uniform convergence in RX. We study the quantitative behavior of sequences in H approximating points in . As a particular case we obtain the results known about angelicity for these Cp(X) spaces obtained by Orihuela. We indeed prove our results for spaces C(X,Z) (hence for Banach-valued functions) and we give examples that show when our estimates are sharp.     

Unitary part of a contraction

For a contraction A on a Hilbert space H, we define the index j(A) (resp., k(A)) as the smallest nonnegative integer j (resp., k) such that ker(I−Aj∗Aj) (resp., ker(I−Ak*Ak)∩ker(I−AkAk∗)) equals the subspace of H on which the unitary part of A acts. We show that if , then j(A)?n (resp., k(A)?⌈n/2⌉), and the equality holds if and only if A is of class Sn (resp., one of the three conditions is true: (1) A is of class Sn, (2) n is even and A is completely nonunitary with ‖An−2‖=1 and ‖An−1‖<1, and (3) n is even and A=U⊕A^′, where U is unitary on a one-dimensional space and A^′ is of class Sn−1).