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Table Of Integrals Series And Products

Real and Complex Analysis

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Methods Of Modern Mathematical Physics

General Theory of Banach Algebra. By C.E. Rickart. Pp. 394. 79s. 1960. (Van Nostrand, London)

Quantum fields, noncommutative spaces, and motives The Riemann zeta function and noncommutative geometry Quantum statistical mechanics and Galois symmetries Endomotives, thermodynamics, and the Weil explicit formula Appendix Bibliography Index.

Noncommutative Geometry, Quantum Fields and Motives

Schauder decompositions and multiplier theorems

Abstract We show that if (E, ∥ · ∥ E ) is a symmetric Banach sequence space then the corresponding space of operators on a separable Hilbert space, defined by if and only if , is a Banach space under the norm . Although this was proved for finite-dimensional spaces by von Neumann in 1937, it has never been established in complete generality in infinite-dimensional spaces; previous proofs have used the stronger hypothesis of full symmetry on E. The proof that is a norm requires the apparently new concept of uniform Hardy-Littlewood majorization; completeness also requires a new proof. We also give the analogous results for operator spaces modelled on a semifinite von Neumann algebra with a normal faithful semi-finite trace.

Symmetric norms and spaces of operators

This text is the first complete study and monograph dedicated to singular traces. For mathematical readers the text offers, due to Nigel Kalton's contribution, a complete theory of traces on symmetrically normed ideals of compact operators. For mathematical physicists and other users of Connes' noncommutative geometry the text offers a complete reference to Dixmier traces and the deeper mathematical features of singular traces. An application section explores the consequences of these features, which previously were not discussed in general texts on noncommutative geometry.

Singular Traces: Theory and Applications

This paper resolves a number of problems in the perturbation theory of linear operators, linked with the 45-year-old conjecure of M. G. Kreĭn. In particular, we prove that every Lipschitz function is operator-Lipschitz in the Schatten–von Neumann ideals S
 α
 , 1 < α < ∞. Alternatively, for every 1 < α < ∞, there is a constant c
 α
 > 0 such that $$ {\left\| {f(a) - f(b)} \right\|_{\alpha }} \leqslant {c_{\alpha }}{\left\| f \right\|_{{{\text{Lip}}\,{1}}}}{\left\| {a - b} \right\|_{\alpha }}, $$
 where f is a Lipschitz function with
 $$ {\left\| f \right\|_{{{\text{Lip}}\,{1}}}}: = \mathop{{\sup }}\limits_{{_{{\lambda \ne \mu }}^{{\lambda, \mu \in \mathbb{R}}}}} \left| {\frac{{f\left( \lambda \right) - f\left( \mu \right)}}{{\lambda - \mu }}} \right| < \infty, $$
 $$ {\left\| \cdot \right\|_{\alpha }} $$
 is the norm is S
 α
 , and a and b are self-adjoint linear operators such that $$ a - b \in {S^{\alpha }} $$
 .

/pdf/operator-lipschitz-functions-in-schatten-von-neumann-classes-135q1odon2.pdf

Operator-Lipschitz functions in Schatten-von Neumann classes

https://openresearch-repository.anu.edu.au/bitstream/1885/23275/2/01_Carey_The_local_index_formula_in_2006.pdf

Fedor Sukochev

Papers

Schauder decompositions and multiplier theorems

Symmetric norms and spaces of operators

Singular Traces: Theory and Applications

Operator-Lipschitz functions in Schatten-von Neumann classes

The local index formula in semifinite Von Neumann algebras I: Spectral flow