Some remarks on spectra of nuclear operators

It was shown by M. I. Zelikin (2007) that the spectrum of a nuclear operator in a separable Hilbert space is central-symmetric iff the spectral traces of all odd powers of the operator equal zero. The criterium can not be extended to the case of general Banach spaces: It follows from Grothendieck-Enflo results that there exists a nuclear operator $U$ in the space $l_1$ with the property that $\operatorname{trace}\, U=1$ and $U^2=0.$ B. Mityagin (2016) has generalized Zelikin's criterium to the case of compact operators (in Banach spaces) some of which powers are nuclear. We give sharp generalizations of Zelikin's theorem (to the cases of subspaces of quotients of $L_p$-spaces) and of Mityagin's result (for the case where the operators are not necessarily compact).Comment: 10 p., accepted for publication in Open Mathematic

A topos perspective on the Kochen-Specker theorem: I. Quantum States as Generalized Valuations

The Kochen-Specker theorem asserts the impossibility of assigning values to quantum quantities in a way that preserves functional relations between them. We construct a new type of valuation which is defined on all operators, and which respects an appropriate version of the functional composition principle. The truth-values assigned to propositions are (i) contextual; and (ii) multi-valued, where the space of contexts and the multi-valued logic for each context come naturally from the topos theory of presheaves. The first step in our theory is to demonstrate that the Kochen-Specker theorem is equivalent to the statement that a certain presheaf defined on the category of self-adjoint operators has no global elements. We then show how the use of ideas drawn from the theory of presheaves leads to the definition of a generalized valuation in quantum theory whose values are sieves of operators. In particular, we show how each quantum state leads to such a generalized valuation.Comment: Clarification of situation for situation for operators with continuous spectr

Several remarks on Pascal automorphism and infinite ergodic theory

We interpret the Pascal-adic transformation as a generalized induced automorphism (over odometer) and formulate the $\sigma$-finite analog of odometer which is also known as "Hajian-Kakutani transformation" (former "Ohio state example"). We shortly suggest a sketch of the theory of random walks on the groups on the base of $\sigma$-finite ergodic theory.Comment: 14 pp,Ref.1

Spectral measures associated with the factorization of the Lebesgue measure on a set via convolution

Let $Q$ be a fundamental domain of some full-rank lattice in ${\Bbb R}^d$ and let $\mu$ and $\nu$ be two positive Borel measures on ${\Bbb R}^d$ such that the convolution $\mu\ast\nu$ is a multiple of $\chi_Q$. We consider the problem as to whether or not both measures must be spectral (i.e. each of their respective associated $L^2$ space admits an orthogonal basis of exponentials) and we show that this is the case when $Q = [0,1]^d$. This theorem yields a large class of examples of spectral measures which are either absolutely continuous, singularly continuous or purely discrete spectral measures. In addition, we propose a generalized Fuglede's conjecture for spectral measures on ${\Bbb R}^1$ and we show that it implies the classical Fuglede's conjecture on ${\Bbb R}^1$

Homotopy-theoretically enriched categories of noncommutative motives

Waldhausen's $K$-theory of the sphere spectrum (closely related to the algebraic $K$-theory of the integers) is a naturally augmented $S^0$-algebra, and so has a Koszul dual. Classic work of Deligne and Goncharov implies an identification of the rationalization of this (covariant) dual with the Hopf algebra of functions on the motivic group for their category of mixed Tate motives over $\Z$. This paper argues that the rationalizations of categories of non-commutative motives defined recently by Blumberg, Gepner, and Tabuada consequently have natural enrichments, with morphism objects in the derived category of mixed Tate motives over $\Z$. We suggest that homotopic descent theory lifts this structure to define a category of motives defined not over $\Z$ but over the sphere ring-spectrum $S^0$.Comment: An attempt at a more readable version. Some reshuffling, a few new references, small notational changes. Thanks to many for comments about foolish blunders and obscuritie