We investigate screen-type boundary value problems for anisotro-pic pseudo-Maxwell’s equations. It is shown that the problems with tangent traces are well posed in tangent Sobolev spaces. The unique solvability result is then obtained by using a potential method and the coercivity result of Costabel for the bilinear form associated with the pseudo-Maxwell’s equation.

For some rearrangement-invariant functional $\Vert \cdot {\Vert }_X$ having the lattice property, we give a characterization of linearity of the set $\{f:\Vert f{\Vert }_X<\infty \}$. Afterwards we apply this general abstract theorem in the case of Orlicz–Lorentz spaces.

In this paper, we characterize the perturbations of the Moore–Penrose metric generalized inverse of closed operator in Banach spaces. Under the condition $R\left(\delta T\right)\subset R\left(T\right)$, $N\left(T\right)\subset N\left(\delta T\right)$, respectively, we get some new results about upper-bound estimates of $\Vert {\overline{T}}^M\Vert$ and $\Vert {\overline{T}}^M-T^M\Vert$.

Let $\mathcal{X}$ be an infinite-dimensional complex Banach space and let $\mathcal{B}\left(\mathcal{X}\right)$ be the algebra of all bounded linear operators. In this article we show that an additive surjective map $\phi$ on $\mathcal{B}\left(\mathcal{X}\right)$ preserving the semi-Fredholm domain in spectrum is an automorphism or an antiautomorphism on $\mathcal{B}\left(\mathcal{X}\right)$.

We show decay bounds of the form

$${\int }_{{\mathbb{R}}^d}\varphi \left(u\right(x,t\left)\right)dx\le C{t}^{-\mu }$$ for integrable and bounded solutions to the nonlocal evolution equation

$$u_t(x,t)={\int }_{{\mathbb{R}}^d}J(x,y)G\left(u\right(y,t)-u(x,t\left)\right)\left(u\right(y,t)-u(x,t\left)\right)dy+f\left(u\right(x,t\left)\right).$$ Here $G$ is a nonnegative and even function, and $f$ verifies $f\left(\xi \right)\xi \le 0$ for all $\xi \ge 0$. We remark that $G$ is not assumed to be homogeneous. The function $\varphi$ and the exponent $\mu$ depend on $G$ via adequate hypotheses, while $J$ is a nonnegative kernel satisfying suitable assumptions.

Using the concept of power-contractive selfmaps of metric spaces, we investigate quasicompact composition operators on certain classes of Lipschitz algebras. As an application of our results, we obtain certain properties of power-contractive selfmaps of plane sets.

Let $A$ be a bounded closed convex set in a Banach space. The boundaries of the wide spherical hull $\eta \left(A\right)$ and the tight spherical hull $\theta \left(A\right)$ are characterized, the existence of diametral points of these three sets are discussed, and a further relation between these three sets is clarified. Moreover, a new characterization of balls is presented.

Let $\Omega$ be a bounded measurable set in ${\mathbb{R}}^n$. The best polynomial approximation operator was recently extended by Cuenya from $L^p$ to $L^{p-1}$.

In this paper, we extend the operator of the best polynomial approximation from $L^{p(\cdot )}(\Omega )$ to the space $L^{p(\cdot )-1}(\Omega )$.

For a locally compact Abelian group $G$, we give a necessary and sufficient condition for shifts of a function $\varphi \in L_2\left(G\right)$ to be a Riesz family. Also, for a finite family $\Phi$ of compactly supported functions in $L_2\left(G\right)$, we show that the shifts of $\Phi$ constitute a Riesz family if and only if the nets $\left(\varphi \right(\xi \eta ){)}_{\eta \in L^{\perp }}$, $\varphi \in \Phi$, are linearly independent for all $\xi \in \hat{G}$.

