Upper triangular matrix operators with diagonal (T_1,T_2), T_2 k-nilpotent

If \(T=\left(\begin{array}{clcr}T_1&\quad C\\ 0&\quad T_2\end{array}\right) \in B(\mathcal{X }_1\oplus \mathcal{X }_2)\) is a Banach space upper triangular operator matrix with diagonal \((T_1, T_2)\) such that \(T_2\) is \(k\)-nilpotent for some integer \(k\ge 1\), then \(T\) inherits a number of its spectral properties, such as SVEP, Bishop’s property \((\beta )\) and the equality of Browder and Weyl spectrum, from those of \(T_1\). This paper studies such operators. The conclusions are then applied to provide a general framework for results pertaining (for example) to Browder, Weyl type theorems and supercyclicity for classes of Hilbert space operators, such as \(k\)-quasi hyponormal, \(k\)-quasi isometric and \(k\)-quasi paranormal operators, defined by a positivity condition.     

The representation type of rational normal scrolls

The goal of this paper is to demonstrate that all non-singular rational normal scrolls \(S(a_0,\ldots ,a_k)\subseteq \mathbb P ^N\), \(N =\sum _{i=0}^k(a_i)+k\), (unless \(\mathbb P ^{k+1}=S(0,\ldots ,0,1)\), the rational normal curve \(S(a)\) in \(\mathbb P ^a\), the quadric surface \(S(1,1)\) in \(\mathbb P ^3\) and the cubic scroll \(S(1,2)\) in \(\mathbb P ^4\)) support families of arbitrarily large rank and dimension of simple Ulrich (and hence indecomposable ACM) vector bundles. Therefore, they are all of wild representation type unless \(\mathbb P ^{k+1}\), \(S(a)\), \(S(1,1)\) and \(S(1,2)\) which are of finite representation type.     

Ergodic Property of Stable-Like Markov Chains

A stable-like Markov chain is a time-homogeneous Markov chain on the real line with the transition kernel \(p(x,\hbox {d}y)=f_x(y-x)\hbox {d}y\), where the density functions \(f_x(y)\), for large \(|y|\), have a power-law decay with exponent \(\alpha (x)+1\), where \(\alpha (x)\in (0,2)\). In this paper, under a certain uniformity condition on the density functions \(f_x(y)\) and additional mild drift conditions, we give sufficient conditions for recurrence in the case when \(0<\liminf _{|x|\longrightarrow \infty }\alpha (x)\), sufficient conditions for transience in the case when \(\limsup _{|x|\longrightarrow \infty }\alpha (x)<2\) and sufficient conditions for ergodicity in the case when \(0<\inf \{\alpha (x):x\in \mathbb {R}\}\). As a special case of these results, we give a new proof for the recurrence and transience property of a symmetric \(\alpha \)-stable random walk on \(\mathbb {R}\) with the index of stability \(\alpha \ne 1\).     

On the graded central polynomials for elementary gradings in matrix algebras

Let \(F\) be a field of characteristic zero. Let \(M_{n}(F)\) be the algebra of all \(n \times n\) matrices over \(F\). We have found, in this article, a generating set for the graded central polynomials of \(M_{n}(F)\) when it is equipped with an elementary grading whose neutral component coincides with the diagonal.     

Few associative triples,isotopisms and groups

Let Q be a quasigroup. For \(\alpha ,\beta \in S_Q\) let \(Q_{\alpha ,\beta }\) be the principal isotope \(x*y = \alpha (x)\beta (y)\). Put \(\mathbf a(Q)= |\{(x,y,z)\in Q^3;\) \(x(yz)) = (xy)z\}|\) and assume that \(|Q|=n\). Then \(\sum _{\alpha ,\beta }\mathbf a(Q_{\alpha ,\beta })/(n!)^2 = n^2(1+(n-1)^{-1})\), and for every \(\alpha \in S_Q\) there is \(\sum _\beta \mathbf a(Q_{\alpha ,\beta })/n! = n(n-1)^{-1}\sum _x(f_x^2-2f_x+n)\ge n^2\), where \(f_x=|\{y\in Q;\) \( y = \alpha (y)x\}|\). If G is a group and \(\alpha \) is an orthomorphism, then \(\mathbf a(G_{\alpha ,\beta })=n^2\) for every \(\beta \in S_Q\). A detailed case study of \(\mathbf a(G_{\alpha ,\beta })\) is made for the situation when \(G = \mathbb Z_{2d}\), and both \(\alpha \) and \(\beta \) are “natural” near-orthomorphisms. Asymptotically, \(\mathbf a(G_{\alpha ,\beta })>3n\) if G is an abelian group of order n. Computational results: \(\mathbf a(7) = 17\) and \(\mathbf a(8) \le 21\), where \(\mathbf a(n) = \min \{\mathbf a(Q);\) \( |Q|=n\}\). There are also determined minimum values for \(\mathbf a(G_{\alpha ,\beta })\), G a group of order \(\le 8\).     

