Related papers: A simple proof of the closed graph theorem

We prove an abstract theorem on keeping the compactness property of a linear operator after interpolation in Banach spaces. No analytical presentation of operators, spaces and interpolation functor is required. We use only some little-known…

We analyse possibility to extend a quantum operation (sub-unital normal CP linear map on the algebra $B(H)$ of bounded operators on a separable Hilbert space $H$) to the space of all operators on $H$ relatively bounded w.r.t. a given…

We say that a metric graph is uniformly bounded if the degrees of all vertices are uniformly bounded and the lengths of edges are pinched between two positive constants; a metric space is approximable by a uniform graph if there is one…

The purpose of this paper is to generalize a very famous result on products of normal operators, due to I. Kaplansky. The context of generalization is that of bounded hyponormal and unbounded normal operators on complex separable Hilbert…

The Bandwidth theorem of B\"ottcher, Schacht and Taraz gives a condition on the minimum degree of an $n$-vertex graph $G$ that ensures $G$ contains every $r$-chromatic graph $H$ on $n$ vertices of bounded degree and of bandwidth $o(n)$,…

The primary purpose of this paper is to show the existence of normal square and nth roots of some classes of bounded operators on Hilbert spaces. Two interesting simple results hold. Namely, under simple conditions we show that if any…

Michael Handel proved in [7] the existence of a fixed point for an orientation preserving homeomorphism of the open unit disk that can be extended to the closed disk, provided that it has points whose orbits form an oriented cycle of links…

A cornerstone of extremal graph theory due to Erd\H{o}s and Stone states that the edge density which guarantees a fixed graph $F$ as subgraph also asymptotically guarantees a blow-up of $F$ as subgraph. It is natural to ask whether this…

A well-known theorem of Paulsen says that if $\mathcal{A}$ is a unital operator algebra and $\phi:\mathcal{A}\to B(\mathcal{H})$ is a unital completely bounded homomorphism, then $\phi$ is similar to a completely contractive map $\phi'$.…

The usual Laurent expansion of the analytic tensors on the complex plane is generalized to any closed and orientable Riemann surface represented as an affine algebraic curve. As an application, the operator formalism for the $b-c$ systems…

This note is devoted to two classical theorems: the open mapping theorem for analytic functions (OMT) and the fundamental theorem of algebra (FTA). We present a new proof of the first theorem, and then derive the second one by a simple…

The closure of a graph $G$ is the graph $G^*$ obtained from $G$ by repeatedly adding edges between pairs of non-adjacent vertices whose degree sum is at least $n$, where $n$ is the number of vertices of $G$. The well-known Closure Lemma…

The object of this paper is to prove a version of the Beurling-Helson-Lowdenslager invariant subspace theorem for operators on certain Banach spaces of functions on a multiply connected domain in the complex plane. The norms for these…

Our main result is a theorem saying that a bounded operator $A$ on a Hilbert space belongs to a certain set associated with its self-commutator $[A^*,A]$, provided that $A-zI$ can be approximated by invertible operators for all complex…

Let $X$ and $Y$ be Banach spaces, $A\,:\,X\rightarrow Y$ and $B,\,C\,:\,Y\rightarrow X$ be bounded linear operators. We prove that if $A(BA)^2=ABACA=ACABA=(AC)^2A,$ then $$\sigma_{*}(AC)\setminus\{0\}=\sigma_{*}(BA)\setminus\{0\}$$ where…

We find first structural background information about the reasons that for any C*-algebra $A$ and any two Hilbert $A$-modules $M \subseteq N$ with $M^\perp=\{0\}$, every bounded $A$-linear map $N \to A$ (or $N \to N)$ vanishing on $M$ might…

We show the following geometric generalization of a classical theorem of W.H. Gottschalk and G.A. Hedlund: a skew action induced by a cocycle of (affine) isometries of a Hilbert space over a minimal dynamics has a continuous invariant…

A $biased\ graph$ is a pair $(G,\mathcal{B})$, where $G$ is a graph and $\mathcal{B}$ is a collection of `balanced' circuits of $G$ such that no $\Theta$-subgraph of $G$ contains precisely two balanced circuits. We prove a Ramsey-type…

In this paper, we characterize all closed linear operators in a separable Hilbert space which are unitarily equivalent to an integral bi-Carleman operator in $L_2(R)$ with bounded and arbitrarily smooth kernel on $R^2$. In addition, we give…

We give necessary and sufficient conditions for a bounded operator defined between complex Hilbert spaces to be absolutely norm attaining. We discuss structure of such operators in the case of self-adjoint and normal operators separately.…