Phase transition on the Toeplitz algebra of the affine semigroup over the natural numbers

We show that the group of orientation-preserving affine transformations of the rational numbers is quasi-lattice ordered by its subsemigroup N?N^×. The associated Toeplitz C^∗-algebra T(N?N^×) is universal for isometric representations which are covariant in the sense of Nica. We give a presentation of T(N?N^×) in terms of generators and relations, and use this to show that the C^∗-algebra QN recently introduced by Cuntz is the boundary quotient of in the sense of Crisp and Laca. The Toeplitz algebra T(N?N^×) carries a natural dynamics σ, which induces the one considered by Cuntz on the quotient QN, and our main result is the computation of the KMSβ (equilibrium) states of the dynamical system (T(N?N^×),R,σ) for all values of the inverse temperature β. For β∈[1,2] there is a unique KMSβ state, and the KMS₁ state factors through the quotient map onto QN, giving the unique KMS state discovered by Cuntz. At β=2 there is a phase transition, and for β>2 the KMSβ states are indexed by probability measures on the circle. There is a further phase transition at β=∞, where the KMS_∞ states are indexed by the probability measures on the circle, but the ground states are indexed by the states on the classical Toeplitz algebra T(N).     

Fredholmness of Toeplitz operators and corona problems

A meromorphic analogue to the corona problem is formulated and studied and its solutions are characterized as being left-invertible in a space of meromorphic functions. The Fredholmness of Toeplitz operators with symbol G∈(L_∞(R))^2×2 is shown to be equivalent to that of a Toeplitz operator with scalar symbol , provided that the Riemann-Hilbert problem admits a solution such that the meromorphic corona problems with data are solvable. The Fredholm properties are characterized in terms of and the corresponding meromorphic left-inverses. Partial index estimates for the symbols and Fredholmness criteria are established for several classes of Toeplitz operators.     

On the unitary equivalence of absolutely continuous parts of self-adjoint extensions

The classical Weyl-von Neumann theorem states that for any self-adjoint operator A₀ in a separable Hilbert space H there exists a (non-unique) Hilbert-Schmidt operator C=C^? such that the perturbed operator A₀+C has purely point spectrum. We are interesting whether this result remains valid for non-additive perturbations by considering the set ExtA of self-adjoint extensions of a given densely defined symmetric operator A in H and some fixed . We show that the ac-parts and of and A₀ are unitarily equivalent provided that the resolvent difference is compact and the Weyl function M(⋅) of the pair {A,A₀} admits weak boundary limits M(t):=w-lim_y→+0M(t+iy) for a.e. t∈R. This result generalizes the classical Kato-Rosenblum theorem. Moreover, it demonstrates that for such pairs {A,A₀} the Weyl-von Neumann theorem is in general not true in the class ExtA.     

Heat flow, BMO, and the compactness of Toeplitz operators

In this paper, it is shown that the Berezin-Toeplitz operator Tg is compact or in the Schatten class Sp of the Segal-Bargmann space for 1?p<∞ whenever (vanishes at infinity) or , respectively, for some s with , where is the heat transform of g on Cn. Moreover, we show that compactness of Tg implies that is in C₀(Cn) for all and use this to show that, for g∈BMO¹(Cn), we have is in C₀(Cn) for some s>0 only if is in C₀(Cn) for alls>0. This “backwards heat flow” result seems to be unknown for g∈BMO¹ and even g∈L^∞. Finally, we show that our compactness and vanishing “backwards heat flow” results hold in the context of the weighted Bergman space , where the “heat flow” is replaced by the Berezin transform Bα(g) on for α>−1.     

Compact operators and Toeplitz algebras on multiply-connected domains

If Ω is a smoothly bounded multiply-connected domain in the complex plane and S belongs to the Toeplitz algebra τ of the Bergman space of Ω, we show that S is compact if and only if its Berezin transform vanishes at the boundary of Ω. We also show that every element S in T, the C^?-subalgebra of τ generated by Toeplitz operators with symbols in H^∞(Ω), has a canonical decomposition for some R in the commutator ideal CT; and S is in CT iff the Berezin transform vanishes identically on the set M₁ of trivial Gleason parts.     

