Eigenvalue Distribution of Compact Operators

Eigenvalue Distribution of Compact Operators
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Let B X be the algebra of bounded operators on a complex Banach space X. Viewing B X as an algebra over R, we study the structure of those irreducible subalgebras which contain nonzero compact operators. In particular, irreducible algebras of trace-class operators with real trace are characterized. This yields an extension of Brauer-type results on matrices to operators in infinite dimensions, answering the question: is an irreducible semigroup of compact operators with real spectra realizable, i.

Indeed there is often repulsion between the eigenvalues. Ciuperca and Z. We characterize nuclearity using a weak uniqueness theorem, where the invariant is the Cuntz semigroup. The spectrum of a Haar unitary random matrix is a determinantal point process on the unit circle, while the spectra of its submatrices are determinantal processes in the unit disc. We derive formulas for the moments of the norm of the trace of such submatrices.

This leads to connections between random matrix theory, the enumeration of lattice walks confined to Weyl chambers, and Toeplitz determinants of Bessel functions. We define a version of generalized Bunce-Deddens algebras as certain crossed products by discrete amenable residually finite groups, and we describe some of their properties.

It has been an open question for some time whether there is a maximal abelian subalgebra in a UHF algebra which is isomorphic to C X for a connected space X.

We present, by examples, an index theory appropriate to algebras without trace. As a corollary we obtain a noncommutative geometry interpretation for Araki's notion of relative entropy in these examples. This is joint work with Alan Carey and Adam Rennie. On local-to-global properties of semigroups of operators Heydar Radjavi , University of Waterloo. Let S be a semigroup of operators with no common, nontrivial, invariant subspaces. What can be said about S if there exists a nonzero linear functional f on all operators whose restriction to S takes only real values or, more restrictively, only positive values?

Even when confined to compact operators, not all the questions have trivial answers. Elliott and Luis Santiago. All such traces form a noncancellative topological cone. I will present some basic properties of this cone that are relevant to classification questions. Uniform continuity over locally compact quantum groups Volker Runde, University of Alberta.

This definition covers both the usual left uniformly continuous functions on a locally compact group and Granirer's uniformly continuous functionals on the Fourier algebra. We use this to partially answer an open problem by Bedos-Tuset: if G is co-amenable, then the existence of a left invariant mean on M C0 G is sufficient for G to be amenable. I will report on some ideas to measure the distance between operators respectively their distributions in terms of resolvents.

Estimating distances to semicircular elements will feature prominently. One consequence of this study has been the discovery that isomorphism classes of Hilbert modules over such algebras can, in the setting of stable rank one, be classified by the Cuntz semigroup. When the natural partial order on the Cuntz semigroup is determined by states, this classification can be realised in terms of K-theory and traces. We show that there is no reasonable way to classify von Neumann factors on a separable Hilbert space by an assignment of invariants which are "countable structures", e.

We also show that the isomorphism relation of factors is complete analytic. In particular, it is not Borel. In this talk, I will discuss a number of results concerning proper actions and their generalized fixed point algebras.

Optics Express

These results are most efficiently stated by showing that our constructions are functorial. This is work in progress with Astrid an Huef and Iain Raeburn. U is a symmetric space, where the transitively operating Lie group consists of all biholomorphic automorphisms of U. This talk has two objectives, an explicit calculation, for all vector fields on U, of the invariant connection and, using results previously obtained with D.

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In general relativity, such a structure is responsible for the concept of causality. Both questions are related since the invariance of the cone fields is intimately connected to the behavior of parallel transport along geodesics. Our results actually cover a much broader class of infinite dimensional symmetric spaces. We show that the symmetric space we are dealing with can be defined in terms of the automorphism group of this structure. For the underlying invariant operator space Finsler structure, the analogous result holds.

A short survey of Burnside type theorems by Bamdad R. A version of a celebrated theorem of Burnside asserts that Mn F is the only irreducible subalgebra of Mn F provided that the field F is algebraically closed. In other words, Burnside's theorem characterizes all irreducible subalgebras of Mn F whenever F is algebraically closed. In view of this, by a Burnside type theorem for certain irreducible subalgebras of matrices, we mean a result which characterizes such subalgebras. In this talk, we present a simple proof of Burnside's theorem. Freund, G. Golub, and N.

Freund and N. Method Appl.

Operators, Eigen values and eigen functions -- Lecture 13 -- Quantum Mechanics

Christ and H. Saad and M. Johnson and R. Ashcroft and N. Mermin, Solid State Physics, 1st ed. SaundersCollege, , Ch. Brongersma and P. Kik, eds. Springer, , pp. Inan and R. Wiley, Palik, ed. Lide, ed. CRC, At the final stage of our work, we were made aware of a related work by M.

Kordy, E. Cherkaev, and P. Citing articles from OSA journals and other participating publishers are listed here. Alert me when this article is cited.

[] Estimating the number of eigenvalues of linear operators on Banach spaces

Click here to see a list of articles that cite this paper. The height of the column on each interval represents the number of the eigenvalues in the interval.

Three systems of linear equations discretized from Eq. Notice that all the columns almost vanish only after four iteration steps. In each figure, a solid line represents a polynomial; an open dot on the horizontal axis indicates the smallest root; solid dots indicate the other roots; dashed lines show the slopes of the polynomial at the roots. The three polynomials have the same roots except for their smallest roots: the smallest root in a becomes smaller positive and negative roots in b and c , respectively.

The vertical scale of the plot is magnified as the iteration proceeds. The figures in the first row describe the three systems. The directions of wave propagation are shown by red arrows, beside which the vacuum wavelengths used are indicated. For all three systems, the waves are excited by electric current sources J strictly inside the simulation domain.

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The plots in the second row show the convergence behavior of QMR. Depending on the sign of s , T 0 has very different eigenvalue distributions in terms of the multiplicity of the eigenvalue 0 and the definiteness of T 0. Table 2 Specification of the finite-difference grids used for the three systems in Fig. Slot uses a nonuniform grid with smoothly varying grid cell size.