The perturbed eigenvalue problem $-\Delta u-{\Delta }_{p}u=\lambda V\left(x\right)u$, with $p\in (1,N)\setminus \left\{2\right\}$ and $V$ a weight function which takes nonnegative values and may have singular points, is studied in an Orlicz–Sobolev setting on general open sets from ${\mathbb{R}}^N$ with $N\ge 3$. The analysis of these problems leads to a full characterization of the set of parameters $\lambda$ for which the problem possesses nontrivial solutions as being an unbounded open interval.

Let $\Phi$ and $\Psi$ be Young functions, and let $L^{\Phi }(\Omega )$ and $L^{\Psi }(\Omega )$ be corresponding Orlicz spaces on a measure space $(\Omega ,\mu )$. Our aim in this paper is to prove that, under mild conditions on $\Phi$ and $\Psi$, the multiplication from $L^{\Phi }(\Omega )\times L^{\Psi }(\Omega )$ onto $L^1(\Omega )$ is uniformly open. This generalizes an interesting recent result due to M. Balcerzak, A. Majchrzycki, and F. Strobin in 2013.

One of the most fundamental properties of the field of values of an operator is the inclusion of the spectrum within its closure. Obtaining information on the spectrum of products of operators in terms of this spectral inclusion region is a demanding issue. Stating general results seems difficult; however, conclusions can be derived in some special instances. In this paper, we show that the field of values of products of Laurent operators is easily related to the product of their fields of values, and the same occurs for certain classes of Laurent operators with matrix symbols. The results also apply to the class of infinite upper (lower) triangular Toeplitz matrices.

Let $H$ be a compact subgroup of a locally compact group $G$, and let $\mu$ be a strongly quasi-invariant Radon measure on the homogeneous space $G/H$. In this article, we show that every element of $\widehat{G/H}$, the character space of $G/H$, determines a nonzero multiplicative linear functional on $L^1(G/H,\mu )$. Using this, we prove that for all $\varphi \in \widehat{G/H}$, the right $\varphi$-amenability of $L^1(G/H,\mu )$ and the right $\varphi$-amenability of $M(G/H)$ are both equivalent to the amenability of $G$. Also, we show that $L^1(G/H,\mu )$, as well as $M(G/H)$, is right $\varphi$-contractible if and only if $G$ is compact. In particular, when $H$ is the trivial subgroup, we obtain the known results on group algebras and measure algebras.

We give a very simple proof of a result by Dafni that states that the weak${}^{\ast }$-convergence is true in the local Hardy space $h^1\left({\mathbb{R}}^d\right)$.

We introduce and study the class of extremally rich JB${}^{\ast }$-triples. We establish new results to determine the distance from an element $a$ in an extremally rich JB${}^{\ast }$-triple $E$ to the set ${\partial }_e\left(E_1\right)$ of all extreme points of the closed unit ball of $E$. More concretely, we prove that

$$dist(a,{\partial }_e(E_1\left)\right)=max\hspace{0.17em}\{1,\Vert a\Vert -1\},$$ for every $a\in E$ which is not Brown–Pedersen quasi-invertible. As a consequence, we determine the form of the $\lambda$-function of Aron and Lohman on the open unit ball of an extremally rich JB${}^{\ast }$-triple $E$ by showing that $\lambda \left(a\right)=1/2$ for every non-BP quasi-invertible element $a$ in the open unit ball of $E$. We also prove that for an extremally rich JB${}^{\ast }$-triple $E$, the quadratic conorm ${\gamma }^q(\cdot )$ is continuous at a point $a\in E$ if and only if either $a$ is not von Neumann regular (i.e., ${\gamma }^q\left(a\right)=0$) or $a$ is Brown–Pedersen quasi-invertible.

There are known results showing a canonical association between Lipschitz cross-norms (norms on the Lipschitz tensor product of a metric space and a Banach space) and ideals of Lipschitz maps from a metric space to a dual Banach space. We extend this association, relating Lipschitz cross-norms to ideals of Lipschitz maps taking values in general Banach spaces. To do that, we prove a Lipschitz version of the representation theorem for maximal operator ideals. As a consequence, we obtain linear characterizations of some ideals of (nonlinear) Lipschitz maps between metric spaces.

