Free biholomorphic functions and operator model theory,II

This is a continuation of the paper entitled "Free biholomorphic functions and operator model theory", in our attempt to transfer the free analogue of Nagy-Foias theory from the unit ball [B(\cH)^n]_1 to other noncommutative domains and varieties in B(\cH)^n, using appropriate maps. The present paper treats the completely non-coisometric case and the case of noncommutative varieties.Comment: 37 page

Joint similarity to operators in noncommutative varieties

In this paper we solve several problems concerning joint similarity to n-tuples of operators in noncommutative varieties in [B(\cH)^n]_1 associated with positive regular free holomorphic functions in $n$ noncommuting variables and with sets of noncommutative polynomials in $n$ indeterminates, where B(\cH) is the algebra of all bounded linear operators on a Hilbert space \cH. In particular, if $f=X_1+...+X_n$ and \cP=\{0\}, the elements of the corresponding variety can be seen as noncommutative multivariable analogues of Agler's $m$-hypercontractions.Comment: 35 pages, corrected typo

A Jost-Pais-type reduction of Fredholm determinants and some applications

We study the analog of semi-separable integral kernels in \cH of the type {equation*} K(x,x')={cases} F_1(x)G_1(x'), & a<x'< x< b, \\ F_2(x)G_2(x'), & a<x<x'<b, {cases} {equation*} where $-\infty\leq a<b\leq \infty$, and for a.e.\ $x \in (a,b)$, F_j (x) \in \cB_2(\cH_j,\cH) and G_j(x) \in \cB_2(\cH,\cH_j) such that $F_j(\cdot)$ and $G_j(\cdot)$ are uniformly measurable, and {equation*} \|F_j(\cdot)\|_{\cB_2(\cH_j,\cH)} \in L^2((a,b)), \; \|G_j (\cdot)\|_{\cB_2(\cH,\cH_j)} \in L^2((a,b)), \quad j=1,2, {equation*} with \cH and \cH_j, $j=1,2$, complex, separable Hilbert spaces. Assuming that $K(\cdot, \cdot)$ generates a trace class operator \bsK in L^2((a,b);\cH), we derive the analog of the Jost-Pais reduction theory that succeeds in proving that the Fredholm determinant {\det}_{L^2((a,b);\cH)}(\bsI - \alpha \bsK), \alpha \in \bbC, naturally reduces to appropriate Fredholm determinants in the Hilbert spaces \cH (and \cH_1 \oplus \cH_2). Explicit applications of this reduction theory are made to Schr\"odinger operators with suitable bounded operator-valued potentials. In addition, we provide an alternative approach to a fundamental trace formula first established by Pushnitski which leads to a Fredholm index computation of a certain model operator.Comment: 50 pages; some typos remove

Composition operators on noncommutative Hardy spaces

In this paper we initiate the study of composition operators on the noncommutative Hardy space $H^2_{\bf ball}$. Several classical results about composition operators (boundedness, norm estimates, spectral properties, compactness, similarity) have free analogues in our noncommutative multivariable setting. The most prominent feature of this paper is the interaction between the noncommutative analytic function theory in the unit ball of B(\cH)^n, the operator algebras generated by the left creation operators on the full Fock space with $n$ generators, and the classical complex function theory in the unit ball of \CC^n.Comment: 39 page

Design of Electronic Commerce

Tato bakalářská práce obsahuje návrh elektronického obchodu pro firmu B&Ch music s.r.o. Obsahuje analýzu elektronického obchodování, analýzu firmy a potřeb firmy, a následně návrh elektronického obchodu.This bachelor thesis includes design of electronic commerce for B&Ch music Ltd company. It includes analysis of E-commerce, analysis of B&Ch Music Ltd and thier needs, and finally design of E-commerce.