On the best constant in Hyers–Ulam stability of some positive linear operators

We obtain the infimum of the Hyers–Ulam stability constants for Stancu, Bernstein and Kantorovich operators and prove that in a class of certain positive linear operators this infimum for Bernstein operator has a minimality property.     

Perturbations of simple eigenvectors¶of linear operators

Residual bounds for perturbed simple eigenvectors of linear operators are derived. Received: 27 January 1999     

Correlation functions of gauged linear σ-model

This is the second paper in a series following Tian and Xu(2015), on the construction of a mathematical theory of the gauged linear σ-model(GLSM). In this paper, assuming the existence of virtual moduli cycles and their certain properties, we define the correlation function of GLSM for a fixed smooth rigidified r-spin curve.     

L p -error estimates for positive linear operators using the second-order τ-modulus

ПустьМ[а, b] — множество вещественных функци йf, измеримых и ограниче нных на отрезке [а, b], иL: M[a, b] → M[a, b] — лин ейный положительный оператор. В статье пол учены оценки погрешностей аппроксимации вL _p -нор ме, выраженные в термина х так называемого усредненного локаль ного модуля гладкост и, илиτ-модуля. ПустьL облад ает свойствами: (e _i ,i(t):=t ⁱ дляi=0,1,2)Le ₀ =e ₀ ,Le ₁ =e ₁ ,Le ₂ (х)=х ² +β(х), x∈[a,b] и i)A ≦ min (1, (b-a)²/4) для 1 <p<∞, ii)A ≦ min (e ^?1 , (b — a)²) дляр=1 приA:=∥β∥_t8. Тогда выполнены оцен ки: $$\begin{gathered} \left\| {f - Lf} \right\|_p \leqq C\frac{p}{{p - 1}}\tau _2 (f;\sqrt A )_p 1< p< \infty \hfill \\ \left\| {f - Lf} \right\|_1 \leqq C\tau _2 \left( {f;\sqrt {A\ln \frac{1}{A}} } \right)_1 , \hfill \\ \end{gathered} $$ где положительные по стоянныеС не завися т от оператораL, функцииfи параметрар.     

On an estimate of Calderón-Zygmund operators by dyadic positive operators

Given a general dyadic grid D and a sparse family of cubes S = {Q _j ^k ∈ D, define a dyadic positive operator A _D,S by $${A_{D,S}}f(x) = \sum\limits_{j,k} {{f_{Q_j^k}}{\chi _{Q_j^k}}} (x)$$ . Given a Banach function space X(? ⁿ ) and the maximal Calderón-Zygmund operator ${T_\natural }$ , we show that $${\left\| {{T_\natural}f} \right\|_X} \leqslant c(T,n)\mathop {\sup }\limits_{D,S} {\left\| {{A_{D,S}}|f|} \right\|_X}$$ This result is applied to weighted inequalities. In particular, it implies (i) the “twoweight conjecture” by D. Cruz-Uribe and C. Pérez in full generality; (ii) a simplification of the proof of the “A ₂ conjecture”; (iii) an extension of certain mixed A _p ?A _r estimates to general Calderón-Zygmund operators; (iv) an extension of sharp A ₁ estimates (known for T ) to the maximal Calderón-Zygmund operator $\natural $ .     

Σ-convergence of nonlinear monotone operators in perforated domains with holes of small size

This paper is devoted to the homogenization beyond the periodic setting, of nonlinear monotone operators in a domain in ℝ ^N with isolated holes of size ɛ² (ɛ > 0 a small parameter). The order of the size of the holes is twice that of the oscillations of the coefficients of the operator, so that the problem under consideration is a reiterated homogenization problem in perforated domains. The usual periodic perforation of the domain and the classical periodicity hypothesis on the coefficients of the operator are here replaced by an abstract assumption covering a great variety of behaviors such as the periodicity, the almost periodicity and many more besides. We illustrate this abstract setting by working out a few concrete homogenization problems. Our main tool is the recent theory of homogenization structures.     

Representation for the W-weighted Drazin inverse of linear operators

In this paper we study the W-weighted Drazin inverse of the bounded linear operators between Banach spaces and its representation theorem. Based on this representation, utilizing the spectral theory of Banach space operators, we derive an approximating expression of the W-weighted Drazin inverse and an error bound. Also, a perturbation theorem for the W-weighted Drazin inverse is uniformly obtained from the representation theorem.     

Some remarks on Birkhoff–James orthogonality of linear operators

We study Birkhoff–James orthogonality of compact (bounded) linear operators between Hilbert spaces and Banach spaces. Applying the notion of semi-inner-products in normed linear spaces and some related geometric ideas, we generalize and improve some of the recent results in this context. In particular, we obtain a characterization of Euclidean spaces and also prove that it is possible to retrieve the norm of a compact (bounded) linear operator (functional) in terms of its Birkhoff–James orthogonality set. We also present some best approximation type results in the space of bounded linear operators.     

