Related papers: Compact perturbations and consequent hereditarily …

The composition operator $C_{\phi_a}f=f\circ\phi_a$ on the Hardy-Hilbert space $H^2(\mathbb{D})$ with affine symbol $\phi_a(z)=az+1-a$ and $0<a<1$ has the property that the Invariant Subspace Problem for complex separable Hilbert spaces…

Finite rank perturbations of diagonalizable normal operators acting boundedly on infinite dimensional, separable, complex Hilbert spaces are considered from the standpoint of view of the existence of invariant subspaces. In particular, if…

For Schr\"odinger operator $H=-\Delta+ V({\mathbf x})\cdot$, acting in the space $L_2(\mathbb R^d)\,(d\ge 3)$, necessary and sufficient conditions for semi-boundedness and discreteness of its spectrum.are obtained without assumption that…

In this paper we consider a stronger property than the Bishop-Phelps-Bollob\'{a}s property for various classes of operators on a complex Hilbert space. The Bishop-Phelps-Bollob\'as {\it point} property for some class $\mathcal{A} \subset…

Let (A_0,A_1) and (B_0,B_1) be Banach couples such that A_0 is contained in A_1 and (B_0,B_1) satisfies Arne Persson's approximation condition (H). Let T:A_1 --> B_1 be a possibly nonlinear Lipschitz mapping which also maps A_0 into B_0 and…

We establish a spectral duality for certain unbounded operators in Hilbert space. The class of operators includes discrete graph Laplacians arising from infinite weighted graphs. The problem in this context is to establish a practical…

We observe that local embedding problems for certain Hardy and Bergman spaces of Dirichlet series are equivalent to boundedness of a class of composition operators. Following this, we perform a careful study of such composition operators…

Let $U_2({\cal H})$ be the Banach-Lie group of unitary operators in the Hilbert space ${\cal H}$ which are Hilbert-Schmidt perturbations of the identity 1. In this paper we study the geometry of the unitary orbit $$\{upu^*: u\in U_2({\cal…

The paper studies properties of acoustic operators in bounded Lipschitz domains $\Omega$ with m-dissipative generalized impedance boundary conditions. We prove that such acoustic operators have a compact resolvent if and only if the…

Suppose $\alpha$ is an orientation preserving diffeomorphism (shift) of $\mR_+=(0,\infty)$ onto itself with the only fixed points $0$ and $\infty$. We establish sufficient conditions for the Fredholmness of the singular integral operator \[…

Let $T$ be a self-adjoint operator in a Hilbert space $H$ with domain $\mathcal D(T)$. Assume that the spectrum of $T$ is confined in the union of disjoint intervals $\Delta_k =[\alpha_{2k-1},\alpha_{2k}]$, $k\in \mathbb{Z}$, and $$…

Let $H$ be a Schr\"odinger operator on $\R^n$. Under a polynomial decay condition for the kernel of its spectral operator, we show that the Besov spaces and Triebel-Lizorkin spaces associated with $H$ are well defined. We further give a…

Let $A$ be a, not necessarily closed, linear relation in a Hilbert space $\sH$ with a multivalued part $\mul A$. An operator $B$ in $\sH$ with $\ran B\perp\mul A^{**}$ is said to be an operator part of $A$ when $A=B \hplus (\{0\}\times \mul…

Let ${\mathcal B}(H)$ denote the Banach algebra of all bounded linear operators on a complex Hilbert space $H$ with $\dim H\geq 3$, and let $\mathcal A$ and $\mathcal B$ be subsets of ${\mathcal B}(H)$ which contain all rank one operators.…

In this article, we show that if $A$ is a maximal monotone operator on a Hilbert space $H$ with $0$ in the range $\textrm{Rg}(A)$ of $A$, then for every $0<s<1$, the Dirichlet problem associated with the Bessel-type equation $$…

If $A \colon D(A) \subset \mathcal{H} \to \mathcal{H}$ is an unbounded Fredholm operator of index $0$ on a Hilbert space $\mathcal{H}$ with a dense domain $D(A)$, then its spectrum is either discrete or the entire complex plane. This…

Consider a non-negative, self-adjoint, maximally subelliptic operator on a compact manifold. We show that the spectral multiplier is a singular integral operator under an appropriate Mihlin-H\"ormander type condition. We establish the…

Let $a$ and $b$ be elements of an ordered normed algebra $\mathcal A$ with unit $e$. Suppose that the element $a$ is positive and that for some $\varepsilon>0$ there exists an element $x\in \mathcal A$ with $\|x\|\leq \varepsilon$ such that…

Given a complex, separable Hilbert space $\mathcal{H}$, we characterize those operators for which $\| P T (I-P) \| = \| (I-P) T P \|$ for all orthogonal projections $P$ on $\mathcal{H}$. When $\mathcal{H}$ is finite-dimensional, we also…

Column and row operator spaces - which we denote by COL and ROW, respectively - over arbitrary Banach spaces were introduced by the first-named author; for Hilbert spaces, these definitions coincide with the usual ones. Given a locally…