Description of extended eigenvalues and extended eigenvectors of integration operators on the Wiener algebra

In the present paper we consider the Volterra integration operator V   on the Wiener algebra W(D)$W(\mathbb{D})$ of analytic functions on the unit disc D$\mathbb{D}$ of the complex plane C$\mathbb{C}$. A complex number λ$\lambda$ is called an extended eigenvalue of V if there exists a nonzero operator A   satisfying the equation AV=λVA$\mathit{AV}=\lambda \mathit{VA}$. We prove that the set of all extended eigenvalues of V   is precisely the set C?{0}$\mathbb{C}?\{0\}$, and describe in terms of Duhamel operators and composition operators the set of corresponding extended eigenvectors of V$V$. The similar result for some weighted shift operator on _?p${?}_p$ spaces is also obtained.     

A spectral radius type formula for approximation numbers of composition operators

For approximation numbers _an(_Cφ)$a_n(C_{\phi })$ of composition operators _Cφ$C_{\phi }$ on weighted analytic Hilbert spaces, including the Hardy, Bergman and Dirichlet cases, with symbol φ   of uniform norm <1, we prove that _limn→∞?[_an(_Cφ)]^1/n=e^{−1/Cap[φ(D)]}${\lim }_{n\to \infty }?{[a_n(C_{\phi })]}^{1/n}={\mathrm{e}}^{-1/\mathrm{Cap}[\phi (\mathbb{D})]}$, where Cap[φ(D)]$\mathrm{Cap}[\phi (\mathbb{D})]$ is the Green capacity of φ(D)$\phi (\mathbb{D})$ in D$\mathbb{D}$. This formula holds also for H^p$H^p$ with 1≤p<∞$1\le p<\infty$.     

Logarithmic bump conditions and the two-weight boundedness of Calderón–Zygmund operators

We prove that if a pair of weights (u,v)$(u,v)$ satisfies a sharp _Ap$A_p$-bump condition in the scale of all log bumps or certain loglog bumps, then Haar shifts map L^p(v)$L^p(v)$ into L^p(u)$L^p(u)$ with a constant quadratic in the complexity of the shift. This in turn implies the two weight boundedness for all Calderón–Zygmund operators. This gives a partial answer to a long-standing conjecture. We also give a partial result for a related conjecture for weak-type inequalities. To prove our main results we combine several different approaches to these problems; in particular we use many of the ideas developed to prove the _A2$A_2$ conjecture. As a byproduct of our work we also disprove a conjecture by Muckenhoupt and Wheeden on weak-type inequalities for the Hilbert transform. This is closely related to the recent counterexamples of Reguera, Scurry and Thiele.     

Inequivalent representations of matroids over prime fields

It is proved that for each prime field GF(p)$\mathit{GF}(p)$, there is an integer _np$n_p$ such that a 4-connected matroid has at most _np$n_p$ inequivalent representations over GF(p)$\mathit{GF}(p)$. We also prove a stronger theorem that obtains the same conclusion for matroids satisfying a connectivity condition, intermediate between 3-connectivity and 4-connectivity that we term “k-coherence”.     

Alternative approaches to asymptotic behaviour of eigenvalues of some unbounded Jacobi matrices

In this article we calculate the asymptotic behaviour of the point spectrum for some special self-adjoint unbounded Jacobi operators J   acting in the Hilbert space l²=l²(N)$l^2=l^2(\mathbb{N})$. For given sequences of positive numbers _λn${\lambda }_n$ and real _qn$q_n$ the Jacobi operator is given by J=SW+WS^*+Q$J=\text{<italic>SW</italic>}+{\mathit{WS}}^{*}+Q$, where Q=diag(_qn)$Q=\mathrm{diag}(q_n)$ and W=diag(_λn)$W=\mathrm{diag}({\lambda }_n)$ are diagonal operators, S is the shift operator and the operator J   acts on the maximal domain. We consider a few types of the sequences {_qn}$\{q_n\}$ and {_λn}$\{{\lambda }_n\}$ and present three different approaches to the problem of the asymptotics of eigenvalues of various classes of J's. In the first approach to asymptotic behaviour of eigenvalues we use a method called successive diagonalization, the second approach is based on analytical models that can be found for some special J's and the third method is based on an abstract theorem of Rozenbljum.     

Quasi-Banach spaces of almost universal disposition

We show that for each p∈(0,1]$p\in (0,1]$ there exists a separable p  -Banach space _Gp${\mathbb{G}}_p$ of almost universal disposition, that is, having the following extension property: for each ε>0$\epsilon >0$ and each isometric embedding g:X→Y$g:X\to Y$, where Y is a finite-dimensional p-Banach space and X   is a subspace of _Gp${\mathbb{G}}_p$, there is an ε  -isometry f:Y→_Gp$f:Y\to {\mathbb{G}}_p$ such that x=f(g(x))$x=f(g(x))$ for all x∈X$x\in X$.     

Kadec–Klee property and fixed points

Let X be a reflexive Banach space which does not have the Kadec–Klee property. Then there exists a compact mapping f   from the unit ball _BX$B_X$ of X   to the dual space X^?$X^{?}$ such that _infx∈BX‖f(x)‖>0${\inf }_{x\in B_X}\Vert f(x)\Vert >0$ and 〈f(x),x〉<‖f(x)‖$〈f(x),x〉<\Vert f(x)\Vert$ for every x∈_BX$x\in B_X$.     

