Operators on Hilbert Spaces
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Some Properties of the Essential Numerical Range on Banach Spaces
Pure Mathematical Sciences, Vol. 6, 2017, no. 1, 105 - 111 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/pms.2017.7912 Some Properties of the Essential Numerical Range on Banach Spaces L. N. Muhati, J. O. Bonyo and J. O. Agure Department of Pure and Applied Mathematics Maseno University, P.O. Box 333 - 40105, Maseno, Kenya Copyright c 2017 L. N. Muhati, J. O. Bonyo and J. O. Agure. This article is dis- tributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract We study the properties of the essential algebraic numerical range as well as the essential spatial numerical range for Banach space operators. Mathematics Subject Classification: 47A10; 47A05; 47A53 Keywords: Essential algebraic numerical range, Essential spatial numer- ical range, Calkin algebra 1 Introduction Let A be a complex normed algebra with a unit and let A∗ denote its dual space. For a comprehensive theory on normed algebras, we refer to [4]. We define the algebraic numerical range of an element a 2 A by V (a; A) = ff(a): f 2 A∗; f(1) = 1 = kfkg. Let X denote a complex Banach space and L(X) be the Banach algebra of all bounded linear operators acting on X. We de- note the algebraic numerical range of any T 2 L(X) by V (T; L(X)). For T 2 L(X), the spatial numerical range is defined by W (T ) = fhT x; x∗i : x 2 X; x∗ 2 X∗; kxk = 1 = kx∗k = hx; x∗ig. -
Numerical Range of Operators, Numerical Index of Banach Spaces, Lush Spaces, and Slicely Countably Determined Banach Spaces
Numerical range of operators, numerical index of Banach spaces, lush spaces, and Slicely Countably Determined Banach spaces Miguel Martín University of Granada (Spain) June 5, 2009 Preface This text is not a book and it is not in its nal form. This is going to be used as classroom notes for the talks I am going to give in the Workshop on Geometry of Banach spaces and its Applications (sponsored by Indian National Board for Higher Mathematics) to be held at the Indian Statistical Institute, Bangalore (India), on 1st - 13th June 2009. Miguel Martín Contents Preface 3 Basic notation7 1 Numerical Range of operators. Surjective isometries9 1.1 Introduction.....................................9 1.2 The exponential function. Isometries....................... 15 1.3 Finite-dimensional spaces with innitely many isometries............ 17 1.3.1 The dimension of the Lie algebra..................... 19 1.4 Surjective isometries and duality......................... 20 2 Numerical index of Banach spaces 23 2.1 Introduction..................................... 23 2.2 Computing the numerical index.......................... 23 2.3 Numerical index and duality............................ 28 2.4 Banach spaces with numerical index one..................... 32 2.5 Renorming and numerical index.......................... 34 2.6 Finite-dimensional spaces with numerical index one: asymptotic behavior.. 38 2.7 Relationship to the Daugavet property....................... 39 2.8 Smoothness, convexity and numerical index 1 .................. 43 3 Lush spaces 47 3.1 Examples of lush spaces.............................. 49 5 3.2 Lush renormings.................................. 52 3.3 Some reformulations of lushness.......................... 53 3.4 Lushness is not equivalent to numerical index 1 ................. 55 3.5 Stability results for lushness............................ 56 4 Slicely countably determined Banach spaces 59 4.1 Slicely countably determined sets........................ -
Numerical Range and Functional Calculus in Hilbert Space
