Numerical Range: (In) a Matrix Nutshell
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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. -
THREE STEPS on an OPEN ROAD Gilbert Strang This Note Describes
Inverse Problems and Imaging doi:10.3934/ipi.2013.7.961 Volume 7, No. 3, 2013, 961{966 THREE STEPS ON AN OPEN ROAD Gilbert Strang Massachusetts Institute of Technology Cambridge, MA 02139, USA Abstract. This note describes three recent factorizations of banded invertible infinite matrices 1. If A has a banded inverse : A=BC with block{diagonal factors B and C. 2. Permutations factor into a shift times N < 2w tridiagonal permutations. 3. A = LP U with lower triangular L, permutation P , upper triangular U. We include examples and references and outlines of proofs. This note describes three small steps in the factorization of banded matrices. It is written to encourage others to go further in this direction (and related directions). At some point the problems will become deeper and more difficult, especially for doubly infinite matrices. Our main point is that matrices need not be Toeplitz or block Toeplitz for progress to be possible. An important theory is already established [2, 9, 10, 13-16] for non-Toeplitz \band-dominated operators". The Fredholm index plays a key role, and the second small step below (taken jointly with Marko Lindner) computes that index in the special case of permutation matrices. Recall that banded Toeplitz matrices lead to Laurent polynomials. If the scalars or matrices a−w; : : : ; a0; : : : ; aw lie along the diagonals, the polynomial is A(z) = P k akz and the bandwidth is w. The required index is in this case a winding number of det A(z). Factorization into A+(z)A−(z) is a classical problem solved by Plemelj [12] and Gohberg [6-7]. -
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. -
Polynomials and Hankel Matrices
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Polynomials and Hankel Matrices Miroslav Fiedler Czechoslovak Academy of Sciences Institute of Mathematics iitnci 25 115 67 Praha 1, Czechoslovakia Submitted by V. Ptak ABSTRACT Compatibility of a Hankel n X n matrix W and a polynomial f of degree m, m < n, is defined. If m = n, compatibility means that HC’ = CfH where Cf is the companion matrix of f With a suitable generalization of Cr, this theorem is gener- alized to the case that m < n. INTRODUCTION By a Hankel matrix [S] we shall mean a square complex matrix which has, if of order n, the form ( ai +k), i, k = 0,. , n - 1. If H = (~y~+~) is a singular n X n Hankel matrix, the H-polynomial (Pi of H was defined 131 as the greatest common divisor of the determinants of all (r + 1) x (r + 1) submatrices~of the matrix where r is the rank of H. In other words, (Pi is that polynomial for which the nX(n+l)matrix I, 0 0 0 %fb) 0 i 0 0 0 1 LINEAR ALGEBRA AND ITS APPLICATIONS 66:235-248(1985) 235 ‘F’Elsevier Science Publishing Co., Inc., 1985 52 Vanderbilt Ave., New York, NY 10017 0024.3795/85/$3.30 236 MIROSLAV FIEDLER is the Smith normal form [6] of H,. It has also been shown [3] that qN is a (nonzero) polynomial of degree at most T. It is known [4] that to a nonsingular n X n Hankel matrix H = ((Y~+~)a linear pencil of polynomials of degree at most n can be assigned as follows: f(x) = fo + f,x + . -
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 -
(Hessenberg) Eigenvalue-Eigenmatrix Relations∗
(HESSENBERG) EIGENVALUE-EIGENMATRIX RELATIONS∗ JENS-PETER M. ZEMKE† Abstract. Explicit relations between eigenvalues, eigenmatrix entries and matrix elements are derived. First, a general, theoretical result based on the Taylor expansion of the adjugate of zI − A on the one hand and explicit knowledge of the Jordan decomposition on the other hand is proven. This result forms the basis for several, more practical and enlightening results tailored to non-derogatory, diagonalizable and normal matrices, respectively. Finally, inherent properties of (upper) Hessenberg, resp. tridiagonal matrix structure are utilized to construct computable relations between eigenvalues, eigenvector components, eigenvalues of principal submatrices and products of lower diagonal elements. Key words. Algebraic eigenvalue problem, eigenvalue-eigenmatrix relations, Jordan normal form, adjugate, principal submatrices, Hessenberg matrices, eigenvector components AMS subject classifications. 15A18 (primary), 15A24, 15A15, 15A57 1. Introduction. Eigenvalues and eigenvectors are defined using the relations Av = vλ and V −1AV = J. (1.1) We speak of a partial eigenvalue problem, when for a given matrix A ∈ Cn×n we seek scalar λ ∈ C and a corresponding nonzero vector v ∈ Cn. The scalar λ is called the eigenvalue and the corresponding vector v is called the eigenvector. We speak of the full or algebraic eigenvalue problem, when for a given matrix A ∈ Cn×n we seek its Jordan normal form J ∈ Cn×n and a corresponding (not necessarily unique) eigenmatrix V ∈ Cn×n. Apart from these constitutional relations, for some classes of structured matrices several more intriguing relations between components of eigenvectors, matrix entries and eigenvalues are known. For example, consider the so-called Jacobi matrices. -
