Narrow Results

In this paper, we introduce the notion of ∗-$K$-operator frame as a generalization of the notion of $K$-operator frame and we study the corresponding frame operator. It is completed by a result on the frame operator of the tensor product of two frame operators.

In this paper, we consider oblique dual frames and particularly, we obtain some of their characterizations. Among other things, special attention is devoted to the study of the effect of perturbations of frame sequences on their oblique duals. In particular, as a surprising result, we show that a frame sequence is uniquely determined by the set of its oblique dual.

Let ${𝒞}$ be a small well-powered category which has pullbacks. The aim of this paper is to study and characterize two (weak) topologies on presheaf category $\widehat{{𝒞}}$, one of these is constructed by means of a subfunctor of the Yoneda functor which corresponds to an ideal of ${𝒞}$, called (weak) ideal topology, and another constructed by a dominion. Naturally, these (weak) topologies of $\widehat{{𝒞}}$ introduce two (weak) Grothendieck topologies on ${𝒞}$. To find and study their relations, using the induced presheaf $M$ given by the fixed dominion ${ℳ}$ on ${𝒞}$ we construct an action of $M$ on the subobject classifier $\mathrm{\Omega }$ of $\widehat{{𝒞}}$. Then, we investigate which one of (weak) Grothendieck topologies corresponds one-to-one to the ideals of ${𝒞}$ and which one is a sub $M$-set of $\mathrm{\Omega }$. Moreover, among other things, we give some conditions under which the Grothendieck topologies on ${𝒞}$ associated to the double negation topology $\neg \neg$ and two topologies $j_M$ and $j^{{ℐ}}$ (mentioned above) are sub $M$-sets of $\mathrm{\Omega }$.