The curvature invariant for a class of homogeneous operators

For an operator T in the class B-n(), introduced by Cowen and Douglas, the simultaneous unitary equivalence class of the curvature and the covariant derivatives up to a certain order of the corresponding bundle E-T determine the unitary equivalence class of the operator T. In a subsequent paper, the authors ask if the simultaneous unitary equivalence class of the curvature and these covariant derivatives are necessary to determine the unitary equivalence class of the operator T is an element of B-n(). Here we show that some of the covariant derivatives are necessary. Our examples consist of homogeneous operators in B-n(). For homogeneous operators, the simultaneous unitary equivalence class of the curvature and all its covariant derivatives at any point w in the unit disc are determined from the simultaneous unitary equivalence class at 0. This shows that it is enough to calculate all the invariants and compare them at just one point, say 0. These calculations are then carried out in number of examples. One of our main results is that the curvature along with its covariant derivative of order (0, 1) at 0 determines the equivalence class of generic homogeneous Hermitian holomorphic vector bundles over the unit disc.

Wigner distributions for finite-state systems without redundant phase-point operators

We set up Wigner distributions for N-state quantum systems following a Dirac-inspired approach. In contrast to much of the work in this study, requiring a 2N x 2N phase space, particularly when N is even, our approach is uniformly based on an N x N phase-space grid and thereby avoids the necessity of having to invoke a `quadrupled' phase space and hence the attendant redundance. Both N odd and even cases are analysed in detail and it is found that there are striking differences between the two. While the N odd case permits full implementation of the marginal property, the even case does so only in a restricted sense. This has the consequence that in the even case one is led to several equally good definitions of the Wigner distributions as opposed to the odd case where the choice turns out to be unique.

Only Connect?: Aspects of Intertextuality in Zadie Smith's On Beauty

Tutkielma käsittelee intertekstuaalisuuden eri muotoja Zadie Smithin romaanissa On Beauty (suom. Kauneudesta). Tutkimuksen tarkoitus on osoittaa kuinka oleellisesti intertekstuaalisuuden teoria on vaikuttanut kirjallisuustieteen metodeihin ja postmoderniin kirjallisuuskäsitykseen, sekä käsitellä sen soveltuvuutta nykykirjallisuuden tutkimiseen analysoimalla teorian sisäistä monimuotoisuutta ja ristiriitoja. Tutkimusmateriaalina käytetään Smithin romaanin lisäksi E. M. Forsterin romaania Howards End (suom. Talo jalavan varjossa), johon On Beauty tietoisesti viittaa. Teoreettisena viitekehyksenä tutkielmassa toimii Gérard Genetten teoksessa Palimpsests sekä Roland Barthesin esseessä Tekijän kuolema esille tuodut kirjallisuusteoreettiset käsitykset. Valittu metodologia antaa mahdollisuuden hahmottaa intertekstuaalisuus kahdella eri tavalla: Genetten strukturalistinen lähestymistapa soveltuu teosten välisten viittaussuhteiden tutkimiseen, kun taas Barthesin jälkistrukturalistinen diskurssi auttaa ymmärtämään tekstienvälisyyden osana merkityksen jatkuvaa epävakautta. Tutkielman ensimmäinen osio keskittyy analysoimaan lähiluvun keinoin romaanien On Beauty ja Howards End välistä strukturalistista suhdetta vertailemalla teosten eroja ja yhtäläisyyksiä Genetten intertekstuaalisuusteorian valossa. Vertailussa kiinnetetään erityisesti huomiota teosten juoneen, rakenteeseen, aikaan ja paikkaan, sekä uudelleenkirjoitusten yleiseen tendenssiin päivittää alkuperäistä tarinaa kohdeyleisölle paremmin sopivaksi. Toisessa osiossa tutkimusta esille nousee jälkistrukturalistinen näkemys intertekstuaalisuudesta osana lukijan tuottaman merkityksen tulkinnanvaraisuutta. Osiossa käsitellään Rembrandtin taideteosten roolia Smithin romaanissa ja analysoidaan hahmojen tulkintoja sekä suhtautumista Rembrandtin tuotantoon Barthesin teoreettisten käsitteiden kautta. Keskeiseksi analyysin kohteeksi nousee lukija sekä lukijan tuottamat tulkinnat ja niiden merkitys Smithin romaanin tematiikassa. Tutkielmassa osoitetaan kuinka intertekstuaalisuus ei ole niin yksinkertainen termi kuin sen laaja käyttö niin kirjallisuustieteessä kuin mediassakin antaa ymmärtää, sekä selvitetään intertekstuaalisuuden teorian kehitystä 60-luvulta nykypäivään. Vaikka strukturalistisessa muodossa käsite soveltuu etenkin kahden toisiinsa kytkeytyneen teoksen tutkimiseen, vertaileva analyysi kuitenkin osoittaa, että On Beauty ei ole pelkkä uudelleenkirjoitus, vaan romaanin tulkintaan tarvitaan myös jälkistrukturalistisen dekonstruktion käsitteitä, jotta laajemmat tekstuaalisuuden verkostot aukeavat lukijalle. Romaanissa esiintyvä taitelijakuva myös osoittaa, että Smith itse on hyvin tietoinen kirjallisuusteoreettisesta keskustelusta.