On a Weighted Singular Integral Operator with Shifts and Slowly Oscillating Data

Let \(\alpha ,\beta \) be orientation-preserving diffeomorphism (shifts) of \(\mathbb {R}_+=(0,\infty )\) onto itself with the only fixed points \(0\) and \(\infty \) and \(U_\alpha ,U_\beta \) be the isometric shift operators on \(L^p(\mathbb {R}_+)\) given by \(U_\alpha f=(\alpha ')^{1/p}(f\circ \alpha )\), \(U_\beta f=(\beta ')^{1/p}(f\circ \beta )\), and \(P_2^\pm =(I\pm S_2)/2\) where

$$\begin{aligned} (S_2 f)(t):=\frac{1}{\pi i}\int \limits _0^\infty \left( \frac{t}{\tau }\right) ^{1/2-1/p}\frac{f(\tau )}{\tau -t}\,d\tau , \quad t\in \mathbb {R}_+, \end{aligned}$$

is the weighted Cauchy singular integral operator. We prove that if \(\alpha ',\beta '\) and \(c,d\) are continuous on \(\mathbb {R}_+\) and slowly oscillating at \(0\) and \(\infty \), and

$$\begin{aligned} \limsup _{t\rightarrow s}|c(t)|<1, \quad \limsup _{t\rightarrow s}|d(t)|<1, \quad s\in \{0,\infty \}, \end{aligned}$$

then the operator \((I-cU_\alpha )P_2^++(I-dU_\beta )P_2^-\) is Fredholm on \(L^p(\mathbb {R}_+)\) and its index is equal to zero. Moreover, its regularizers are described.

     

Convergence of the solutions of the discounted equation: the discrete case

We derive a discrete version of the results of Davini et al. (Convergence of the solutions of the discounted Hamilton–Jacobi equation. Invent Math, 2016). If M is a compact metric space, \(c : M\times M \rightarrow \mathbb {R}\) a continuous cost function and \(\lambda \in (0,1)\), the unique solution to the discrete \(\lambda \)-discounted equation is the only function \(u_\lambda : M\rightarrow \mathbb {R}\) such that

$$\begin{aligned} \forall x\in M, \quad u_\lambda (x) = \min _{y\in M} \lambda u_\lambda (y) + c(y,x). \end{aligned}$$

We prove that there exists a unique constant \(\alpha \in \mathbb {R}\) such that the family of \(u_\lambda +\alpha /(1-\lambda )\) is bounded as \(\lambda \rightarrow 1\) and that for this \(\alpha \), the family uniformly converges to a function \(u_0 : M\rightarrow \mathbb {R}\) which then verifies

$$\begin{aligned} \forall x\in X, \quad u_0(x) = \min _{y\in X}u_0(y) + c(y,x)+\alpha . \end{aligned}$$

The proofs make use of Discrete Weak KAM theory. We also characterize \(u_0\) in terms of Peierls barrier and projected Mather measures.

     

Gradient Estimates for Commutators of Square Roots of Elliptic Operators with Complex Bounded Measurable Coefficients

Let \(L=-\mathrm{div}(A\nabla )\) be a second order divergence form elliptic operator and A an accretive \(n\times n\) matrix with bounded measurable complex coefficients in \({\mathbb R}^n\). Let \(\nabla b\in L^n({\mathbb R}^n)\,(n>2)\). In this paper, we prove that the commutator generated by b and the square root of L, which is defined by \([b,\sqrt{L}]f(x)=b(x)\sqrt{L}f(x)-\sqrt{L}(bf)(x)\), is bounded from the homogenous Sobolev space \({\dot{L}}_1^2({\mathbb R}^n)\) to \(L^2({\mathbb R}^n)\).     