Perfectness and imperfectness of unit disk graphs on triangular lattice points

Given a finite set of 2-dimensional points P⊆R² and a positive real d, a unit disk graph, denoted by (P,d), is an undirected graph with vertex set P such that two vertices are adjacent if and only if the Euclidean distance between the pair is less than or equal to d. Given a pair of non-negative integers m and n, P(m,n) denotes a subset of 2-dimensional triangular lattice points defined by where . Let T_m,n(d) be a unit disk graph defined on a vertex set P(m,n) and a positive real d. Let be the kth power of T_m,n(1).In this paper, we show necessary and sufficient conditions that [ is perfect] and/or [ is perfect], respectively. These conditions imply polynomial time approximation algorithms for multicoloring (T_m,n(d),w) and .     

Explicit formulas for Laplace transforms of certain functionals of some time inhomogeneous diffusions

We consider a process given by the SDE , t∈[0,T), with initial condition , where T∈(0,∞], α∈R, (Bt)_t∈[0,T) is a standard Wiener process, b:[0,T)→R?{0} and σ:[0,T)→(0,∞) are continuously differentiable functions. Assuming , t∈[0,T), with some K∈R, we derive an explicit formula for the joint Laplace transform of and for all t∈[0,T) and for all α∈R. Our motivation is that the maximum likelihood estimator (MLE) of α can be expressed in terms of these random variables. As an application, we show that in case of α=K, K≠0,

     

The oscillation of differential transforms - entire Jacobi expansions

Let L=(1−x²)D²−((β−α)−(α+β+2)x)D with , and . Let f∈C^∞[−1,1], , with normalized Jacobi polynomials and the Cn decrease sufficiently fast. Set Lk=L(Lk−1), k?2. Let ρ>1. If the number of sign changes of (Lkf)(x) in (−1,1) is O(k1/(ρ+1)), then f extends to be an entire function of logarithmic order . For Legendre expansions, the result holds with replaced with .     

On the Laplacian spectral radii of trees

Let Δ(T) and μ(T) denote the maximum degree and the Laplacian spectral radius of a tree T, respectively. Let T_n be the set of trees on n vertices, and . In this paper, we determine the two trees which take the first two largest values of μ(T) of the trees T in when . And among the trees in , the tree which alone minimizes the Laplacian spectral radius is characterized. We also prove that for two trees T₁ and T₂ in , if Δ(T₁)>Δ(T₂) and , then μ(T₁)>μ(T₂). As an application of these results, we give a general approach about extending the known ordering of trees in T_n by their Laplacian spectral radii.     

A functional equation characterizing the second derivative

Consider an operator T:C²(R)→C(R) and isotropic maps A₁,A₂:C¹(R)→C(R) such that the functional equation

     

On a class of analytic functions with missing coefficients

Let T_n(A,B,α) denote the class of functions of the form:

     

An explicit method for finding common solutions of variational inequalities and systems of equilibrium problems and fixed points of an infinite family of nonexpansive mappings

Let H be a Hilbert space and C be a nonempty closed convex subset of H, {T_i}_i∈N be a family of nonexpansive mappings from C into H, G_i:C×C→R be a finite family of equilibrium functions (i∈{1,2,…,K}), A be a strongly positive bounded linear operator with a coefficient and -Lipschitzian, relaxed (μ,ν)-cocoercive map of C into H. Moreover, let , {α_n} satisfy appropriate conditions and ; we introduce an explicit scheme which defines a suitable sequence as follows:

     

On the existence of local strong solutions for the Navier-Stokes equations in completely general domains

There are only very few results on the existence of unique local in time strong solutions of the Navier-Stokes equations for completely general domains Ω⊆R³, although domains with edges and corners, bounded or unbounded, are very important in applications. The reason is that the L^q-theory for the Stokes operator A is available in general only in the Hilbert space setting, i.e., with q=2. Our main result for a general domain Ω is optimal in a certain sense: Consider an initial value and a zero external force. Then the condition is sufficient and necessary for the existence of a unique local strong solution u∈L⁸(0,T;L⁴(Ω)) in some interval [0,T), 0<T≤∞, with u(0)=u₀, satisfying Serrin’s condition . Note that Fujita-Kato’s sufficient condition u₀∈D(A^1/4) is strictly stronger and therefore not optimal.     

