فهرست مطالب

  • Volume:13 Issue:3, 2019
  • تاریخ انتشار: 1398/05/09
  • تعداد عناوین: 12
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  • Rank-one perturbations and Anderson-type Hamiltonians
    Constanze Liaw* Pages 507-523

    Motivated by applications of the discrete random Schrödinger operator, mathematical physicists and analysts began studying more general Anderson-type Hamiltonians; that is, the family of self-adjoint operators Hω=H+Vω on a separable Hilbert space H, where the perturbation is given by Vω=∑nωn(⋅,φn)φn with a sequence {φn}⊂H and independent identically distributed random variables ωn. We show that the essential parts of Hamiltonians associated to any two realizations of the random variable are (almost surely) related by a rank-one perturbation. This result connects one of the least trackable perturbation problem (with almost surely noncompact perturbations) with one where the perturbation is “only” of rank-one perturbations. The latter presents a basic application of model theory. We also show that the intersection of the essential spectrum with open sets is almost surely either the empty set, or it has nonzero Lebesgue measure.

    Keywords: rank-one perturbations, Anderson-type Hamiltonian, Krein–Lifshits spectral shift, discrete random Schrödinger operator
  • Abelian theorems for distributional Kontorovich–Lebedev and Mehler–Fock transforms of general order
    Benito J. González, Emilio R. Negrín Pages 524-537
  • Factorized sectorial relations, their maximal-sectorial extensions, and form sums
    Seppo Hassi, Adrian Sandovici, Henk de Snoo Pages 538-564

    In this paper we consider sectorial operators, or more generally, sectorial relations and their maximal-sectorial extensions in a Hilbert space H. Our particular interest is in sectorial relations S, which can be expressed in the factorized form S=T∗(I+iB)TorS=T(I+iB)T∗, where B is a bounded self-adjoint operator in a Hilbert space K and T:H→K (or T:K→H, respectively) is a linear operator or a linear relation which is not assumed to be closed. Using the specific factorized form of S, a description of all the maximal-sectorial extensions of S is given, along with a straightforward construction of the extreme extensions SF, the Friedrichs extension, and SK, the Kreĭn extension of S, which uses the above factorized form of S. As an application of this construction, we also treat the form sum of maximal-sectorial extensions of two sectorial relations.

    Keywords: sectorial relation, Friedrichs extension, Kreĭn extension, extremal extension form sum
  • The Bass and topological stable ranks for algebras of almost periodic functions on the real line, II
    Raymond Mortini*, Amol Sasane Pages 565-581

    Let ΛΛ be either a subgroup of the integers ZZ, a semigroup in NN, or Λ=QΛ=Q (resp., Q+Q+). We determine the Bass and topological stable ranks of the algebras APΛ={f∈AP:σ(f)⊆Λ}APΛ={f∈AP:σ(f)⊆Λ} of almost periodic functions on the real line and with Bohr spectrum in ΛΛ. This answers a question in the first part of this series of articles under the same heading, where it was shown that, in contrast to the present situation, these ranks were infinite for each semigroup ΛΛ of real numbers for which the QQ-vector space generated by ΛΛ had infinite dimension.

    Keywords: almost periodic functions, Bass stable rank, topological stable rank, bounded analytic functions, reducibility of function pairs
  • Generalized quasidiagonality for extensions
    P. W. Ng*, Tracy Robin Pages 582-598

    We generalize the notion of quasidiagonality, for extensions, allowing for the case where the canonical ideal has few projections. We prove that the pointwise-norm limit of generalized quasidiagonal extensions is generalized quasidiagonal. We also provide a K-theory sufficient condition for generalized quasidiagonality of certain extensions of simple continuous-scale C∗-algebras, including certain continuous-scale hereditary C∗-subalgebras of the stabilized Jiang–Su algebra.

