Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userIterative actions of normal operators
306
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Let $A$ be a normal operator in a Hilbert space $mathcal{H}$, and let $mathcal{G} subset mathcal{H}$ be a countable set of vectors. We investigate the relations between $A$, $mathcal{G}$ , and $L$ that makes the system of iterations ${A^ng: gin mathcal{G},;0leq n< L(g)}$ complete, Bessel, a basis, or a frame for $mathcal{H}$. The problem is motivated by the dynamical sampling problem and is connected to several topics in functional analysis, including, frame theory and spectral theory. It also has relations to topics in applied harmonic analysis including, wavelet theory and time-frequency analysis.
rate research
Read More
In this note the notions of trace compatible operators and infinitesimal spectral flow are introduced. We define the spectral shift function as the integral of infinitesimal spectral flow. It is proved that the spectral shift function thus defined is absolutely continuous and Kreins formula is established. Some examples of trace compatible affine spaces of operators are given.
Let $H_V=-Delta +V$ be a Schrodinger operator on an arbitrary open set $Omega$ of $mathbb R^d$, where $d geq 3$, and $Delta$ is the Dirichlet Laplacian and the potential $V$ belongs to the Kato class on $Omega$. The purpose of this paper is to show $L^p$-boundedness of an operator $varphi(H_V)$ for any rapidly decreasing function $varphi$ on $mathbb R$. $varphi(H_V)$ is defined by the spectral theorem. As a by-product, $L^p$-$L^q$-estimates for $varphi(H_V)$ are also obtained.
We study generalized polar decompositions of densely defined, closed linear operators in Hilbert spaces and provide some applications to relatively (form) bounded and relatively (form) compact perturbations of self-adjoint, normal, and m-sectorial operators.
We describe here the higher rank numerical range, as defined by Choi, Kribs and Zyczkowski, of a normal operator on an infinite dimensional Hilbert space in terms of its spectral measure. This generalizes a result of Avendano for self-adjoint operators. An analogous description of the numerical range of a normal operator by Durszt is derived for the higher rank numerical range as an immediate consequence. It has several interesting applications. We show using Durszts example that there exists a normal contraction $T$ for which the intersection of the higher rank numerical ranges of all unitary dilations of $T$ contains the higher rank numerical range of $T$ as a proper subset. Finally, we strengthen and generalize a result of Wu by providing a necessary and sufficient condition for the higher rank numerical range of a normal contraction being equal to the intersection of the higher rank numerical ranges of all possible unitary dilations of it.
Let $G$ be a compact Lie group acting smoothly on a smooth, compact manifold $M$, let $P in psi^m(M; E_0, E_1)$ be a $G$--invariant, classical pseudodifferential operator acting between sections of two vector bundles $E_i to M$, $i = 0,1$, and let $alpha$ be an irreducible representation of the group $G$. Then $P$ induces a map $pi_alpha(P) : H^s(M; E_0)_alpha to H^{s-m}(M; E_1)_alpha$ between the $alpha$-isotypical components. We prove that the map $pi_alpha(P)$ is Fredholm if, and only if, $P$ is {em transversally $alpha$-elliptic}, a condition defined in terms of the principal symbol of $P$ and the action of $G$ on the vector bundles $E_i$.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