A continuum of unusual self-adjoint linear partial differential operators

In an earlier publication a linear operator _THar$T_{\mathrm{Har}}$ was defined as an unusual self-adjoint extension generated by each linear elliptic partial differential expression, satisfying suitable conditions on a bounded region Ω$\Omega$ of some Euclidean space. In this present work the authors define an extensive class of _THar$T_{\mathrm{Har}}$-like self-adjoint operators on the Hilbert function space _L2(Ω);$L_2(\Omega );$ but here for brevity we restrict the development to the classical Laplacian differential expression, with Ω$\Omega$ now the planar unit disk. It is demonstrated that there exists a non-denumerable set of such _THar$T_{\mathrm{Har}}$-like operators (each a self-adjoint extension generated by the Laplacian), each of which has a domain in _L2(Ω)$L_2(\Omega )$ that does not lie within the usual Sobolev Hilbert function space W²(Ω)$W^2(\Omega )$. These _THar$T_{\mathrm{Har}}$-like operators cannot be specified by conventional differential boundary conditions on the boundary of ∂Ω$\mathrm{\partial }\Omega$, and may have non-empty essential spectra.     

Completeness,precompactness and compactness in finite-dimensional asymmetrically normed lattices

If (X,‖⋅‖)$(X,\Vert \cdot \Vert )$ is a real normed lattice, then p(x)=‖x⁺‖$p(x)=\Vert x^{+}\Vert$ defines an asymmetric norm on X. We characterise the left-K   sequentially complete, precompact and compact subsets of (R^m,p)$({\mathbb{R}}^m,p)$.     

An approximation method for strictly pseudocontractive mappings

Let K$K$ be a closed convex subset of a q$q$-uniformly smooth separable Banach space, T:K→K$T:K\to K$ a strictly pseudocontractive mapping, and f:K→K$f:K\to K$ an L$L$-Lispschitzian strongly pseudocontractive mapping. For any t∈(0,1)$t\in (0,1)$, let _xt$x_t$ be the unique fixed point of tf+(1-t)T$\mathit{tf}+(1-t)T$. We prove that if T$T$ has a fixed point, then {_xt}$\{x_t\}$ converges to a fixed point of T$T$ as t$t$ approaches to 0.     

Completely positive interpolations of compact,trace-class and Schatten-p class operators

Extending Li and Poon's results on interpolation problems for matrices, we give characterizations of the existence of a completely positive linear map _Φcp${\Phi }_{\mathrm{cp}}$ between compact (or Schatten-p class) operators sending a particular operator A to another B. It is shown that such a map exists if a multiple of the numerical range of A contains the numerical range of B. Given two commutative families of compact (or Schatten-p   class) operators {_Aα}$\{A_{\alpha }\}$ and {_Bα}$\{B_{\alpha }\}$, we provide sufficient and necessary conditions to ensure that we can choose a completely positive interpolation _Φcp${\Phi }_{\mathrm{cp}}$ to preserve trace and/or approximate units such that _Φcp(_Aα)=_Bα${\Phi }_{\mathrm{cp}}(A_{\alpha })=B_{\alpha }$ for all α.     

Fredholm operators,evolution semigroups,and periodic solutions of nonlinear periodic systems

Let X be a Banach space and L the generator of the evolution semigroup associated with the τ  -periodic evolutionary process _{{U(t,s)}t≥s}${\{U(t,s)\}}_{t\ge s}$ on the space _Pτ(X)$P_{\tau }(X)$ of all τ-periodic continuous X  -valued functions. We give criteria for the existence of periodic solutions to nonlinear systems of the form Lp=−?F(p,?)$Lp=-?F(p,?)$ under the condition that 1 is a normal eigenvalue of the monodromy operator U(τ,0)$U(\tau ,0)$. The proof is based on a new decomposition of the space _Pτ(X)$P_{\tau }(X)$ by constructing a right inverse of L.     

Rigidity of symmetric products

Given a metric continuum X, we consider the following hyperspaces of X  : 2^X$2^X$, _Cn(X)$C_n(X)$ and _Fn(X)$F_n(X)$ (n∈N$n\in \mathbb{N}$). Let _F1(X)={{x}:x∈X}$F_1(X)=\{\{x\}:x\in X\}$. A hyperspace K(X)$K(X)$ of X   is said to be rigid provided that for every homeomorphism h:K(X)→K(X)$h:K(X)\to K(X)$ we have that h(_F1(X))=_F1(X)$h(F_1(X))=F_1(X)$. In this paper we study under which conditions a continuum X   has a rigid hyperspace _Fn(X)$F_n(X)$.     

Some Liouville-type theorems for harmonic functions on Finsler manifolds

In this paper, we give some Liouville-type theorems for L^p$L^p$(p∈R)$(p\in \mathbb{R})$ harmonic (resp. subharmonic, superharmonic) functions on forward complete Finsler manifolds. Moreover, we derive a gradient estimate for harmonic functions on a closed Finsler manifold. As an application, one obtains that any harmonic function on a closed Finsler manifold with nonnegative weighted Ricci curvature _RicN${\mathit{Ric}}_N$(N∈(n,∞))$(N\in (n,\infty ))$ must be constant.