    Keywords: C∗-algebras, multiplier algebras, extension theory, Weyl–von Neumann–Berg theorem, KK-theory
  • On the property IR of Friis and Rørdam
    Lawrence G. Brown* Pages 599-611

    Lin solved a longstanding problem as follows. For each ϵ>0ϵ>0, there is δ>0δ>0 such that, if hh and kk are self-adjoint contractive n×nn×n matrices and ∥hk−kh∥<δ‖hk−kh‖<δ, then there are commuting self-adjoint matrices h'h' and k'k' such that ∥h'−h∥‖h'−h‖, ∥k'−k∥<ϵ‖k'−k‖<ϵ. Here δδ depends only on ϵϵand not on nn. Friis and Rørdam greatly simplified Lin’s proof by using a property they called IRIR. They also generalized Lin’s result by showing that the matrix algebras can be replaced by any C∗C∗-algebras satisfying IRIR. The purpose of this paper is to study the property IRIR. One of our results shows how IRIR behaves for C∗C∗-algebra extensions. Other results concern nonstable KK-theory. One shows that IRIR (at least the stable version) implies a cancellation property for projections which is intermediate between the strong cancellation satisfied by C∗C∗-algebras of stable rank 11 and the weak cancellation defined in a 2014 paper by Pedersen and the author.

    Keywords: C∗-algebras, invertible, extension, nonstable K-theory
  • Kernels of Hankel operators on the Hardy space over the bidisk
    Kei Ji Izuchi*, Kou Hei Izuchi, and Yuko Izuchi Pages 612-626
  • The polar decomposition for adjointable operators on Hilbert C∗-modules and n-centered operators
    Na Liu, Wei Luo, Qingxiang Xu* Pages 627-646

    Let n be any natural number. The n-centered operator is introduced for adjointable operators on Hilbert C∗-modules. Based on the characterizations of the polar decomposition for the product of two adjointable operators, n-centered operators, centered operators as well as binormal operators are clarified, and some results known for the Hilbert space operators are improved. It is proved that for an adjointable operator T, if T is Moore–Penrose invertible and is n-centered, then its Moore–Penrose inverse is also n-centered. A Hilbert space operator T is constructed such that T is n-centered, whereas it fails to be (n+1)-centered.

    Keywords: Hilbert C∗-module, polar decomposition, centered operator, n-centered operator, binormal operator
  • On the structure of universal functions for classes Lp[0,1)2,p∈(0,1), with respect to the double Walsh system
    Martin Grigoryan, Artsrun Sargsyan* Pages 647-674
  • Boundedness of Cesàro and Riesz means in variable dyadic Hardy spaces
    Kristóf Szarvas, Ferenc Weisz* Pages 675-696

    We consider two types of maximal operators. We prove that, under some conditions, each maximal operator is bounded from the classical dyadic martingale Hardy space Hp to the classical Lebesgue space Lp and from the variable dyadic martingale Hardy space Hp(⋅) to the variable Lebesgue space Lp(⋅). Using this, we can prove the boundedness of the Cesàro and Riesz maximal operator from Hp(⋅) to Lp(⋅) and from the variable Hardy–Lorentz space Hp(⋅),q to the variable Lorentz space Lp(⋅),q. As a consequence, we can prove theorems about almost everywhere and norm convergence.

    Keywords: variable Hardy spaces, variable Hardy–Lorentz spaces, Cesàro means, Riesz means, Cesàro, Riesz maximal operator, boundedness
  • Riesz transforms, Cauchy–Riemann systems, and Hardy-amalgam spaces
    Al, Tarazi Assaubay, Jorge J. Betancor, Alejandro J. Castro*, Juan C. Fariña Pages 697-725

    In this article we study Hardy spaces Hp,q(Rd), 0<p,q<∞, modeled over amalgam spaces (Lp,ℓq)(Rd). We characterize Hp,q(Rd) by using first-order classical Riesz transforms and compositions of first-order Riesz transforms, depending on the values of the exponents p and q. Also, we describe the distributions in Hp,q(Rd) as the boundary values of solutions of harmonic and caloric Cauchy–Riemann systems. We remark that caloric Cauchy–Riemann systems involve fractional derivatives in the time variable. Finally, we characterize the functions in L2(Rd)∩Hp,q(Rd) by means of Fourier multipliers mθ with symbol θ(⋅/|⋅|), where θ∈C∞(Sd−1) and Sd−1 denotes the unit sphere in Rd.

    Keywords: Hardy spaces, amalgam spaces, Cauchy–Riemann equations, Riesz transforms
  • Positivity of 2×2 block matrices of operators
    Mohammad Sal Moslehian*, Mohsen Kian, Qingxiang Xu Pages 726-743

    We review some significant generalizations and applications of the celebrated Douglas theorem on equivalence of factorization, range inclusion, and majorization of operators. We then apply it to find a characterization of the positivity of 2×2 block matrices of operators on Hilbert spaces. Finally, we describe the nature of such block matrices and provide several ways for showing their positivity.

    Keywords: positive operator, block matrix, Hilbert space