Numerical range and functional calculus in Hilbert space Michel Crouzeix Abstract We prove an inequality related to polynomial functions of a square matrix, involving the numerical range of the matrix. We also show extensions valid for bounded and also unbounded operators in Hilbert spaces, which allow the development of a functional calculus. Mathematical subject classification : 47A12 ; 47A25 Keywords : functional calculus, numerical range, spectral sets. 1 Introduction In this paper H denotes a complex Hilbert space equipped with the inner product . , . and h i corresponding norm v = v, v 1/2. We denote its unit sphere by Σ := v H; v = 1 . For k k h i H { ∈ k k } a bounded linear operator A (H) we use the operator norm A and denote by W (A) its ∈ L k k numerical range : A := sup A v , W (A) := Av,v ; v ΣH . k k v ΣH k k {h i ∈ } ∈ Recall that the numerical range is a convex subset of C and that its closure contains the spectrum of A. We keep the same notation in the particular case where H = Cd and A is a d d matrix. × The main aim of this article is to prove that the inequality p(A) 56 sup p(z) (1) k k≤ z W (A) | | ∈ holds for any matrix A Cd,d and any polynomial p : C C (independently of d). ∈ → It is remarkable that the completely bounded version of this inequality is also valid. More precisely, we consider now matrix-valued polynomials P : C Cm,n, i.e., for z C, P (z) = → ∈ (p (z)) is a matrix, with each entry p (.) a polynomial C C. -
Numerical Range of Square Matrices a Study in Spectral Theory
Numerical Range of Square Matrices A Study in Spectral Theory Department of Mathematics, Linköping University Erik Jonsson LiTH-MAT-EX–2019/03–SE Credits: 16 hp Level: G2 Supervisor: Göran Bergqvist, Department of Mathematics, Linköping University Examiner: Milagros Izquierdo, Department of Mathematics, Linköping University Linköping: June 2019 Abstract In this thesis, we discuss important results for the numerical range of general square matrices. Especially, we examine analytically the numerical range of complex-valued 2 × 2 matrices. Also, we investigate and discuss the Gershgorin region of general square matrices. Lastly, we examine numerically the numerical range and Gershgorin regions for different types of square matrices, both contain the spectrum of the matrix, and compare these regions, using the calculation software Maple. Keywords: numerical range, square matrix, spectrum, Gershgorin regions URL for electronic version: urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-157661 Erik Jonsson, 2019. iii Sammanfattning I denna uppsats diskuterar vi viktiga resultat för numerical range av gene- rella kvadratiska matriser. Speciellt undersöker vi analytiskt numerical range av komplexvärda 2 × 2 matriser. Vi utreder och diskuterar också Gershgorin områden för generella kvadratiska matriser. Slutligen undersöker vi numeriskt numerical range och Gershgorin områden för olika typer av matriser, där båda innehåller matrisens spektrum, och jämför dessa områden genom att använda beräkningsprogrammet Maple. Nyckelord: numerical range, kvadratisk matris, spektrum, Gershgorin områden URL för elektronisk version: urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-157661 Erik Jonsson, 2019. v Acknowledgements I’d like to thank my supervisor Göran Bergqvist, who has shown his great interest for linear algebra and inspired me for writing this thesis. -
Morphophoric Povms, Generalised Qplexes, Ewline and 2-Designs
Morphophoric POVMs, generalised qplexes, and 2-designs Wojciech S lomczynski´ and Anna Szymusiak Institute of Mathematics, Jagiellonian University,Lojasiewicza 6, 30-348 Krak´ow, Poland We study the class of quantum measurements with the property that the image of the set of quantum states under the measurement map transforming states into prob- ability distributions is similar to this set and call such measurements morphophoric. This leads to the generalisation of the notion of a qplex, where SIC-POVMs are re- placed by the elements of the much larger class of morphophoric POVMs, containing in particular 2-design (rank-1 and equal-trace) POVMs. The intrinsic geometry of a generalised qplex is the same as that of the set of quantum states, so we explore its external geometry, investigating, inter alia, the algebraic and geometric form of the inner (basis) and the outer (primal) polytopes between which the generalised qplex is sandwiched. In particular, we examine generalised qplexes generated by MUB-like 2-design POVMs utilising their graph-theoretical properties. Moreover, we show how to extend the primal equation of QBism designed for SIC-POVMs to the morphophoric case. 1 Introduction and preliminaries Over the last ten years in a series of papers [2,3, 30{32, 49, 68] Fuchs, Schack, Appleby, Stacey, Zhu and others have introduced first the idea of QBism (formerly quantum Bayesianism) and then its probabilistic embodiment - the (Hilbert) qplex, both based on the notion of symmetric informationally complete positive operator-valued measure -