Higher power moments of the Riesz mean error term of symmetric square L-function

Let be the error term of the Riesz mean of the symmetric square L-function. We give the higher power moments of and show that if there exists a real number A₀:=A₀(ρ)>3 such that , then we can derive asymptotic formulas for , 3?h<A₀, h∈N. Particularly, we get asymptotic formulas for , h=3,4,5 unconditionally.     

Linnik-type problems for automorphic L-functions

Let m?2 be an integer, and π an irreducible unitary cuspidal representation for GLm(AQ), whose attached automorphic L-function is denoted by L(s,π). Let be the sequence of coefficients in the Dirichlet series expression of L(s,π) in the half-plane Rs>1. It is proved in this paper that, if π is such that the sequence is real, then there are infinitely many sign changes in the sequence , and the first sign change occurs at some , where Qπ is the conductor of π, and the implied constant depends only on m and ε. This generalizes the previous results for GL₂. A result of the same quality is also established for , the sequence of coefficients in the Dirichlet series expression of in the half-plane Rs>1.     

Lengths of central linear generalized polynomials in matrix algebras

Let D be a division algebra finite-dimensional over its center C and let A=M_m(D), the m×m matrix ring over D. By the length of a linear generalized polynomial (GP) ?(X), we mean the least positive integer n such that ?(X) can be represented in the form for some . We denote by L(?)=n the length of ?. By a central linear GP for A we mean a nonzero linear GP with central values on A. In this paper we characterize all central linear GPs for A and determine the lengths of all central linear GPs for A.     

Fixed points of dual quantum operations

Let ?A be a normal completely positive map on B(H) with Kraus operators . Denote M the subset of normal completely positive maps by . In this note, the relations between the fixed points of ?A and are investigated. We obtain that , where K(H) is the set of all compact operators on H and is the dual of ?A∈M. In addition, we show that the map is a bijection on M.     

Haagerup property for C-algebras

We define the Haagerup property for C^?-algebras A and extend this to a notion of relative Haagerup property for the inclusion B⊆A, where B is a C^?-subalgebra of A. Let Γ be a discrete group and Λ a normal subgroup of Γ, we show that the inclusion A_?α,rΛ⊆A_?α,rΓ has the relative Haagerup property if and only if the quotient group Γ/Λ has the Haagerup property. In particular, the inclusion has the relative Haagerup property if and only if Γ/Λ has the Haagerup property; has the Haagerup property if and only if Γ has the Haagerup property. We also characterize the Haagerup property for Γ in terms of its Fourier algebra A(Γ).     

Well-posedness of the fourth-order perturbed Schrödinger type equation in non-isotropic Sobolev spaces

In this paper we study the Cauchy problem of the non-isotropically perturbed fourth-order nonlinear Schrödinger type equation: ((x₁,x₂,…,xn)∈Rn, t?0), where a is a real constant, 1?d<n is an integer, g(x,|u|)u is a nonlinear function which behaves like α|u|u for some constant α>0. By using Kato method, we prove that this perturbed fourth-order Schrödinger type equation is locally well-posed with initial data belonging to the non-isotropic Sobolev spaces provided that s₁,s₂ satisfy the conditions: s₁?0, s₂?0 for or for with some additional conditions. Furthermore, by using non-isotropic Sobolev inequality and energy method, we obtain some global well-posedness results for initial data belonging to non-isotropic Sobolev spaces .     

Further results on iterative methods for computing generalized inverses

In this paper, we reconsider the iterative method X_k=X_k−1+βY(I−AX_k−1), k=1,2,…,β∈C?{0} for computing the generalized inverse over Banach spaces or the generalized Drazin inverse a^d of a Banach algebra element a, reveal the intrinsic relationship between the convergence of such iterations and the existence of or a^d, and present the error bounds of the iterative methods for approximating or a^d. Moreover, we deduce some necessary and sufficient conditions for iterative convergence to or a^d.