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Register a new userA Topological Approach to Unitary Spectral Flow via Continuous Enumeration of Eigenvalues
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It is a well-known result of T.,Kato that given a continuous path of square matrices of a fixed dimension, the eigenvalues of the path can be chosen continuously. In this paper, we give an infinite-dimensional analogue of this result, which naturally arises in the context of the so-called unitary spectral flow. This provides a new approach to spectral flow, which seems to be missing from the literature. It is the purpose of the present paper to fill in this gap.
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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.
Suppose we want to find the eigenvalues and eigenvectors for the linear operator L, and suppose that we have solved this problem for some other nearby operator K. In this paper we show how to represent the eigenvalues and eigenvectors of L in terms of the corresponding properties of K.
This article constructs a surface whose Neumann-Poincare (NP) integral operator has infinitely many eigenvalues embedded in its essential spectrum. The surface is a sphere perturbed by smoothly attaching a conical singularity, which imparts essential spectrum. Rotational symmetry allows a decomposition of the operator into Fourier components. Eigenvalues of infinitely many Fourier components are constructed so that they lie within the essential spectrum of other Fourier components and thus within the essential spectrum of the full NP operator. The proof requires the perturbation to be sufficiently small, with controlled curvature, and the conical singularity to be sufficiently flat.
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.
The main object of the paper is a symmetric system $J y-B(t)y=lD(t) y$ defined on an interval $cI=[a,b) $ with the regular endpoint $a$. Let $f(cd,l)$ be a matrix solution of this system of an arbitrary dimension and let $(Vf)(s)=intlimits_cI f^*(t,s)D(t)f(t),dt$ be the Fourier transform of the function $f(cd)in L_D^2(cI)$. We define a pseudospectral function of the system as a matrix-valued distribution function $s(cd)$ of the dimension $n_s$ such that $V$ is a partial isometry from $L_D^2(cI)$ to $L^2(s;bC^{n_s})$ with the minimally possible kernel. Moreover, we find the minimally possible value of $n_s$ and parameterize all spectral and pseudospectral functions of every possible dimensions $n_s$ by means of a Nevanlinna boundary parameter. The obtained results develop the results by Arov and Dym; A.~Sakhnovich, L.~Sakhnovich and Roitberg; Langer and Textorius.
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