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. -
On Invariance of the Numerical Range and Some Classes of Operators In
ON INVARIANCE OF THE NUMERICAL RANGE AND SOME CLASSES OF OPERATORS IN HILBERT SPACES By Arthur Wanyonyi Wafula A thesis submitted in fulfillment of the requirements for the award of the Degree of Doctor of Philosophy in Pure Mathematics School of Mathematics,University of Nairobi 15 April 2013 G^ o a op h University of NAIROBI Library 0430140 4 0.1 DECLARATION This thesis is my orginal work and has not been presented for a degree in any other University S ignat ure.....D ate . .....A* / ° 5 / 7'tTL. m h I Arthur Wanyonyi Wanambisi Wafula This thesis has been submitted for examination with our approval as University super visors; Signature Date 2- Signature P rof. J. M. Khalagai Prof. G.P.Pokhariyal 1 0.2 DEDICATION This work is dedicated to my family ; namely, my wife Anne Atieno and my sons Davies Magudha and Mishael Makali. n 0.3 ACKNOWLEDGEMENT My humble acknowledgement go to the Almighty God Jehovah who endows us withdhe gift of the Human brain that enables us to be creative and imaginative. I would also like to express my sincere gratitude to my Supervisor Prof.J.M.Khalagai for tirelessly guiding me in my research.I also wish to thank Prof.G.P.Pokhariyal for never ceasing to put me on my toes. I sincerely wish to thank the Director of the School of Mathematics DrJ .Were for his invaluable support. Furthermore,! am indebted to my colleagues Mrs.E.Muriuki, Dr.Mile, Dr.I.Kipchichir, Mr R.Ogik and Dr.N.'Owuor who have been a source of inspiration. -
Mathematische Annalen Digital Object Identifier (DOI) 10.1007/S002080100153
Math. Ann. (2001) Mathematische Annalen Digital Object Identifier (DOI) 10.1007/s002080100153 Pfaff τ-functions M. Adler · T. Shiota · P. van Moerbeke Received: 20 September 1999 / Published online: ♣ – © Springer-Verlag 2001 Consider the two-dimensional Toda lattice, with certain skew-symmetric initial condition, which is preserved along the locus s =−t of the space of time variables. Restricting the solution to s =−t, we obtain another hierarchy called Pfaff lattice, which has its own tau function, being equal to the square root of the restriction of 2D-Toda tau function. We study its bilinear and Fay identities, W and Virasoro symmetries, relation to symmetric and symplectic matrix integrals and quasiperiodic solutions. 0. Introduction Consider the set of equations ∂m∞ n ∂m∞ n = Λ m∞ , =−m∞(Λ ) ,n= 1, 2,..., (0.1) ∂tn ∂sn on infinite matrices m∞ = m∞(t, s) = (µi,j (t, s))0≤i,j<∞ , M. Adler∗ Department of Mathematics, Brandeis University, Waltham, MA 02454–9110, USA (e-mail: [email protected]) T. Shiota∗∗ Department of Mathematics, Kyoto University, Kyoto 606–8502, Japan (e-mail: [email protected]) P. van Moerbeke∗∗∗ Department of Mathematics, Universit´e de Louvain, 1348 Louvain-la-Neuve, Belgium Brandeis University, Waltham, MA 02454–9110, USA (e-mail: [email protected] and @math.brandeis.edu) ∗ The support of a National Science Foundation grant # DMS-98-4-50790 is gratefully ac- knowledged. ∗∗ The support of Japanese ministry of education’s grant-in-aid for scientific research, and the hospitality of the University of Louvain and Brandeis University are gratefully acknowledged. -
Uncertainty Relations and Joint Numerical Ranges
Konrad Szyma´nski Uncertainty relations and joint numerical ranges MSc thesis under the supervision of Prof. Karol Zyczkowski˙ arXiv:1707.03464v1 [quant-ph] 11 Jul 2017 MARIANSMOLUCHOWSKIINSTITUTEOFPHYSICS, UNIWERSYTETJAGIELLO NSKI´ KRAKÓW,JULY 2 0 1 7 Contents Quantum states 1 Joint Numerical Range 5 Phase transitions 11 Uncertainty relations 19 Conclusions and open questions 23 Bibliography 25 iii To MS, AS, unknow, DS, JJK, AM, FF, KZ,˙ SM, P, N. Introduction Noncommutativity lies at the heart of quantum theory and provides a rich set of mathematical and physical questions. In this work, I address this topic through the concept of Joint Numerical Range (JNR) — the set of simultaneously attainable expectation values of multiple quantum observables, which in general do not commute. The thesis is divided into several chapters: Quantum states Geometry of the set of quantum states of size d is closely related to JNR, which provides a nice tool to analyze the former set. The problem of determining the intricate structure of this set is known to quickly become hard as the dimensionality grows (approximately as d2). The difficulty stems from the nonlinear constraints put on the set of parameters. In this chapter I briefly introduce the formalism of density matrices, which will prove useful in the later sections. Joint Numerical Range Joint Numerical Range of a collection of k operators, or JNR for short, is an object capturing the notion of simultaneous measurement of averages — expectation values of multiple observables. This is precisely the set of values which are simultaneously attainable for fixed observables (F1, F2,..., Fk) over a given quantum state r 2 Md: if we had a function taking quantum state and returning the tuple of average values: E(r) = (hF1ir , hF1ir ,..., hFkir),(1) Joint Numerical Range L(F1,..., Fk) is precisely E[Md].