Let $\mathcal{H}$ be a complex separable Hilbert space, and let $A$ be a bounded self-adjoint operator on $\mathcal{H}$. Consider the orthonormal basis $\mathcal{B}=\{e_1,e_2,\dots \}$ and the projection $P_n$ of $\mathcal{H}$ onto the finite-dimensional subspace spanned by the first $n$ elements of $\mathcal{B}$. The finite-dimensional truncations $A_n=P_{n}A{P}_n$ shall be regarded as a sequence of finite matrices by restricting their domains to $P_n\left(\mathcal{H}\right)$ for each $n$. Many researchers used the sequence of eigenvalues of $A_n$ to obtain information about the spectrum of $A$. But in many situations, these $A_n$’s need not be simple enough to make the computations easier. The natural question Can we use some simpler sequence of matrices $B_n$ instead of $A_n$? is addressed in this article. The notion of preconditioners and their convergence in the sense of eigenvalue clustering are used to study the problem. The connection between preconditioners and compact perturbations of operators is identified here. The usage of preconditioners in the spectral gap prediction problems is also discussed. The examples of Toeplitz and block Toeplitz operators are considered as an application of these results. Finally, some future possibilities are discussed.

In this article, necessary and sufficient conditions for the cone nonnegativity of Moore–Penrose inverses of unbounded Gram operators are derived. These conditions include statements on acuteness of certain closed convex cones in infinite-dimensional real Hilbert spaces.

In this paper, we prove that, for any positive definite matrices $A,B$, and real numbers $\nu ,\mu ,p$ with $-1\le p<1$ and $0<\nu \le \mu <1$, we have

$$\frac{\nu }{\mu }(A{\nabla }_{\mu }B-A{♯}_{p,\mu }B)\le A{\nabla }_{\nu }B-A{♯}_{p,\nu }B\le \frac{1-\nu }{1-\mu }(A{\nabla }_{\mu }B-A{♯}_{p,\mu }B),$$ where ${\nabla }_{\nu }$ and ${♯}_{p,\nu }$ stand for weighted arithmetic and power mean, respectively. In the special cases when $p=0,1$, this inequality can be considered as a generalization of harmonic-arithmetic and geometric-arithmetic means inequalities and their reverses.

Applying this inequality, some inequalities for the Heinz mean and determinant inequalities related to weighted power means are obtained.

Let $C[0,t]$ denote a generalized Wiener space, the space of real-valued continuous functions on the interval $[0,t]$, and define a random vector $Z_n:C[0,t]\to {\mathbb{R}}^n$ by $$Z_n\left(x\right)=\left({\int }_0^{t_1}h\right(s\left)dx\right(s),\dots ,{\int }_0^{t_n}h(s\left)dx\right(s\left)\right),$$ where $0<t_1<\cdots <t_n=t$ is a partition of $[0,t]$ and $h\in L_2[0,t]$ with $h\ne 0$ almost everywhere. Using a simple formula for a generalized conditional Wiener integral on $C[0,t]$ with the conditioning function $Z_n$, we evaluate the generalized analytic conditional Wiener and Feynman integrals of the cylinder function $$G\left(x\right)=f\left(\right(e,x\left)\right)\varphi \left(\right(e,x\left)\right)$$ for $x\in C[0,t]$, where $f\in L_p\left(\mathbb{R}\right)(1\le p\le \infty )$, $e$ is a unit element in $L_2[0,t]$, and $\varphi$ is the Fourier transform of a measure of bounded variation over $\mathbb{R}$. We then express the generalized analytic conditional Feynman integral of $G$ as two kinds of limits of nonconditional generalized Wiener integrals with a polygonal function and cylinder functions using a change-of-scale transformation. The choice of a complete orthonormal subset of $L_2[0,t]$ used in the transformation is independent of $e$.

For a generic set in the Teichmüller space, we construct a covariant functor with the range in a category of the $\mathit{AF}$-algebras; the functor maps isomorphic Riemann surfaces to the stably isomorphic $\mathit{AF}$-algebras. In the special case of genus one, one gets a functor between the category of complex tori and the Effros–Shen algebras.