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.

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Convexity of the Joint Numerical Range

Joint numerical ranges of operators in semi-Hilbertian spaces

We show that p -wA(s,t) operators S,T ∗ (s + t 1 , 0 < p 1) with ker(S) ⊆ ker(S∗) and ker(T ∗) ⊆ ker(T ) satisfy Fuglede-Putnam theorem, i.e., SX = XT for some X implies S∗X = XT ∗ . Also, we show that two quasisimilar p -wA(s,t) operators S,T (s+ t 1 , 0 < p 1) with ker(S) ⊆ ker(S∗) and ker(T ) ⊆ ker(T ∗) have equal spectra and essential spectra. Mathematics subject classification (2010): 47B20, 47A62.

/pdf/fuglede-putnam-theorem-and-quasisimilarity-of-class-p-wa-s-t-w8zcrqt15p.pdf

Fuglede-Putnam theorem and quasisimilarity of class p-wA(s,t) operators

Inspired by recent works on (m, C)-isometric and [m, C]-isometric operators on Hilbert spaces studied respectively in Chō et al. (Complex Anal. Oper. Theory 10:1679–1694, 2016; Filomat 31:7, 2017), in this paper we introduce the class of (m, C)-isometries for tuple of commuting operators. This is a generalization of the class of (m, C)-isometric operators. A commuting tuples of operators $$\mathbf{\large T}=(T_1,\ldots ,T_d)\in {\mathcal {B}}^{(d)}({\mathcal {H}})$$
 is said to be (m, C)-isometric tuple if $$\begin{aligned} {{\mathcal {Q}}}_{m}(\mathbf{T}):=\sum _{0\le k\le m}(-1)^{m-k}\left( {\begin{array}{c}m\\ k\end{array}}\right) \left( \sum _{|\beta |=k}\frac{k!}{\beta !}\mathbf{\large T}^{*\beta }C\mathbf{\large T}^{\beta }C\right) =0 \end{aligned}$$
 for some positive integer m and some conjugation C. We consider a multivariable generalization of these single variable (m, C)-isometric operators and explore some of their basic properties.

On $$\varvec{(m,C)}$$ ( m , C ) -Isometric Commuting Tuples of Operators on a Hilbert Space

In this paper, we prove that if 
$$T\in {\mathcal {L}({\mathcal {H}})}$$

 is complex symmetric, then its generalized mean transform 
$${\widehat{T}}(t)~ (t\not =0)$$

 of T is also complex symmetric. Next, we consider complex symmetry property of the mean transform 
$${\widehat{T}}(0)$$

 of truncated weighted shift operators. Finally, we study properties of the generalized mean transform of skew complex symmetric operators.

On the generalized mean transforms of complex symmetric operators

In this paper, we introduce complex isosymmetric and $(m,n,C)$-isosymmetric operators on a Hilbert space $\mathcal H$ and study properties of such operators. In particular, we prove that if $T \in {\mathcal B}(\mathcal H)$ is an $(m,n,C)$-isosymmetric operator and $N$ is a $k$-nilpotent operator such that $T$ and $N$ are $C$-doubly commuting, then $T + N$ is an $(m+2k-2, n+2k-1,C)$-isosymmetric operator. Moreover, we show that if $T$ is $(m,n,C)$-isosymmetric and if $S$ is $(m',D)$-isometric and $n'$-complex symmetric with a conjugation $D$, then $T \otimes S$ is $(m+m'-1,n+n'-1,C \otimes D)$-isosymmetric.

Complex isosymmetric operators

Let T = X + iY be the Cartesian decomposition of an invertible operator T on a Hilbert space with trace class self-commutator [T ⁄ ;T]: Carey{Pincus introduced the principal function g and proved a trace formula associated with the Cartesian decomposition T = X+iY: Applying the ordered C 1 -functional calculus for (X;Y ) to their trace formula, we deflne the principal function g P and prove a trace formula associated with the polar decomposition T = UjTj: Using this formula, we show that g(x;y) = g P (e iµ ;r) almost everywhere x + iy = re iµ on C:

/pdf/trace-formulae-associated-with-the-polar-decomposition-of-1c6xkby3rw.pdf

Muneo Chō

Papers

Fuglede-Putnam theorem and quasisimilarity of class p-wA(s,t) operators

On $$\varvec{(m,C)}$$ ( m , C ) -Isometric Commuting Tuples of Operators on a Hilbert Space

On the generalized mean transforms of complex symmetric operators

Complex isosymmetric operators

Trace formulae associated with the polar decomposition of operators