A bounded linear operator $S$ on a Banach space $X$ is called an $m$-left generalized inverse of an operator $T$ for a positive integer $m$ if

$$T{\sum }_{j=0}^m(-1{)}^j(\begin{array}{c}\hfill m\hfill \\ \hfill j\hfill \end{array})S^{m-j}{T}^{m-j}=0,$$ and it is called an $m$-right generalized inverse of $T$ if

$$S{\sum }_{j=0}^m(-1{)}^j(\begin{array}{c}\hfill m\hfill \\ \hfill j\hfill \end{array})T^{m-j}{S}^{m-j}=0.$$ If $T$ is both an $m$-left and an $m$-right generalized inverse of $T$, then it is said to be an $m$-generalized inverse of $T$.

This paper has two purposes. The first is to extend the notion of generalized inverse to $m$-generalized inverse of an operator on Banach spaces and to give some structure results. The second is to generalize some properties of $m$-partial isometries on Hilbert spaces to the class of $m$-left generalized invertible operators on Banach spaces. In particular, we study some cases in which a power of an $m$-left generalized invertible operator is again $m$-left generalized invertible.

This paper extends a recent matrix trace inequality of Bourin–Lee to semifinite von Neumann algebras. This provides a generalization of the Lieb–Thirring-type inequality in von Neumann algebras due to Kosaki. Some new inequalities, even in the matrix case, are also given for the Heinz means.

For Banach spaces $X$ and $Y$, let $K_{w^{\ast }}(X^{\ast },Y)$ denote the space of all $w^{\ast }-w$ continuous compact operators from $X^{\ast }$ to $Y$ endowed with the operator norm. A Banach space $X$ has the $\mathit{BD}$ property if every limited subset of $X$ is relatively weakly compact. We prove that if $X$ has the Gelfand–Phillips property and $Y$ has the $\mathit{BD}$ property, then $K_{w^{\ast }}(X^{\ast },Y)$ has the $\mathit{BD}$ property.

We show that every dominated linear operator from a Banach–Kantorovich space over an atomless Dedekind-complete vector lattice to a sequence Banach lattice ${\ell }_p(\Gamma )$ or $c_0(\Gamma )$ is narrow. As a consequence, we obtain that an atomless Banach lattice cannot have a finite-dimensional decomposition of a certain kind. Finally, we show that the order-narrowness of a linear dominated operator $T$ from a lattice-normed space $V$ to the Banach space with a mixed norm $(W,F)$ over an order-continuous Banach lattice $F$ implies the order-narrowness of its exact dominant |$T$|.

This article is devoted to introducing new geometric constants that quantify the difference between Roberts orthogonality and Birkhoff orthogonality in normed planes. We start by characterizing Roberts orthogonality in two different ways: via bisectors of two points and the use of certain linear transformations. Each of these characterizations yields one of those geometric constants that we study.

We prove that a Banach space $X$ has the bounded approximation property if and only if, for every separable Banach space $Z$ and every injective operator $T$ from $Z$ to $X$, there exists a net $\left(S_{\alpha }\right)$ of finite-rank operators from $Z$ to $X$ with $\Vert S_{\alpha }\Vert \le {\lambda }_T$ such that ${lim\hspace{0.17em}}_{\alpha }\Vert S_{\alpha }z-Tz\Vert =0$ for every $z\in Z$.

In this article, we consider the $\phi$-Gateaux derivative of the norm in spaces of compact operators in such a way as to extend the Kečkić theorem. Our main result determines the $\phi$-Gateaux derivative of the $\mathcal{K}(X;Y)$ norm.

We construct dense Banach subalgebras $A$ of the null sequence algebra $c_0$ which are spectral-invariant but do not satisfy the $D_1$-condition $\Vert ab{\Vert }_A\le K(\Vert a{\Vert }_{\infty }\Vert b{\Vert }_A+\Vert a{\Vert }_A\Vert b{\Vert }_{\infty })$ for all $a,b\in A$. The sequences in $A$ vanish in a skewed manner with respect to an unbounded function $\sigma :\mathbb{N}\to [1,\infty )$.