To say that this is the best book on the quantum theory of fields is no praise, since to my knowledge it is the only book on this subject But it is a very good and most useful book The original was written in German and appeared in 1942 This is a translation with some minor changes A few remarks have been added, concerning meson theory and nuclear forces, also footnotes referring to modern work in this field, and finally an appendix on the symmetrization of the energy momentum tensor according to Belinfante Quantum Theory of Fields Prof Gregor Wentzel Translated from the German by Charlotte Houtermans and J M Jauch Pp ix + 224, (New York and London: Interscience Publishers, Inc, 1949) 36s

Quantum Theory of Fields

Introduction to Functional Analysis.

This is lecture notes for several courses on Functional Analysis at School of Mathematics of University of Leeds. They are based on the notes of Dr. Matt Daws, Prof. Jonathan R. Partington and Dr. David Salinger used in the previous years. Some sections are borrowed from the textbooks, which I used since being a student myself. However all misprints, omissions, and errors are only my responsibility. I am very grateful to Filipa Soares de Almeida, Eric Borgnet, Pasc Gavruta for pointing out some of them. Please let me know if you find more. The notes are available also for download in PDF. The suggested textbooks are [1,6,8,9]. The other nice books with many interesting problems are [3, 7]. Exercises with stars are not a part of mandatory material but are nevertheless worth to hear about. And they are not necessarily difficult, try to solve them! CONTENTS List of Figures 3 Notations and Assumptions 4 Integrability conditions 4 1. Motivating Example: Fourier Series 4 1.1. Fourier series: basic notions 4 1.2. The vibrating string 8 1.3. Historic: Joseph Fourier 10 2. Basics of Linear Spaces 11 2.1. Banach spaces (basic definitions only) 12 2.2. Hilbert spaces 14 2.3. Subspaces 16 2.4. Linear spans 19 3. Orthogonality 20 3.1. Orthogonal System in Hilbert Space 21 3.2. Bessel’s inequality 23 3.3. The Riesz–Fischer theorem 25 3.4. Construction of Orthonormal Sequences 26 3.5. Orthogonal complements 28 4. Fourier Analysis 29 Date: 16th October 2017. 1

Introduction to Functional Analysis. By Angus E. Taylor. Pp. xvi. 423. $12.50. 1958. (John Wiley and Sons)

Let A=(A1, . . ., Am) be an m-tuple of n × n Hermitian matrices. For $1 \le k \le n$, the $k${\rm th} joint numerical range of A is defined by $$W_k(A) = \{ ({\rm \tr}(X^*A_1X), \dots, {\rm \tr}(X^*A_mX) ): X \in {\bf C}^{n\times k}, X^*X = I_k \}.$$ We consider linearly independent families of Hermitian matrices {A1, . . . , Am} so that Wk(A) is convex. It is shown that m can reach the upper bound 2k(n-k)+1. A key idea in our study is relating the convexity of Wk(A) to the problem of constructing rank k orthogonal projections under linear constraints determined by A. The techniques are extended to study the convexity of other generalized numerical ranges and the corresponding matrix construction problems.

/pdf/convexity-of-the-joint-numerical-range-33sk0ug53w.pdf

Convexity of the Joint Numerical Range

Joint numerical ranges of operators in semi-Hilbertian spaces

In this paper we introduce an [m,C]-isometric operator T on a complex Hilbert space H and study its spectral properties. We show that if T is an [m,C]-isometric operator and N is an n-nilpotent operator, respectively, then T + N is an [m + 2n − 2,C]-isometric operator. Finally we give a short proof of Duggal’s result for tensor product of m-isometries and give a similar result for [m,C]-isometric operators.

/pdf/c-isometric-operators-3ry1tz5rqg.pdf

(∞,C)-isometric operators

Spectrum of class pwA(s,t) operators

In this paper, we introduce a polynomially normal operator on a complex Hilbert space, extending the notation of n-normal and normal operators. Several basic properties of polynomially normal operator are firstly presented. We show some spectral properties of polynomially normal operators under new assumption in literature. Precisely, we prove that 
$$\sigma (T)=\sigma _a(T)$$

, 
$$\ker (T-z) \, \bot \, \ker (T - w)$$

 if z and w are distinct eigen-values of T and others results. Thus, we generalize some results for n-normal and normal operators.

Polynomially normal operators

A Banach space operator T ∈ B(X ) is hereditarily polaroid, T ∈ (HP), if the isolated points of the spectrum of every part Tp of the operator are poles of the resolvent of Tp; T is hereditarly normaloid, T ∈ (HN ), if every part Tp of T is normaloid. Let (HNP) denote the class of operators T ∈ B(X ) such that T ∈ (HP)∩ (HN ). (HNP) operators such that the Berberian–Quigley extension T◦ of T is also in (HNP) satisfy Bishop’s property (β). Given Hilbert space operators A, B∗ ∈ B(H), let dAB ∈ B(B(H)) stands for either of the elementary operators δAB(X) = AX−XB and 4AB(X) = AXB−X. If A, B∗ ∈ (HP) satisfy property (β), and the eigenspaces corresponding to distinct eigenvalues of A (resp., B∗) are orthogonal, then f(dAB) satisfies Weyl’s theorem and f(dAB) ∗ satisfies a-Weyl’s theorem for every function f which is analytic on a neighbourhood of σ(dAB). Finally, we characterize perturbations of dAB by quasinilpotent and algebraic operators A, B ∈ B(H).

/pdf/bishop-s-property-b-and-an-elementary-operator-2xd57dj74z.pdf

Bishop's property (β) and an elementary operator

/pdf/trace-formulae-for-p-hyponormal-operators-1hdy0uf0ib.pdf

Muneo Chō

Papers

(∞,C)-isometric operators

Spectrum of class pwA(s,t) operators

Polynomially normal operators

Bishop's property (β) and an elementary operator

Trace formulae for p-hyponormal operators