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

Let T be an M-hyponormal operator acting on infinite dimensional separable Hilbert space and let $E: = \frac{1}{{2\pi i}}{\int_{\partial D} {(\lambda - T)^{{ - 1}} d\lambda } }$ be the Riesz idempotent for λ0, where D is a closed disk of center λ0 which contains no other points of σ (T) In this note we show that E is self-adjoint and $E(\mathcal{H}) = N(T - \lambda _{0} ) = N(T^{*} - \overline{{\lambda _{0} }} )$ As an application, if T is an algebraically M-hyponormal operator then we prove : (i) Weyl’s theorem holds for f(T) for every $f \in H(\sigma(T));$ (ii) a-Browder’s theorem holds for f(S) for every $S \prec T$ and f ∈ H(σ(S)); (iii) the the spectral mapping theorem holds for the Weyl spectrum of T and for the essential approximate point spectrum of T

Riesz Idempotent and Algebraically M -hyponormal Operators

Let 
$$ T = U{\left| T \right|} $$

 be an invertible class A operator such that 
$$ [T^{{*\,}} ,T] \in C_{1} $$

 . Then we show that 
$$ {\text{supp(gT)}} \subseteq \sigma {\text{(T)}} $$

 , where gT is the principal function of T. Moreover, we show that if T is pure, then 
$$ {\text{supp(gT)}} = \sigma {\text{(T)}} $$

 .

A Remark on Support of the Principal Function for Class A Operators

Let T = (T1, · · · ,Td) and A be a commuting d -tuple of operators and a positive operator on a complex Hilbert space, respectively. We introduce an (A,m) -symmetric commuting tuple of operators and characterize the joint approximate point spectrum of (A,m) -symmetric commuting tuple T . Next we introduce an (A,m) -expansive symmetric commuting tuple of operators and show basic properties of (A,m) -expansive symmetric commuting tuple. Mathematics subject classification (2010): 47A05, 47A10, 47A11.

/pdf/a-m-symmetric-commuting-tuples-of-operators-on-a-hilbert-2kjkbvqcez.pdf

(A,m)-Symmetric commuting tuples of operators on a Hilbert space

n-Tuples of operators satisfying σT(AB)=σT(BA)

In this paper, we study the structure of $${\Delta_{m}(T)}$$
 defined by the following: $${\Delta_m(T):=\sum_{j=0}^{m}(-1)^{m-j}
\begin{pmatrix} m \cr j \end{pmatrix}{T^{\ast}}^{j}C{T}^{m-j}C}$$
 . In particular, we prove that if $${m}$$
 is even, then $${\Delta_m(T)}$$
 is complex symmetric with the conjugation $${C}$$
 , and if $${m}$$
 is odd, then $${\Delta_m(T)}$$
 is skew complex symmetric with the conjugation $${C}$$
 . Moreover, we investigate the conditions for (
 $${m+1}$$
 )-complex symmetric operators to be $${m}$$
 -complex symmetric operators and characterize the spectrum of $${\Delta_m(T)}$$
 . Finally, we show that if $${T\in{\mathcal L({\mathcal H})}}$$
 is Hermitian or $${\Delta_1(T)}$$
 is $${p}$$
 -hyponormal, then $${\Delta_2(T)=0}$$
 implies $${\Delta_1(T)=0}$$
 .

Muneo Chō

Papers

Riesz Idempotent and Algebraically M -hyponormal Operators

A Remark on Support of the Principal Function for Class A Operators

(A,m)-Symmetric commuting tuples of operators on a Hilbert space

n-Tuples of operators satisfying σT(AB)=σT(BA)

On m -Complex Symmetric Operators II