Quantitative convergence theorems for a class of Bernstein–Durrmeyer operators preserving linear functions

We supplement recent results on a class of Bernstein–Durrmeyer operators preserving linear functions. This is done by discussing two limiting cases and proving quantitative Voronovskaya-type assertions involving the first-order and second-order moduli of smoothness. The results generalize and improve earlier statements for Bernstein and genuine Bernstein–Durrmeyer operators.     

q-Szász operators which preserve x 2

In the present paper we construct q-Szász operators that preserve the third test function e ₂. Rate of global convergence is obtained in the frame of weighted spaces. Furthermore, we obtain a Voronovskaja type theorem for these operators.     

Which sets are images of disk-algebra functions?

We give a characterization of those compact sets in the plane with finitely many holes that are images of disk-algebra functions. We also show that the image of the closed unit disk via a polynomial is, in general, not polynomially convex.     

When do several linear operators share an invariant cone?

We establish a criterion for a finite family of matrices to possess a common invariant cone. The criterion reduces the problem of existence of an invariant cone to equality of two special numbers that depend on the family. In spite of theoretical simplicity, the practical use of the criterion may be difficult. We show that the problem of existence of a common invariant cone for four matrices with integral entries is algorithmically undecidable. Corollaries of the criterion, which give sufficient and necessary conditions, are derived. Finally, we introduce a “co-directional number” of several matrices. We prove that this parameter is close to zero iff there is a small perturbation of matrices, after which they get an invariant cone. An algorithm for its computation is presented.     

Which subnormal Toeplitz operators are either normal or analytic?

We study subnormal Toeplitz operators on the vector-valued Hardy space of the unit circle, along with an appropriate reformulation of P.R. Halmos?s Problem 5: Which subnormal block Toeplitz operators are either normal or analytic? We extend and prove Abrahamse?s theorem to the case of matrix-valued symbols; that is, we show that every subnormal block Toeplitz operator with bounded type symbol (i.e., a quotient of two bounded analytic functions), whose analytic and co-analytic parts have the “left coprime factorization”, is normal or analytic. We also prove that the left coprime factorization condition is essential. Finally, we examine a well-known conjecture, of whether every subnormal Toeplitz operator with finite rank self-commutator is normal or analytic.     

Are unbounded linear operators computable on the average for Gaussian measures?

Traub and Werschulz [Complexity and Information, Cambridge University Press, New York, 1999] ask whether every linear operator S:⊆X→Y$S:\subseteq X\to Y$ is “computable on the average” w.r.t. a Gaussian measure on X. The question is inspired by an analogous result in information-based complexity on the average-case solvability of linear approximation problems. We give several interpretations of Traub and Werschulz’ question within the framework of type-2 theory of effectivity. We have negative answers to all of these interpretations but the one with minimal requirements on the algorithm's uniformness. On our way to these results, we give an effective version of the Mourier–Prokhorov characterization of Gaussian measures on separable Hilbert spaces.     

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.     

Perturbations of simple eigenvectors¶of linear operators

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

Approximation of integrable functions by general linear operators of their Fourier series at the Lebesgue points

Pointwise estimates of the deviation T _n,A,B f(⋅)−f(⋅) in terms of moduli of continuity [`(w)]_·f\bar{w}_{\cdot}f and w _⋅ f are proved. Analog results on norm approximation with remarks and corollaries are also given. In the results essentially weaker conditions than those in [2, Theorem 1, p. 437] are used.     

Li–Yorke chaos translation set for linear operators

In order to study Li–Yorke chaos by the scalar perturbation for a given bounded linear operator T on a Banach space X, we introduce the Li–Yorke chaos translation set of T, which is defined by \(S_{LY}(T)=\{\lambda \in {\mathbb {C}};\lambda +T \text { is Li--Yorke chaotic}\}\). In this paper, some operator classes are considered, such as normal operators, compact operators, shift operators, and so on. In particular, we show that the Li–Yorke chaos translation set of the Kalisch operator on the Hilbert space \(\mathcal {L}^2[0,2\pi ]\) is a simple point set \(\{0\}\).     

On Bayes linear unbiased estimation of estimable functions for the singular linear model

The unique Bayes linear unbiased estimator (Bayes LUE) of estimable functions is derived for the singular linear model. The superiority of Bayes LUE over ordinary best linear unbiased estimator is investigated under mean square error matrix (MSEM) criterion.     

Commutators of BMO functions and degenerate Schr?dinger operators with certain nonnegative potentials

Let ${\mathcal{L}f(x)=-\frac{1}{\omega}\sum_{i,j} \partial_i(a_{i,j}(\cdot)\partial_jf)(x)+V(x)f(x)}$ with the non-negative potential V belonging to reverse H?lder class with respect to the measure ??(x)dx, where ??(x) satisfies the A ₂ condition of Muckenhoupt and a _i,j (x) is a real symmetric matrix satisfying ${\lambda^{-1}\omega(x)|\xi|^2\le \sum^n_{i,j=1}a_{i,j}(x)\xi_i\xi_j\le\lambda\omega(x)|\xi|^2. }$ We obtain some estimates for ${V^{\alpha}\mathcal{L}^{-\alpha}}$ on the weighted L ^p spaces and we study the weighted L ^p boundedness of the commutator ${[b, V^{\alpha} \mathcal{L}^{-\alpha}]}$ when ${b\in BMO_\omega}$ and 0?<??? ?? 1.     

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.     

Hyers–Ulam stability of derivations and linear functions

In this paper the following implication is verified for certain basic algebraic curves: if the additive real function f approximately (i.e., with a bounded error) satisfies the derivation rule along the graph of the algebraic curve in consideration, then f can be represented as the sum of a derivation and a linear function. When, instead of the additivity of f, it is assumed that, in addition, the Cauchy difference of f is bounded, a stability theorem is obtained for such characterizations of derivations.     

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.     

Commutators of BMO functions and degenerate Schr?dinger operators with certain nonnegative potentials

Let Lf(x)=-\frac1w?_i,j ?_i(a_i,j(·)?_jf)(x)+V(x)f(x){\mathcal{L}f(x)=-\frac{1}{\omega}\sum_{i,j} \partial_i(a_{i,j}(\cdot)\partial_jf)(x)+V(x)f(x)} with the non-negative potential V belonging to reverse H?lder class with respect to the measure ω(x)dx, where ω(x) satisfies the A ₂ condition of Muckenhoupt and a _i,j (x) is a real symmetric matrix satisfying l^-1w(x)|x|² ￡ ?ⁿ_i,j=1a_i,j(x)x_ix_j ￡ lw(x)|x|².{\lambda^{-1}\omega(x)|\xi|^2\le \sum^n_{i,j=1}a_{i,j}(x)\xi_i\xi_j\le\lambda\omega(x)|\xi|^2. } We obtain some estimates for V^aL^-a{V^{\alpha}\mathcal{L}^{-\alpha}} on the weighted L ^p spaces and we study the weighted L ^p boundedness of the commutator [b, V^a L^-a]{[b, V^{\alpha} \mathcal{L}^{-\alpha}]} when b ? BMO_w{b\in BMO_\omega} and 0 < α ≤ 1.     

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.