Families of conditionally approximately convex functions

Let \((G,+)\) be an Abelian topological group, which is also a \(T_{0}\)-space and a Baire space simultaneously, D be an open connected subset of G and \(\alpha : D-D \rightarrow {\mathbb R}\) be a function continuous at zero and such that \(\alpha (0)=0\). We show that if \((f_n)\) is a sequence of continuous functions \(f_n : D \rightarrow {\mathbb R}\) such that \(f_n(z) \le \frac{1}{2} f_n(x)+\frac{1}{2}f(y)+\alpha (x-y)\) for \(n\in {\mathbb N}\) and \(x,y,z\in D\) such that \(2z=x+y\) and if \((f_n)\) is pointwise convergent [bounded] then it is convergent uniformly on compact subsets of D [in the case when G is additionally a separable space, it contains a subsequence which is convergent on compact subsets of D].     

Short-Time Nonlinear Effects in the Exciton-Polariton System

In the exciton-polariton system, a linear dispersive photon field is coupled to a nonlinear exciton field. Short-time analysis of the lossless system shows that, when the photon field is excited, the time required for that field to exhibit nonlinear effects is longer than the time required for the nonlinear Schrödinger equation, in which the photon field itself is nonlinear. When the initial condition is scaled by \(\epsilon ^\alpha \), it is found that the relative error committed by omitting the nonlinear term in the exciton-polariton system remains within \(\epsilon \) for all times up to \(t=C\epsilon ^\beta \), where \(\beta =(1-\alpha (p-1))/(p+2)\). This is in contrast to \(\beta =1-\alpha (p-1)\) for the nonlinear Schrödinger equation. The result is proved for solutions in \(H^s(\mathbb {R}^n)\) for \(s>n/2\). Numerical computations indicate that the results are sharp and also hold in \(L^2(\mathbb {R}^n)\).     

Weak type (1, 1) of some operators for the Laplacian with drift

Let \(v = (v_1, \ldots , v_n)\) be a vector in \(\mathbb {R}^n {\setminus } \{ 0 \}\). Consider the Laplacian on \(\mathbb {R}^n\) with drift \(\Delta _{v} = \sum _{i = 1}^n \Big ( \frac{\partial ^2}{\partial x_i^2} + 2 v_i \frac{\partial }{\partial x_i} \Big )\) and the measure \(d\mu (x) = e^{2 \langle v, x \rangle } dx\), with respect to which \(\Delta _{v}\) is self-adjoint. Let d and \(\nabla \) denote the Euclidean distance and the gradient operator on \(\mathbb {R}^n\). Consider the space \((\mathbb {R}^n, d, d\mu )\), which has the property of exponential volume growth. We obtain weak type (1, 1) for the Riesz transform \(\nabla (- \Delta _{v} )^{-\frac{1}{2}}\) and for the heat maximal operator, with respect to \(d\mu \). Further, we prove that the uncentered Hardy–Littlewood maximal operator is bounded on \(L^p\) for \(1 < p \le +\infty \) but not of weak type (1, 1) if \(n \ge 2\).     

Thue’s inequalities and the hypergeometric method

We establish upper bounds for the number of primitive integer solutions to inequalities of the shape \(0<|F(x, y)| \le h\), where \(F(x , y) =(\alpha x + \beta y)^r -(\gamma x + \delta y)^r \in \mathbb {Z}[x ,y]\), \(\alpha \), \(\beta \), \(\gamma \) and \(\delta \) are algebraic constants with \(\alpha \delta -\beta \gamma \ne 0\), and \(r \ge 5\) and h are integers. As an important application, we pay special attention to binomial Thue’s inequalities \(|ax^r - by^r| \le c\). The proofs are based on the hypergeometric method of Thue and Siegel and its refinement by Evertse.     

