In this paper the hyponormal operators on Krein spaces are introduced. We state conditions for the hyponormality of bounded operators focusing, in particular, on those operators $T$ for which there exists a fundamental decomposition $𝕂={𝕂}^{+}\oplus {𝕂}^{-}$ of the Krein space $𝕂$ with ${𝕂}^{+}$ and ${𝕂}^{-}$ invariant under $T$.

Let $F$ be a finite field of characteristic $p$ and $K$ a field which contains a primitive $p$th root of unity and $\mathrm{char}K\ne p$. Suppose that a classical group $G$ acts on the $F$-vector space $V$. Then it can induce the actions on the vector space $V\oplus V$ and on the group algebra $K[V\oplus V]$, respectively. In this paper we determine the structure of $G$-invariant ideals of the group algebra $K[V\oplus V]$, and establish the relationship between the invariant ideals of $K\left[V\right]$ and the vector invariant ideals of $K[V\oplus V]$, if $G$ is a unitary group or orthogonal...

We study the space of p-compact operators, ${}_p$, using the theory of tensor norms and operator ideals. We prove that ${}_p$ is associated to $/d_p$, the left injective associate of the Chevet-Saphar tensor norm $d_p$ (which is equal to $g_{p^{\text{'}}}^{\text{'}}$). This allows us to relate the theory of p-summing operators to that of p-compact operators. Using the results known for the former class and appropriate hypotheses on E and F we prove that ${}_p(E;F)$ is equal to ${}_q(E;F)$ for a wide range of values of p and q, and show that our results...

We introduce the definition of $p$-limited completely continuous operators, $1\le p<\infty$. The question of whether a space of operators has the property that every $p$-limited subset is relative compact when the dual of the domain and the codomain have this property is studied using $p$-limited completely continuous evaluation operators.

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...

Let $L\left(H\right)$ denote the algebra of operators on a complex infinite dimensional Hilbert space $H$. For $A,B\in L\left(H\right)$, the generalized derivation ${\delta }_{A,B}$ and the elementary operator ${\Delta }_{A,B}$ are defined by ${\delta }_{A,B}\left(X\right)=AX-XB$ and ${\Delta }_{A,B}\left(X\right)=AXB-X$ for all $X\in L\left(H\right)$. In this paper, we exhibit pairs $(A,B)$ of operators such that the range-kernel orthogonality of ${\delta }_{A,B}$ holds for the usual operator norm. We generalize some recent results. We also establish some theorems on the orthogonality of the range and the kernel of ${\Delta }_{A,B}$ with respect to the wider class of unitarily invariant...

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 < ∞. ...

Let $R$ be a commutative ring with nonzero identity, let $ℐ\left(ℛ\right)$ be the set of all ideals of $R$ and $\delta :ℐ\left(ℛ\right)\to ℐ\left(ℛ\right)$ an expansion of ideals of $R$ defined by $I\mapsto \delta \left(I\right)$. We introduce the concept of $(\delta ,2)$-primary ideals in commutative rings. A proper ideal $I$ of $R$ is called a $(\delta ,2)$-primary ideal if whenever $a,b\in R$ and $ab\in I$, then $a^2\in I$ or $b^2\in \delta \left(I\right)$. Our purpose is to extend the concept of $2$-ideals to $(\delta ,2)$-primary ideals of commutative rings. Then we investigate the basic properties of $(\delta ,2)$-primary ideals and also discuss the relations among $(\delta ,2)$-primary, $\delta$-primary...

Let $f_i$ be polynomials in $n$ variables without a common zero. Hilbert’s Nullstellensatz says that there are polynomials $g_i$ such that $\sum g_i{f}_i=1$. The effective versions of this result bound the degrees of the $g_i$ in terms of the degrees of the $f_j$. The aim of this paper is to generalize this to the case when the $f_i$ are replaced by arbitrary ideals. Applications to the Bézout theorem, to Łojasiewicz–type inequalities and to deformation theory are also discussed.

A classification of weakly compact multiplication operators on $L\left(L_p\right),$1<p<$,isgiven.ThisanswersaquestionraisedbySaksmanandTylliin1992.Theclassificationinvolvestheconceptof$p$-strictlysingularoperators,andwealsoinvestigatethestructureofgeneral$p$-strictlysingularoperatorson$Lp$.Themainresultisthatifanoperator$T$on$Lp$,$1<p<2$,is$p$-strictlysingularand$T|X$isanisomorphismforsomesubspace$X$of$Lp$,then$X$embedsinto$Lr$forall$r<2$,but$X$neednotbeisomorphictoaHilbertspace.$It is also shown that if $T$ is convolution by a biased coin on $L_p$ of the Cantor group, $1\le p<2$, and $T_{|X}$ is an isomorphism for some reflexive subspace $X$ of $L_p$, then $X$ is isomorphic to a Hilbert space. The case $p=1$ answers a question asked by Rosenthal in 1976.

We consider real Banach spaces X for which the quotient algebra (X)/ℐn(X) is finite-dimensional, where ℐn(X) stands for the ideal of inessential operators on X. We show that these spaces admit a decomposition as a finite direct sum of indecomposable subspaces $X_i$ for which $\left(X_i\right)/ℐn\left(X_i\right)$ is isomorphic as a real algebra to either the real numbers ℝ, the complex numbers ℂ, or the quaternion numbers ℍ. Moreover, the set of subspaces $X_i$ can be divided into subsets in such a way that if $X_i$ and $X_j$ are in different...

A nonzero $R$-module $M$ is atomic if for each two nonzero elements $a,b$ in $M$, both cyclic submodules $Ra$ and $Rb$ have nonzero isomorphic submodules. In this article it is shown that for an infinite $P$-space $X$, the factor rings $C(X,\mathbb{Z})/C_F(X,\mathbb{Z})$ and $C_c\left(X\right)/C_F\left(X\right)$ have no atomic ideals. This fact generalizes a result published in paper by A. Mozaffarikhah, E. Momtahan, A. R. Olfati and S. Safaeeyan (2020), which says that for an infinite set $X$, the factor ring ${\mathbb{Z}}^X/{\mathbb{Z}}^{\left(X\right)}$ has no atomic ideal. Another result is that for each infinite...

Let $K$ be a number field defined by an irreducible polynomial $F\left(X\right)\in \mathbb{Z}\left[X\right]$ and ${\mathbb{Z}}_K$ its ring of integers. For every prime integer $p$, we give sufficient and necessary conditions on $F\left(X\right)$ that guarantee the existence of exactly $r$ prime ideals of ${\mathbb{Z}}_K$ lying above $p$, where $\overline{F}\left(X\right)$ factors into powers of $r$ monic irreducible polynomials in ${𝔽}_p\left[X\right]$. The given result presents a weaker condition than that given by S. K. Khanduja and M. Kumar (2010), which guarantees the existence of exactly $r$ prime ideals of ${\mathbb{Z}}_K$ lying above $p$....