Quantum Errors and Disturbances: Response to Busch, Lahti and Werner
entropy Article Quantum Errors and Disturbances: Response to Busch, Lahti and Werner David Marcus Appleby Centre for Engineered Quantum Systems, School of Physics, The University of Sydney, Sydney, NSW 2006, Australia; [email protected]; Tel.: +44-734-210-5857 Academic Editors: Gregg Jaeger and Andrei Khrennikov Received: 27 February 2016; Accepted: 28 April 2016; Published: 6 May 2016 Abstract: Busch, Lahti and Werner (BLW) have recently criticized the operator approach to the description of quantum errors and disturbances. Their criticisms are justified to the extent that the physical meaning of the operator definitions has not hitherto been adequately explained. We rectify that omission. We then examine BLW’s criticisms in the light of our analysis. We argue that, although the BLW approach favour (based on the Wasserstein two-deviation) has its uses, there are important physical situations where an operator approach is preferable. We also discuss the reason why the error-disturbance relation is still giving rise to controversies almost a century after Heisenberg first stated his microscope argument. We argue that the source of the difficulties is the problem of interpretation, which is not so wholly disconnected from experimental practicalities as is sometimes supposed. Keywords: error disturbance principle; uncertainty principle; quantum measurement; Heisenberg PACS: 03.65.Ta 1. Introduction The error-disturbance principle remains highly controversial almost a century after Heisenberg wrote the paper [1], which originally suggested it. It is remarkable that this should be so, since the disagreements concern what is arguably the most fundamental concept of all, not only in physics, but in empirical science generally: namely, the concept of measurement accuracy. -
Numerical Ranges and Applications in Quantum Information
Numerical Ranges and Applications in Quantum Information by Ningping Cao A Thesis presented to The University of Guelph In partial fulfilment of requirements for the degree of Doctor of Philosophy in Mathematics & Statistics Guelph, Ontario, Canada © Ningping Cao, August, 2021 ABSTRACT NUMERICAL RANGES AND APPLICATIONS IN QUANTUM INFORMATION Ningping Cao Advisor: University of Guelph, 2021 Dr. Bei Zeng Dr. David Kribs The numerical range (NR) of a matrix is a concept that first arose in the early 1900’s as part of efforts to build a rigorous mathematical framework for quantum mechanics andother challenges at the time. Quantum information science (QIS) is a new field having risen to prominence over the past two decades, combining quantum physics and information science. In this thesis, we connect NR and its extensions with QIS in several topics, and apply the knowledge to solve related QIS problems. We utilize the insight offered by NRs and apply them to quantum state inference and Hamiltonian reconstruction. We propose a new general deep learning method for quantum state inference in chapter 3, and employ it to two scenarios – maximum entropy estimation of unknown states and ground state reconstruction. The method manifests high fidelity, exceptional efficiency, and noise tolerance on both numerical and experimental data. A new optimization algorithm is presented in chapter 4 for recovering Hamiltonians. It takes in partial measurements from any eigenstates of an unknown Hamiltonian H, then provides an estimation of H. Our algorithm almost perfectly deduces generic and local generic Hamiltonians. Inspired by hybrid classical-quantum error correction (Hybrid QEC), the higher rank (k : p)-matricial range is defined and studied in chapter 5. -
The Notion of Observable and the Moment Problem for ∗-Algebras and Their GNS Representations
