In this paper, we introduce subspace-frequently hypercyclic operators. We show that these operators are subspace-hypercyclic and there are subspace-hypercyclic  operators that are not subspace-frequently hypercyclic. There is a criterion like to subspace-hypercyclicity criterion that implies subspace-frequent hypercyclicity and if an operator $T$ satisfies this criterion, then $Toplus T$ is sub...

In this paper we explore a family of strong completeness properties in GO-spaces defined on sets of real numbers with the usual linear ordering. We show that if τ is any GO-topology on the real line R, then (R, τ) is subcompact, and so is any Gδ-subspace of (R, τ). We also show that if (X, τ) is a subcompact GO-space constructed on a subset X ⊆ R, then X is a Gδ-subset of any space (R, σ) where...

• Random Sampling: X are iid uniform from unit sphere in S`. • Random Subspace: S` are spanned by d iid uniform vectors in R. GRAPH CONNECTIVITY What about the second design objective? • Nashihatkon & Hartley nailed that connectivity is NOT a generic property for SSC in general when d ≥ 4. • What about for LRR? Assume random sampling and random subspace, we have: Proposition 1 Under independent...

In this article‎, ‎we have characterized ideals in $C(X)$ in which‎ ‎every ideal is also an ideal (a $z$-ideal) of $C(X)$‎. ‎Motivated by‎ ‎this characterization‎, ‎we observe that $C_infty(X)$ is a regular‎ ‎ring if and only if every open locally compact $sigma$-compact‎ ‎subset of $X$ is finite‎. ‎Concerning prime ideals‎, ‎it is shown that‎ ‎the sum of every two prime (semiprime) ideals of e...

Let E be an infinite dimensional complex Banach space. We prove the existence of an infinitely generated algebra, an infinite dimensional closed subspace and a dense subspace of entire functions on E whose non-zero elements are functions of unbounded type. We also show that the τδ topology on the space of all holomorphic functions cannot be obtained as a countable inductive limit of Fréchet spa...

An important open problem in operator theory is the invariant subspace problem. Since the problem is solved for all finite dimensional complex vector spaces of dimension at least 2, H denotes a separable Hilbert space whose dimension is infinite. It is enough to think for a contraction T , that is, ‖T‖ ≤ 1 on H . Thus, T ( 6= 0) denotes a contraction in L(H). Let λ always denote an element of t...

A subset Y of a space X is Gδ-dense if it intersects every nonempty Gδ-set. The Gδ-closure of Y in X is the largest subspace of X in which Y is Gδ-dense. The space X has a regular Gδ-diagonal if the diagonal of X is the intersection of countably many regular-closed subsets of X ×X. Consider now these results: (a) [N. Noble, 1972] every Gδ-dense subspace in a product of separable metric spaces i...

A space $Y$ is called an {em extension} of a space $X$, if $Y$ contains $X$ as a dense subspace. Two extensions of $X$ are said to be {em equivalent}, if there is a homeomorphism between them which fixes $X$ point-wise. For two (equivalence classes of) extensions $Y$ and $Y'$ of $X$ let $Yleq Y'$, if there is a continuous function of $Y'$ into $Y$ which fixes $X$ point-wise. An extension $Y$ ...

We show that it is consistent with ZF that there is a dense-in-itself compact metric space (X, d) which has the countable chain condition (ccc), but X is neither separable nor second countable. It is also shown that X has an open dense subspace which is not paracompact and that in ZF the Principle of Dependent Choice, DC, does not imply the disjoint union of metrizable spaces is normal .