An $n\times n$ sign pattern $𝒜$ is said to be potentially nilpotent if there exists a nilpotent real matrix $B$ with the same sign pattern as $𝒜$. Let ${𝒟}_{n,r}$ be an $n\times n$ sign pattern with $2\le r\le n$ such that the superdiagonal and the $(n,n)$ entries are positive, the $(i,1)$ $(i=1,\cdots ,r)$ and $(i,i-r+1)$ $(i=r+1,\cdots ,n)$ entries are negative, and zeros elsewhere. We prove that for $r\ge 3$ and $n\ge 4r-2$, the sign pattern ${𝒟}_{n,r}$ is not potentially nilpotent, and so not spectrally arbitrary.

Let $k$ be a field of characteristic zero and $B$ a $k$-domain. Let $R$ be a retract of $B$ being the kernel of a locally nilpotent derivation of $B$. We show that if $B=R\oplus I$ for some principal ideal $I$ (in particular, if $B$ is a UFD), then $B=R^{\left[1\right]}$, i.e., $B$ is a polynomial algebra over $R$ in one variable. It is natural to ask that, if a retract $R$ of a $k$-UFD $B$ is the kernel of two commuting locally nilpotent derivations of $B$, then does it follow that $B\cong R^{\left[2\right]}$? We give a negative answer to this question. The interest in...

An $n\times n$ ray pattern $𝒜$ is called a spectrally arbitrary ray pattern if the complex matrices in $Q\left(𝒜\right)$ give rise to all possible complex polynomials of degree $n$. In a paper of Mei, Gao, Shao, and Wang (2014) was proved that the minimum number of nonzeros in an $n\times n$ irreducible spectrally arbitrary ray pattern is $3n-1$. In this paper, we introduce a new family of spectrally arbitrary ray patterns of order $n$ with exactly $3n-1$ nonzeros.

A theorem of Burnside asserts that a finite group $G$ is $p$-nilpotent if for some prime $p$ a Sylow $p$-subgroup of $G$ lies in the center of its normalizer. In this paper, let $G$ be a finite group and $p$ the smallest prime divisor of $\left|G\right|$, the order of $G$. Let $P\in {\mathrm{Syl}}_p\left(G\right)$. As a generalization of Burnside’s theorem, it is shown that if every non-cyclic $p$-subgroup of $G$ is self-normalizing or normal in $G$ then $G$ is solvable. In particular, if $P\ncong \langle a,b|a^{p^{n-1}}=1,b^2=1,b^{-1}ab=a^{1+p^{n-2}}\rangle$, where $n\ge 3$ for $p>2$ and $n\ge 4$ for $p=2$, then $G$ is $p$-nilpotent or $p$-closed. ...

Let $G$ be a group. If every nontrivial subgroup of $G$ has a proper supplement, then $G$ is called an $aS$-group. We study some properties of $aS$-groups. For instance, it is shown that a nilpotent group $G$ is an $aS$-group if and only if $G$ is a subdirect product of cyclic groups of prime orders. We prove that if $G$ is an $aS$-group which satisfies the descending chain condition on subgroups, then $G$ is finite. Among other results, we characterize all abelian groups for which every nontrivial quotient group...

For Banach spaces $X$ and $Y$, let $K_{w^{*}}(X^{*},Y)$ denote the space of all $w^{*}-w$ continuous compact operators from $X^{*}$ to $Y$ endowed with the operator norm. A Banach space $X$ has the $wGP$ property if every Grothendieck subset of $X$ is relatively weakly compact. In this paper we study Banach spaces with property $wGP$. We investigate whether the spaces $K_{w^{*}}(X^{*},Y)$ and $X{\otimes }_{ϵ}Y$ have the $wGP$ property, when $X$ and $Y$ have the $wGP$ property.

Let $X$ be a Banach space, $ℬ\left(X\right)$ the algebra of bounded linear operators on $X$ and $(J,\parallel ·{\parallel }_J)$ an admissible Banach ideal of $ℬ\left(X\right)$. For $T\in ℬ\left(X\right)$, let $L_{J,T}$ and $R_{J,T}\in ℬ\left(J\right)$ denote the left and right multiplication defined by $L_{J,T}\left(A\right)=TA$ and $R_{J,T}\left(A\right)=AT$, respectively. In this paper, we study the transmission of some concepts related to recurrent operators between $T\in ℬ\left(X\right)$, and their elementary operators $L_{J,T}$ and $R_{J,T}$. In particular, we give necessary and sufficient conditions for $L_{J,T}$ and $R_{J,T}$ to be sequentially recurrent. Furthermore, we prove that $L_{J,T}$ is recurrent...

Let 1 ≤ p < ∞, $={\left(Xₙ\right)}_{n\in \mathbb{N}}$ be a sequence of Banach spaces and $l_p\left(\right)$ the coresponding vector valued sequence space. Let $={\left(Xₙ\right)}_{n\in \mathbb{N}}$, $={\left(Yₙ\right)}_{n\in \mathbb{N}}$ be two sequences of Banach spaces, $={\left(Vₙ\right)}_{n\in \mathbb{N}}$, Vₙ: Xₙ → Yₙ, a sequence of bounded linear operators and 1 ≤ p,q < ∞. We define the multiplication operator $M_{}:l_p\left(\right)\to l_q\left(\right)$ by $M_{}\left({\left(xₙ\right)}_{n\in \mathbb{N}}\right):={\left(Vₙ\left(xₙ\right)\right)}_{n\in \mathbb{N}}$. We give necessary and sufficient conditions for $M_{}$ to be 2-summing when (p,q) is one of the couples (1,2), (2,1), (2,2), (1,1), (p,1), (p,2), (2,p), (1,p), (p,q); in the last case 1 < p < 2, 1 < q < ∞. ...

