Related papers: Remarks on strictly singular operators

Let $L^{m,p}(\R^n)$ denote the Sobolev space of functions whose $m$-th derivatives lie in $L^p(\R^n)$, and assume that $p>n$. For $E \subset \R^n$, denote by $L^{m,p}(E)$ the space of restrictions to $E$ of functions $F \in L^{m,p}(\R^n)$.…

The purpose of this survey article is a comprehensive study of operator Lipschitz functions. A continuous function $f$ on the real line ${\Bbb R}$ is called operator Lipschitz if $\|f(A)-f(B)\|\le{\rm const}\|A-B\|$ for arbitrary…

Valadier and Hensgen proved independently that the restriction of functional $\phi(x)=\int_{0}^{1}x(t)dt,\,\,x\in L^{\infty}([0,1])$ on the space of continuous functions $C([0,1])$ admits a singular extension back to the whole space…

In this note, we define a bounded variant on the Hilbert projective metric on an infinite dimensional space $E$ and study the contraction properties of the projective maps associated with positive linear operators on $E$. More precisely, we…

The notion of singular reduction operators, i.e., of singular operators of nonclassical (conditional) symmetry, of partial differential equations in two independent variables is introduced. All possible reductions of these equations to…

We show that (for the weak operator topology) the set of unitary operators on a separable infinite-dimensional Hilbert space is residual in the set of all contractions. The analogous result holds for isometries and the strong operator…

Let $X$ be a complex topological vector space with dim$(X)>1$ and $\mathcal{B}(X)$ the space of all continuous linear operators on $X$. In this paper, we extend the concept of supercyclicity of a single operators and strongly continuous…

Given a dissipative operator $A$ on a complex Hilbert space $\mathcal{H}$ such that the quadratic form $f\mapsto \mbox{Im}\langle f,Af\rangle$ is closable, we give a necessary and sufficient condition for an extension of $A$ to still be…

We show that whole-line Schr\"odinger operators with finitely many bound states have no embedded singular spectrum. In contradistinction, we show that embedded singular spectrum is possible even when the bound states approach the essential…

This note deals with the operator $T^*T$, where $T$ is a densely defined operator on a complex Hilbert space. We reprove a recent result of Z. Sebesty\'en and Zs. Tarcsay [13]: If $T^*T$ and $TT^*$ are self-adjoint, then $T$ is closed. In…

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.…

In [9] a question is raised: if a power bounded operator is quasisimilar to a singular unitary operator, is it similar to this unitary operator? For polynomially bounded operators, a positive answer to this question is known [1], [13]. In…

A bounded linear operator $T$ acting on a Banach space $\B$ is called weakly hypercyclic if there exists $x\in \B$ such that the orbit ${T^n x: n=0,1,...}$ is weakly dense in $\B$ and $T$ is called weakly supercyclic if there is $x\in \B$…

This paper concerns three classes of real-valued functions on intervals, operator monotone functions, operator convex functions, and strongly operator convex functions. Strongly operator convex functions were previously treated in [3] and…

Johnson and Zippin recently showed that if $X$ is a weak^*-closed subspace of $\ell_1$ and T:X-> C(K) is any bounded operator then T can extended to a bounded operator $\tilde T:\ell_1\to C(K).$ We give a converse result: if X is a subspace…

In this paper we consider the cone of all positive, bounded operators acting on an infinite dimensional, complex Hilbert space, and examine bijective maps that preserve absolute continuity in both directions. It turns out that these maps…

A space $X$ is said to be hereditarily indecomposable if no two (infinite dimensional) subspaces of $X$ are in a direct sum. In this paper, we show that if $X$ is a complex hereditarily indecomposable Banach space, then every operator from…

Let $L^{m,p}(\R^n)$ be the Sobolev space of functions with $m^{th}$ derivatives lying in $L^p(\R^n)$. Assume that $n< p < \infty$. For $E \subset \R^n$, let $L^{m,p}(E)$ denote the space of restrictions to $E$ of functions in…

We give a generalization of the Jordan canonical form theorem for a class of bounded linear operators on complex separable Hilbert spaces in terms of direct integrals. Precisely, we study the uniqueness of strongly irreducible…

For a linear operator $T$ in a Banach space let $\sigma_p(T)$ denote the point spectrum of $T$, $\sigma_{p[n]}(T)$ for finite $n > 0$ be the set of all $\lambda \in \sigma_p(T)$ such that $\dim \ker (T - \lambda) = n$ and let…