Closed range and nonclosed range adjointable operators on Hilbert $$C^*$$-modules

In this paper, we show that for a positive operator A on a Hilbert \(C^*\)-module \( \mathscr {E} \), the range \( \mathscr {R}(A) \) of A is closed if and only if \( \mathscr {R}(A^\alpha ) \) is closed for all \(\alpha \in (0,1)\cup (1,+\,\infty )\), and this occurs if and only if \( \mathscr {R}(A)=\mathscr {R}(A^\alpha ) \) for all \(\alpha \in (0,1)\cup (1,+\,\infty )\). As an application, we prove that for an adjontable operator A if \(\mathscr {R}(A)\) is nonclosed, then \(\dim \left( \overline{\mathscr {R}(A)}/\mathscr {R}(A)\right) =+\,\infty \). Finally, we show that for an adjointable operator A if \( \overline{\mathscr {R}(A^*) } \) is orthogonally complemented in \( \mathscr {E} \), then under certain coditions there exists an idempotent C and a unique operator X such that \( XAX=X, AXA=CA, AX=C \) and \( XA=P_{A^*} \), where \( P_{A^*} \) is the orthogonal projection of \( \mathscr {E} \) onto \( \overline{\mathscr {R}(A^*)}\).     

Regularity for Inhomogeneous Dirichlet Problems of Some Schrödinger Equations on Domains

Let \(n\ge 3, \Omega \) be a bounded, simply connected and semiconvex domain in \({\mathbb {R}}^n\) and \(L_{\Omega }:=-\Delta +V\) a Schrödinger operator on \(L^2 (\Omega )\) with the Dirichlet boundary condition, where \(\Delta \) denotes the Laplace operator and the potential \(0\le V\) belongs to the reverse Hölder class \(RH_{q_0}({\mathbb {R}}^n)\) for some \(q_0\in (\max \{n/2,2\},\infty ]\). Assume that the growth function \(\varphi :\,{\mathbb {R}}^n\times [0,\infty ) \rightarrow [0,\infty )\) satisfies that \(\varphi (x,\cdot )\) is an Orlicz function and \(\varphi (\cdot ,t)\in {\mathbb {A}}_{\infty }({\mathbb {R}}^n)\) (the class of uniformly Muckenhoupt weights). Let \(H_{\varphi ,\,L_{{\mathbb {R}}^n},\,r}(\Omega )\) be the Musielak–Orlicz–Hardy space whose elements are restrictions of elements of the Musielak–Orlicz–Hardy space, associated with \(L_{{\mathbb {R}}^n}:=-\Delta +V\) on \({\mathbb {R}}^n\), to \(\Omega \). In this article, the authors show that the operators \(VL^{-1}_\Omega \) and \(\nabla ^2L^{-1}_\Omega \) are bounded from \(L^1(\Omega )\) to weak-\(L^1(\Omega )\), from \(L^p(\Omega )\) to itself, with \(p\in (1,2]\), and also from \(H_{\varphi ,\,L_{{\mathbb {R}}^n},\,r}(\Omega )\) to the Musielak–Orlicz space \(L^\varphi (\Omega )\) or to \(H_{\varphi ,\,L_{{\mathbb {R}}^n},\,r}(\Omega )\) itself. As applications, the boundedness of \(\nabla ^2{\mathbb {G}}_D\) on \(L^p(\Omega )\), with \(p\in (1,2]\), and from \(H_{\varphi ,\,L_{{\mathbb {R}}^n},\,r}(\Omega )\) to \(L^\varphi (\Omega )\) or to \(H_{\varphi ,\,L_{{\mathbb {R}}^n},\,r}(\Omega )\) itself is obtained, where \({\mathbb {G}}_D\) denotes the Dirichlet Green operator associated with \(L_\Omega \). All these results are new even for the Hardy space \(H^1_{L_{{\mathbb {R}}^n},\,r}(\Omega )\), which is just \(H_{\varphi ,\,L_{{\mathbb {R}}^n},\,r}(\Omega )\) with \(\varphi (x,t):=t\) for all \(x\in {\mathbb {R}}^n\) and \(t\in [0,\infty )\).     