The notion of observable and the moment problem for ∗-algebras and their GNS representations Nicol`oDragoa, Valter Morettib Department of Mathematics, University of Trento, and INFN-TIFPA via Sommarive 14, I-38123 Povo (Trento), Italy. [email protected], [email protected] February, 17, 2020 Abstract We address some usually overlooked issues concerning the use of ∗-algebras in quantum theory and their physical interpretation. If A is a ∗-algebra describing a quantum system and ! : A ! C a state, we focus in particular on the interpretation of !(a) as expectation value for an algebraic observable a = a∗ 2 A, studying the problem of finding a probability n measure reproducing the moments f!(a )gn2N. This problem enjoys a close relation with the selfadjointness of the (in general only symmetric) operator π!(a) in the GNS representation of ! and thus it has important consequences for the interpretation of a as an observable. We n provide physical examples (also from QFT) where the moment problem for f!(a )gn2N does not admit a unique solution. To reduce this ambiguity, we consider the moment problem n ∗ (a) for the sequences f!b(a )gn2N, being b 2 A and !b(·) := !(b · b). Letting µ!b be a n solution of the moment problem for the sequence f!b(a )gn2N, we introduce a consistency (a) relation on the family fµ!b gb2A. We prove a 1-1 correspondence between consistent families (a) fµ!b gb2A and positive operator-valued measures (POVM) associated with the symmetric (a) operator π!(a). In particular there exists a unique consistent family of fµ!b gb2A if and only if π!(a) is maximally symmetric. -
A Stepwise Planned Approach to the Solution of Hilbert's Sixth Problem. II: Supmech and Quantum Systems
A Stepwise Planned Approach to the Solution of Hilbert’s Sixth Problem. II : Supmech and Quantum Systems Tulsi Dass Indian Statistical Institute, Delhi Centre, 7, SJS Sansanwal Marg, New Delhi, 110016, India. E-mail: [email protected]; [email protected] Abstract: Supmech, which is noncommutative Hamiltonian mechanics (NHM) (developed in paper I) with two extra ingredients : positive ob- servable valued measures (PObVMs) [which serve to connect state-induced expectation values and classical probabilities] and the ‘CC condition’ [which stipulates that the sets of observables and pure states be mutually separating] is proposed as a universal mechanics potentially covering all physical phe- nomena. It facilitates development of an autonomous formalism for quantum mechanics. Quantum systems, defined algebraically as supmech Hamiltonian systems with non-supercommutative system algebras, are shown to inevitably have Hilbert space based realizations (so as to accommodate rigged Hilbert space based Dirac bra-ket formalism), generally admitting commutative su- perselection rules. Traditional features of quantum mechanics of finite parti- cle systems appear naturally. A treatment of localizability much simpler and more general than the traditional one is given. Treating massive particles as localizable elementary quantum systems, the Schr¨odinger wave functions with traditional Born interpretation appear as natural objects for the descrip- tion of their pure states and the Schr¨odinger equation for them is obtained without ever using a classical Hamiltonian or Lagrangian. A provisional set of axioms for the supmech program is given. arXiv:1002.2061v4 [math-ph] 18 Dec 2010 1 I. Introduction This is the second of a series of papers aimed at obtaining a solution of Hilbert’s sixth problem in the framework of a noncommutative geome- try (NCG) based ‘all-embracing’ scheme of mechanics. -
Riesz-Like Bases in Rigged Hilbert Spaces, in Preparation [14] Bonet, J., Fern´Andez, C., Galbis, A
