Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userSpectral averaging for trace compatible operators
156
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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.
rate research
Read More
For a commuting $d$- tuple of operators $boldsymbol T$ defined on a complex separable Hilbert space $mathcal H$, let $big [ !!big [ boldsymbol T^*, boldsymbol T big ]!!big ]$ be the $dtimes d$ block operator $big (!!big (big [ T_j^* , T_ibig ]big )!!big )$ of the commutators $[T^*_j , T_i] := T^*_j T_i - T_iT_j^*$. We define the determinant of $big [ !!big [ boldsymbol T^*, boldsymbol T big ]!!big ]$ by symmetrizing the products in the Laplace formula for the determinant of a scalar matrix. We prove that the determinant of $big [ !!big [ boldsymbol T^*, boldsymbol T big ]!!big ]$ equals the generalized commutator of the $2d$ - tuple of operators, $(T_1,T_1^*, ldots, T_d,T_d^*)$ introduced earlier by Helton and Howe. We then apply the Amitsur-Levitzki theorem to conclude that for any commuting $d$ - tuple of $d$ - normal operators, the determinant of $big [ !!big [ boldsymbol T^*, boldsymbol T big ]!!big ]$ must be $0$. We show that if the $d$- tuple $boldsymbol T$ is cyclic, the determinant of $big [ !!big [ boldsymbol T^*, boldsymbol T big ]!!big ]$ is non-negative and the compression of a fixed set of words in $T_j^* $ and $T_i$ -- to a nested sequence of finite dimensional subspaces increasing to $mathcal H$ -- does not grow very rapidly, then the trace of the determinant of the operator $big [!! big [ boldsymbol T^* , boldsymbol Tbig ] !!big ]$ is finite. Moreover, an upper bound for this trace is given. This upper bound is shown to be sharp for a class of commuting $d$ - tuples. We make a conjecture of what might be a sharp bound in much greater generality and verify it in many examples.
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.
This paper extends Kreins spectral shift function theory to the setting of semifinite spectral triples. We define the spectral shift function under these hypotheses via Birman-Solomyak spectral averaging formula and show that it computes spectral flow.
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.
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.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
