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

Isolated points of spectrum of (p, k)-quasihyponormal operators

In this paper, we shall give an operator transform ${\hat T}$ from class A to the class of hyponormal operators. Then we shall show that $\sigma (T) = \sigma (\hat T)$ and $\sigma_a (T) = \sigma_a (\hat T)$ in case T belongs to class A. Next, as an application of ${\hat T},$ we will show that every class A operator has SVEP and property (β).

An Operator Transform from Class A to the Class of Hyponormal Operators and its Application

In this note we give an example of an ∞-hyponormal operator T whose Aluthge transform $ \widetilde T$ is not (1+ɛ)-hyponormal for any ɛ > 0 and show that the sequence $ \{ \widetilde T^{{(n)}} \} ^{\infty }_{{n = 1}} $ of interated Aluthge transforms of T need not converge in the weak operator topology, which solve two problems in [6].

Muneo Chō

Papers

Isolated points of spectrum of (p, k)-quasihyponormal operators

An Operator Transform from Class A to the Class of Hyponormal Operators and its Application

On Aluthge Transforms of p-hyponormal Operators

Boundary points of joint numerical ranges

Some classes of commuting n-tuples of operators