Order property and modulus of continuity of weak KAM solutions

For the Hamilton–Jacobi equation \(H(x,\partial _xu+c)=\alpha (c)\) with \(x\in \mathbb {T}^2\), it is shown in this paper that, for all \(c\in \alpha ^{-1}(E)\) with \(E>\min \alpha \), the elementary weak KAM solutions can be parameterized so that they are \(\frac{1}{3}\)-Hölder continuous in \(C^0\)-topology.     

The non-abelian tensor square of residually finite groups

Let m, n be positive integers and p a prime. We denote by \(\nu (G)\) an extension of the non-abelian tensor square \(G \otimes G\) by \(G \times G\). We prove that if G is a residually finite group satisfying some non-trivial identity \(f \equiv ~1\) and for every \(x,y \in G\) there exists a p-power \(q=q(x,y)\) such that \([x,y^{\varphi }]^q = 1\), then the derived subgroup \(\nu (G)'\) is locally finite (Theorem A). Moreover, we show that if G is a residually finite group in which for every \(x,y \in G\) there exists a p-power \(q=q(x,y)\) dividing \(p^m\) such that \([x,y^{\varphi }]^q\) is left n-Engel, then the non-abelian tensor square \(G \otimes G\) is locally virtually nilpotent (Theorem B).     

On generalized Ramanujan primes

Let \(\Delta = \sum _{m=0}^\infty q^{(2m+1)^2} \in \mathbf {F}_2[[q]]\) be the reduction mod 2 of the \(\Delta \) series. A modular form of level 1, \(f=\sum _{n\geqslant 0} c(n) \,q^n\), with integer coefficients, is congruent modulo \(2\) to a polynomial in \(\Delta \). Let us set \(W_f(x)=\sum _{n\leqslant x,\ c(n)\text { odd }} 1\), the number of odd Fourier coefficients of \(f\) of index \(\leqslant x\). The order of magnitude of \(W_f(x)\) (for \(x\rightarrow \infty \)) has been determined by Serre in the seventies. Here, we give an asymptotic equivalent for \(W_f(x)\). Let \(p(n)\) be the partition function and \(A_0(x)\) (resp. \(A_1(x)\)) be the number of \(n\leqslant x\) such that \(p(n)\) is even (resp. odd). In the preceding papers, the second-named author has shown that \(A_0(x)\geqslant 0.28 \sqrt{x\;\log \log x}\) for \(x\geqslant 3\) and \(A_1(x)>\frac{4.57 \sqrt{x}}{\log x}\) for \(x\geqslant 7\). Here, it is proved that \(A_0(x)\geqslant 0.069 \sqrt{x}\;\log \log x\) holds for \(x>1\) and that \(A_1(x) \geqslant \frac{0.037 \sqrt{x}}{(\log x)^{7/8}}\) holds for \(x\geqslant 2\). The main tools used to prove these results are the determination of the order of nilpotence of a modular form of level-\(1\) modulo \(2\), and of the structure of the space of those modular forms as a module over the Hecke algebra, which have been given in a recent work of Serre and the second-named author.     

One-sided invertibility of discrete operators and their applications

For \(p\in [1,\infty ]\), we establish criteria for the one-sided invertibility of binomial discrete difference operators \({{\mathcal {A}}}=aI-bV\) on the space \(l^p=l^p(\mathbb {Z})\), where \(a,b\in l^\infty \), I is the identity operator and the isometric shift operator V is given on functions \(f\in l^p\) by \((Vf)(n)=f(n+1)\) for all \(n\in \mathbb {Z}\). Applying these criteria, we obtain criteria for the one-sided invertibility of binomial functional operators \(A=aI-bU_\alpha \) on the Lebesgue space \(L^p(\mathbb {R}_+)\) for every \(p\in [1,\infty ]\), where \(a,b\in L^\infty (\mathbb {R}_+)\), \(\alpha \) is an orientation-preserving bi-Lipschitz homeomorphism of \([0,+\infty ]\) onto itself with only two fixed points 0 and \(\infty \), and \(U_\alpha \) is the isometric weighted shift operator on \(L^p(\mathbb {R}_+)\) given by \(U_\alpha f= (\alpha ^\prime )^{1/p}(f\circ \alpha )\). Applications of binomial discrete operators to interpolation theory are given.