RIESZ-LIKE BASES IN RIGGED HILBERT SPACES GIORGIA BELLOMONTE AND CAMILLO TRAPANI Abstract. The notions of Bessel sequence, Riesz-Fischer sequence and Riesz basis are generalized to a rigged Hilbert space D[t] ⊂H⊂D×[t×]. A Riesz- like basis, in particular, is obtained by considering a sequence {ξn}⊂D which is mapped by a one-to-one continuous operator T : D[t] → H[k · k] into an orthonormal basis of the central Hilbert space H of the triplet. The operator T is, in general, an unbounded operator in H. If T has a bounded inverse then the rigged Hilbert space is shown to be equivalent to a triplet of Hilbert spaces. 1. Introduction Riesz bases (i.e., sequences of elements ξn of a Hilbert space which are trans- formed into orthonormal bases by some bounded{ } operator withH bounded inverse) often appear as eigenvectors of nonself-adjoint operators. The simplest situation is the following one. Let H be a self-adjoint operator with discrete spectrum defined on a subset D(H) of the Hilbert space . Assume, to be more definite, that each H eigenvalue λn is simple. Then the corresponding eigenvectors en constitute an orthonormal basis of . If X is another operator similar to H,{ i.e.,} there exists a bounded operator T withH bounded inverse T −1 which intertwines X and H, in the sense that T : D(H) D(X) and XT ξ = T Hξ, for every ξ D(H), then, as it is → ∈ easily seen, the vectors ϕn with ϕn = Ten are eigenvectors of X and constitute a Riesz basis for . -
Spectral Theory
SPECTRAL THEORY EVAN JENKINS Abstract. These are notes from two lectures given in MATH 27200, Basic Functional Analysis, at the University of Chicago in March 2010. The proof of the spectral theorem for compact operators comes from [Zim90, Chapter 3]. 1. The Spectral Theorem for Compact Operators The idea of the proof of the spectral theorem for compact self-adjoint operators on a Hilbert space is very similar to the finite-dimensional case. Namely, we first show that an eigenvector exists, and then we show that there is an orthonormal basis of eigenvectors by an inductive argument (in the form of an application of Zorn's lemma). We first recall a few facts about self-adjoint operators. Proposition 1.1. Let V be a Hilbert space, and T : V ! V a bounded, self-adjoint operator. (1) If W ⊂ V is a T -invariant subspace, then W ? is T -invariant. (2) For all v 2 V , hT v; vi 2 R. (3) Let Vλ = fv 2 V j T v = λvg. Then Vλ ? Vµ whenever λ 6= µ. We will need one further technical fact that characterizes the norm of a self-adjoint operator. Lemma 1.2. Let V be a Hilbert space, and T : V ! V a bounded, self-adjoint operator. Then kT k = supfjhT v; vij j kvk = 1g. Proof. Let α = supfjhT v; vij j kvk = 1g. Evidently, α ≤ kT k. We need to show the other direction. Given v 2 V with T v 6= 0, setting w0 = T v=kT vk gives jhT v; w0ij = kT vk. -
Quantum Mechanical Observables Under a Symplectic Transformation
Quantum Mechanical Observables under a Symplectic Transformation of Coordinates Jakub K´aninsk´y1 1Charles University, Faculty of Mathematics and Physics, Institute of Theoretical Physics. E-mail address: [email protected] March 17, 2021 We consider a general symplectic transformation (also known as linear canonical transformation) of quantum-mechanical observables in a quan- tized version of a finite-dimensional system with configuration space iso- morphic to Rq. Using the formalism of rigged Hilbert spaces, we define eigenstates for all the observables. Then we work out the explicit form of the corresponding transformation of these eigenstates. A few examples are included at the end of the paper. 1 Introduction From a mathematical perspective we can view quantum mechanics as a science of finite- dimensional quantum systems, i.e. systems whose classical pre-image has configuration space of finite dimension. The single most important representative from this family is a quantized version of the classical system whose configuration space is isomorphic to Rq with some q N, which corresponds e.g. to the system of finitely many coupled ∈ harmonic oscillators. Classically, the state of such system is given by a point in the arXiv:2007.10858v2 [quant-ph] 16 Mar 2021 phase space, which is a vector space of dimension 2q equipped with the symplectic form. One would typically prefer to work in the abstract setting, without the need to choose any particular set of coordinates: in principle this is possible. In practice though, one will often end up choosing a symplectic basis in the phase space aligned with the given symplectic form and resort to the coordinate description.