On the class of positive almost weak Dunford–Pettis operators

We introduce and study the class of almost weak Dunford–Pettis operators and we derive the following interesting consequence: other characterizations of the weak Dunford–Pettis property. After that we characterize pairs of Banach lattices for which the adjoint of almost weak Dunford–Pettis operator is almost Dunford–Pettis. Finally, we establish a necessary and sufficient conditions on the pair of Banach lattices E and F which guarantees that if T : E → F is a positive almost weak Dunford–Pettis then T is almost Dunford–Pettis.     

Interpolation of linear operators in the Köthe dual spaces

Summary In this paper we investigate when the Köthe dual spaces Y and X are interpolation spaces with respect to couples of the Köthe dual spaces (Y₀, Y₁) and (X₀, X₁), respectively, where X and Y are interpolation spaces with respect to given couples (X₀,X₁) and (Y₀, Y₁ of Banach function spaces.     

Absence of discrete spectrum of the Schrödinger operators of positive molecular ions

For multiatomic positive molecular ions withN electrons and chargesZ _i of the nuclei it is shown that there are no bound states forZ _i >N for some . It is also shown that the discrete spectrum of the Schrödinger operator of a molecular ion is empty if for at least some of the nuclei the conditionZ _i N is satisfied, whereas the charges of the other nuclei can be small.St Petersburg Branch of the V. A. Steklov Mathematics Institute, Russian Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 93, No. 1, pp. 94–106, October, 1992.     

Γ-convergence of free-discontinuity problems

We study the Γ-convergence of sequences of free-discontinuity functionals depending on vector-valued functions u which can be discontinuous across hypersurfaces whose shape and location are not known a priori. The main novelty of our result is that we work under very general assumptions on the integrands which, in particular, are not required to be periodic in the space variable. Further, we consider the case of surface integrands which are not bounded from below by the amplitude of the jump of u.We obtain three main results: compactness with respect to Γ-convergence, representation of the Γ-limit in an integral form and identification of its integrands, and homogenisation formulas without periodicity assumptions. In particular, the classical case of periodic homogenisation follows as a by-product of our analysis. Moreover, our result covers also the case of stochastic homogenisation, as we will show in a forthcoming paper.     

The lower estimate for the linear combinations of Bernstein–Kantorovich operators

In this paper we obtain a new strong type of Steckin inequality for the linear combinations of Bernstein–Kantorovich operators, which gives the optimal approximation rate. On the basis of this inequality, we further obtain the lower estimate for these operators.     

On the absence of positive eigenvalues of Schrödinger operators with rough potentials

We prove the absence of positive eigenvalues of Schrödinger operators $ H=-\Delta+V $ on Euclidean spaces $ \mathbb{R}^n $ for a certain class of rough potentials $V$. To describe our class of potentials fix an exponent $q\in[n/2,\infty]$ (or $q\in(1,\infty]$, if $n=2$) and let $\beta(q)=(2q-n)/(2q)$. For the potential $V$ we assume that $V\in L^{n/2}_{{\rm{loc}}}(\mathbb{R}^n)$ (or $V\in L^{r}_{{\rm{loc}}}(\mathbb{R}^n)$, $r>1$, if $n=2$) and$\begin{equation*}$$\lim_{R\to\infty}R^{\beta(q)}||V||_{L^q(R\leq |x|\leq 2R)}=0\,.$$\end{equation*}$Under these assumptions we prove that the operator $H$ does not admit positive eigenvalues. The case $q=\infty$ was considered by Kato [K]. The absence of positive eigenvalues follows from a uniform Carleman inequality of the form$\begin{equation*}$$||W_m u||_{l^a(L^{p(q)})(\mathbb R^n)}\leq C_q||W_m|x|^{\beta(q)}(\Delta+1)u||_{l^a(L^{p(q)})(\mathbb{R}^n)}$$\end{equation*}$for all smooth compactly supported functions $u$ and a suitable sequence of weights $W_m$, where $p(q)$ and $p(q)$ are dual exponents with the property that $1/p(q)-1/p(q)=1/q$.     

On the measure of noncompactness of linear operators in spaces of strongly α-summable and bounded sequences

In this paper, we characterize classes of matrix transformations from BK spaces into spaces of bounded sequences and their subclasses of infinite matrices that define compact operators. Furthermore, using these results and the solvability of certain infinite linear systems we give necessary and sufficient conditions for A to be a compact operator on spaces that are strongly α-bounded or summable. Research is supported by the German DAAD foundation (German Academic Exchange Service), the University of Le Havre and the research project #1232 of the Serbian Ministry of Science, Technology and Development.