Let $\parallel ·\parallel$ be a norm on the algebra ${ℳ}_n$ of all $n\times n$ matrices over $ℂ$. An interesting problem in matrix theory is that “Are there two norms ${\parallel ·\parallel }_1$ and ${\parallel ·\parallel }_2$ on ${ℂ}^n$ such that $\parallel A\parallel =max\{\parallel Ax{\parallel }_2\parallel x{\parallel }_1=1\}$ for all $A\in {ℳ}_n$?” We will investigate this problem and its various aspects and will discuss some conditions under which ${\parallel ·\parallel }_1={\parallel ·\parallel }_2$.

Commutativity and continuity conditions for the Moore-Penrose inverse and the "conorm" are established in a C*-algebra; moreover, spectral permanence and B*-properties for the conorm are proved.

In many recent articles, medians have been used as a replacement of integral averages when the function fails to be locally integrable. A point $x$ in a metric measure space $(X,d,\mu )$ is called a generalized Lebesgue point of a measurable function $f$ if the medians of $f$ over the balls $B(x,r)$ converge to $f\left(x\right)$ when $r$ converges to $0$. We know that almost every point of a measurable, almost everywhere finite function is a generalized Lebesgue point and the same is true for every point of a continuous function. We show...

We establish results on interpolation of Rosenthal operators, Banach-Saks operators, Asplund operators and weakly compact operators by means of generalized Lions-Peetre methods of constants and means. Applications are presented for the K-method space generated by the Calderón-Lozanovskii space parameters.

L-norms and M-norms on Banach lattices, unit-norms and base norms on ordered vector spaces are well known. In this paper m- and $m_{\le }$-norms are introduced on ordered normed spaces. They generalize the notions of the M-norm and the order-unit norm, possess also some interesting properties and can be characterized by means of the constants of reproducibility of cones. In particular, the dual norm of an ordered Banach space with a closed cone turns out to be additive on the dual cone if and only if the...

Let $\Omega \subset {ℝ}^n$, $n\ge 2$, be a bounded connected domain of the class $C^{1,\theta }$ for some $\theta \in (0,1]$. Applying the generalized Moser-Trudinger inequality without boundary condition, the Mountain Pass Theorem and the Ekeland Variational Principle, we prove the existence and multiplicity of nontrivial weak solutions to the problem $$u\in W^1{L}^{\Phi }\left(\Omega \right),-div\left({\Phi }^{\text{'}}\left(\right|\nabla u\left|\right)\frac{\nabla u}{\left|\nabla u\right|}\right)+V\left(x\right){\Phi }^{\text{'}}\left(\right|u\left|\right)\frac{u}{\left|u\right|}=f(x,u)+\mu h\left(x\right)\text{in}\Omega ,\frac{\partial u}{\partial 𝐧}=0\text{on}\partial \Omega ,$$ where $\Phi$ is a Young function such that the space $W^1{L}^{\Phi }\left(\Omega \right)$ is embedded into exponential or multiple exponential Orlicz space, the nonlinearity $f(x,t)$ has the corresponding critical growth, $V\left(x\right)$ is a continuous potential,...

Let n ≥ 2 and let Ω ⊂ ℝn be an open set. We prove the boundedness of weak solutions to the problem $$u\in W_0^1{L}^{\Phi }\left(\Omega \right)and-div\left({\Phi }^{\text{'}}\left(\left|\nabla u\right|\right)\frac{\nabla u}{\left|\nabla u\right|}\right)+V\left(x\right){\Phi }^{\text{'}}\left(\left|u\right|\right)\frac{u}{\left|u\right|}=f\left(x,u\right)+\mu h\left(x\right)in\Omega ,$$ where ϕ is a Young function such that the space W 01 L Φ(Ω) is embedded into an exponential or multiple exponential Orlicz space, the nonlinearity f(x, t) has the corresponding critical growth, V(x) is a continuous potential, h ∈ L Φ(Ω) is a non-trivial continuous function and µ ≥ 0 is a small parameter. We consider two classical cases: the case of Ω being an open bounded set and the case of Ω =...

The generalized non-commutative torus $T_{\varrho }^k$ of rank n is defined by the crossed product $A_{m/k}{\times }_{\alpha ₃}\mathbb{Z}{\times }_{\alpha ₄}...{\times }_{\alpha ₙ}\mathbb{Z}$, where the actions ${\alpha }_i$ of ℤ on the fibre $M_k\left(ℂ\right)$ of a rational rotation algebra $A_{m/k}$ are trivial, and $C*\left(k\mathbb{Z}\times k\mathbb{Z}\right){\times }_{\alpha ₃}\mathbb{Z}{\times }_{\alpha ₄}...{\times }_{\alpha ₙ}\mathbb{Z}$ is a non-commutative torus $A_{\varrho }$. It is shown that $T_{\varrho }^k$ is strongly Morita equivalent to $A_{\varrho }$, and that $T_{\varrho }^k\otimes M_{p^{\infty }}$ is isomorphic to $A_{\varrho }\otimes M_k\left(ℂ\right)\otimes M_{p^{\infty }}$ if and only if the set of prime factors of k is a subset of the set of prime factors of p.