Spectral Invariance of SG Pseudo-Differential Operators on L-P(R-n)

We prove the spectral invariance of SG pseudo-differential operators on L-P(R-n), 1 < p < infinity, by using the equivalence of ellipticity and Fredholmness of SG pseudo-differential operators on L-p(R-n), 1 < p < infinity. A key ingredient in the proof is the spectral invariance of SC pseudo-differential operators on L-2(R-n).

Properties of Toeplitz operators on analytic function spaces : from function symbols to distributions

Toeplitz operators are among the most important classes of concrete operators with applications to several branches of pure and applied mathematics. This doctoral thesis deals with Toeplitz operators on analytic Bergman, Bloch and Fock spaces. Usually, a Toeplitz operator is a composition of multiplication by a function and a suitable projection. The present work deals with generalizing the notion to the case where the function is replaced by a distributional symbol. Fredholm theory for Toeplitz operators with matrix-valued symbols is also considered. The subject of this thesis belongs to the areas of complex analysis, functional analysis and operator theory. This work contains five research articles. The articles one, three and four deal with finding suitable distributional classes in Bergman, Fock and Bloch spaces, respectively. In each case the symbol class to be considered turns out to be a certain weighted Sobolev-type space of distributions. The Bergman space setting is the most straightforward. When dealing with Fock spaces, some difficulties arise due to unboundedness of the complex plane and the properties of the Gaussian measure in the definition. In the Bloch-type spaces an additional logarithmic weight must be introduced. Sufficient conditions for boundedness and compactness are derived. The article two contains a portion showing that under additional assumptions, the condition for Bergman spaces is also necessary. The fifth article deals with Fredholm theory for Toeplitz operators having matrix-valued symbols. The essential spectra and index theorems are obtained with the help of Hardy space factorization and the Berezin transform, for instance. The article two also has a part dealing with matrix-valued symbols in a non-reflexive Bergman space, in which case a condition on the oscillation of the symbol (a logarithmic VMO-condition) must be added.

Descriptions of operators in quantum mechanics

The problem of expressing a general dynamical variable in quantum mechanics as a function of a primitive set of operators is studied from several points of view. In the context of the Heisenberg commutation relation, the Weyl representation for operators and a new Fourier-Mellin representation are related to the Heisenberg group and the groupSL(2,R) respectively. The description of unitary transformations via generating functions is analysed in detail. The relation between functions and ordered functions of noncommuting operators is discussed, and results closely paralleling classical results are obtained.

A classification of homogeneous operators in the Cowen-Douglas class

An explicit construction of all the homogeneous holomorphic Hermitian vector bundles over the unit disc D is given. It is shown that every such vector bundle is a direct sum of irreducible ones. Among these irreducible homogeneous holomorphic Hermitian vector bundles over D, the ones corresponding to operators in the Cowen-Douglas class B-n(D) are identified. The classification of homogeneous operators in B-n(D) is completed using an explicit realization of these operators. We also show how the homogeneous operators in B-n(D) split into similarity classes. (C) 2011 Elsevier Inc. All rights reserved.

Perturbative corrections for staggered four-fermion operators

We present results for one-loop matching coefficients between continuum four-fermion operators, defined in the Naive Dimensional Regularization scheme, and staggered fermion operators of various types. We calculate diagrams involving gluon exchange between quark fines, and ''penguin'' diagrams containing quark loops. For the former we use Landau-gauge operators, with and without O(a) improvement, and including the tadpole improvement suggested by Lepage and Mackenzie. For the latter we use gauge-invariant operators. Combined with existing results for two-loop anomalous dimension matrices and one-loop matching coefficients, our results allow a lattice calculation of the amplitudes for KKBAR mixing and K --> pipi decays with all corrections of O(g2) included. We also discuss the mixing of DELTAS = 1 operators with lower dimension operators, and show that, with staggered fermions, only a single lower dimension operator need be removed by non-perturbative subtraction.

Toeplitz Operators with Special Symbols on Segal-Bargmann Spaces

We study the boundedness of Toeplitz operators on Segal-Bargmann spaces in various contexts. Using Gutzmer's formula as the main tool we identify symbols for which the Toeplitz operators correspond to Fourier multipliers on the underlying groups. The spaces considered include Fock spaces, Hermite and twisted Bergman spaces and Segal-Bargmann spaces associated to Riemannian symmetric spaces of compact type.

Boson Inverse Operators and a New Family of Two-photon Annihilation Operators

We introduce the inverse of the Hermitian operator (acircacirc†) and express the Boson inverse operators acirc-1 and acirc†-1 in terms of the operators acirc, acirc† and (acircacirc†)-1. We show that these Boson inverse operators may be realized by Susskind-Glogower phase operators. In this way, we find a new two-photon annihilation operator and denote it as acirc2(acircacirc†)-1. We show that the eigenstates of this operator have interesting non-classical properties. We find that the eigenstates of the operators (acircacirc†)-1 acirc2, acirc(acircacirc†)-1 acirc and acirc2(acircacirc†)-1 have many similar properties and thus they constitute a family of two-photon annihilation operators.