This paper is concerned with the isomorphic structure of the Banach space ${\ell }_{\infty }/c₀$ and how it depends on combinatorial tools whose existence is consistent with but not provable from the usual axioms of ZFC. Our main global result is that it is consistent that ${\ell }_{\infty }/c₀$ does not have an orthogonal ${\ell }_{\infty }$-decomposition, that is, it is not of the form ${\ell }_{\infty }\left(X\right)$ for any Banach space X. The main local result is that it is consistent that ${\ell }_{\infty }\left(c₀\left(\right)\right)$ does not embed isomorphically into ${\ell }_{\infty }/c₀$, where is the cardinality of the continuum,...

Let $\mathscr{H}$ be a complex Hilbert space, $A$ a positive operator with closed range in $ℬ\left(\mathscr{H}\right)$ and ${ℬ}_A\left(\mathscr{H}\right)$ the sub-algebra of $ℬ\left(\mathscr{H}\right)$ of all $A$-self-adjoint operators. Assume $\phi :{ℬ}_A\left(\mathscr{H}\right)$ onto itself is a linear continuous map. This paper shows that if $\phi$ preserves $A$-unitary operators such that $\phi \left(I\right)=P$ then $\psi$ defined by $\psi \left(T\right)=P\phi \left(PT\right)$ is a homomorphism or an anti-homomorphism and $\psi \left(T^{♯}\right)=\psi {\left(T\right)}^{♯}$ for all $T\in {ℬ}_A\left(\mathscr{H}\right)$, where $P=A^{+}A$ and $A^{+}$ is the Moore-Penrose inverse of $A$. A similar result is also true if $\phi$ preserves $A$-quasi-unitary operators in both directions such that there...

Denote by $C$ the commutator $AB-BA$ of two bounded operators $A$ and $B$ acting on a locally convex topological vector space. If $AC-CA=0$, we show that $C$ is a quasinilpotent operator and we prove that if $AC-CA$ is a compact operator, then $C$ is a Riesz operator.

In this paper, we define the direct sum ${\left({⨁}_{i=1}^n{X}_i\right)}_{ce{s}_p}$ of Banach spaces X₁,X₂,..., and Xₙ and consider it equipped with the Cesàro p-norm when 1 ≤ p < ∞. We show that ${\left({⨁}_{i=1}^n{X}_i\right)}_{ce{s}_p}$ has the H-property if and only if each $X_i$ has the H-property, and ${\left({⨁}_{i=1}^n{X}_i\right)}_{ce{s}_p}$ has the Schur property if and only if each $X_i$ has the Schur property. Moreover, we also show that ${\left({⨁}_{i=1}^n{X}_i\right)}_{ce{s}_p}$ is rotund if and only if each $X_i$ is rotund.

We give existence theorems for weak and strong solutions with trichotomy of the nonlinear differential equation $$\dot{x}\left(t\right)=ℒ\left(t\right)x\left(t\right)+f(t,x\left(t\right)),t\in ℝ\left(\mathrm{P}\right)$$ where $\left\{ℒ\right(t):t\in ℝ\}$ is a family of linear operators from a Banach space $E$ into itself and $f:ℝ\times E\to E$. By $L\left(E\right)$ we denote the space of linear operators from $E$ into itself. Furthermore, for $a<b$ and $d>0$, we let $C\left(\right[-d,0],E)$ be the Banach space of continuous functions from $[-d,0]$ into $E$ and $f^d:[a,b]\times C([-d,0],E)\to E$. Let $\widehat{ℒ}:[a,b]\to L\left(E\right)$ be a strongly measurable and Bochner integrable operator on $[a,b]$ and for $t\in [a,b]$ define ${\tau }_{t}x\left(s\right)=x(t+s)$ for each $s\in [-d,0]$. We prove that, under certain...

The Cauchy problem for the system of linear generalized ordinary differential equations in the J. Kurzweil sense $\mathrm{d}x\left(t\right)=\mathrm{d}{A}_0\left(t\right)·x\left(t\right)+\mathrm{d}{f}_0\left(t\right)$, $x\left(t_0\right)=c_0$ $(t\in I)$ with a unique solution $x_0$ is considered. Necessary and sufficient conditions are obtained for a sequence of the Cauchy problems $\mathrm{d}x\left(t\right)=\mathrm{d}{A}_k\left(t\right)·x\left(t\right)+\mathrm{d}{f}_k\left(t\right)$, $x\left(t_k\right)=c_k$ $(k=1,2,\cdots )$ to have a unique solution $x_k$ for any sufficiently large $k$ such that $x_k\left(t\right)\to x_0\left(t\right)$ uniformly on $I$. Presented results are analogous to the sufficient conditions due to Z. Opial for linear ordinary differential systems....

We take another approach to the well known theorem of Korovkin, in the following situation: X, Y are compact Hausdorff spaces, M is a unital subspace of the Banach space C(X) (respectively, $C_{ℝ}\left(X\right)$) of all complex-valued (resp., real-valued) continuous functions on X, S ⊂ M a complex (resp., real) function space on X, ϕₙ a sequence of unital linear contractions from M into C(Y) (resp., $C_{ℝ}\left(Y\right)$), and ${\varphi }_{\infty }$ a linear isometry from M into C(Y) (resp., $C_{ℝ}\left(Y\right)$). We show, under the assumption that ${\Pi }_N\subset {\Pi }_T$, where ${\Pi }_N$ is...