scholarly journals Operator Symbols and Operator Indices

Symmetry ◽  
2019 ◽  
Vol 12 (1) ◽  
pp. 64
Author(s):  
Vladimir Vasilyev
Keyword(s):  
Banach Spaces ◽  
Differential Operators ◽  
Index Theorem ◽  
Symbolic Calculus ◽  
Smooth Manifolds ◽  
Bounded Operators ◽  
Pseudo Differential Operators ◽  
Operator Symbols ◽  
Special Classes

We suggest a certain variant of symbolic calculus for special classes of linear bounded operators acting in Banach spaces. According to the calculus we formulate an index theorem and give applications to elliptic pseudo-differential operators on smooth manifolds with non-smooth boundaries.

Symbolic Calculus and Boundedness of Multi-parameter and Multi-linear Pseudo-differential Operators

2014 ◽  
Vol 14 (4) ◽  
Author(s):  
Qing Hong ◽  
Guozhen Lu
Keyword(s):  
Partial Differential Equations ◽  
Differential Operators ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Regularity Of Solutions ◽  
Symbolic Calculus ◽  
Pseudo Differential Operators ◽  
Analysis Theory ◽  
Modern Analysis ◽  
Satisfactory Theory

AbstractSince the work of Hörmander on linear pseudo-differential operators, the applications of pseudo-differential operators have played an important role in partial differential equations, harmonic analysis, theory of several complex variables and other branches of modern analysis (e.g., they are used to construct parametrices and establish the regularity of solutions to PDEs such as the ∂̅ problem, etc.). The work of Coifman and Meyer on multi-linear Fourier multipliers and pseudo-differential operators has stimulated further such applications. In [2], the authors developed a fairly satisfactory theory of symbolic calculus for multi-linear pseudo-differential operators. Motivated by this work [2] and L

Continuity and compactness for pseudo-differential operators with symbols in quasi-Banach spaces or Hörmander classes

2017 ◽  
Vol 15 (03) ◽  
pp. 353-389 ◽  
Author(s):  
Joachim Toft
Keyword(s):  
Banach Spaces ◽  
Differential Operators ◽  
Modulation Spaces ◽  
Von Neumann ◽  
Norm Estimates ◽  
Pseudo Differential Operators

We deduce continuity and Schatten–von Neumann properties for operators with matrices satisfying mixed quasi-norm estimates with Lebesgue and Schatten parameters in [Formula: see text]. We use these results to deduce continuity and Schatten–von Neumann properties for pseudo-differential operators with symbols in quasi-Banach modulation spaces, or in appropriate Hörmander classes.

scholarly journals Recent developments on Pseudo-Differential Operators (I)

2015 ◽  
Vol 46 (1) ◽  
pp. 1-30 ◽  
Author(s):  
D.-C. Chang ◽  
W. RUNGROTTHEERA ◽  
B.-W. SCHULZE
Keyword(s):  
Boundary Value Problems ◽  
Differential Calculus ◽  
Differential Operators ◽  
Elliptic Boundary ◽  
Higher Order ◽  
Symbolic Calculus ◽  
Elliptic Boundary Value ◽  
The Past ◽  
Pseudo Differential Operators ◽  
Recent Developments

In recent years the analysis of (pseudo-)differential operators on manifolds with second and higher order corners made considerable progress, and essential new structures have been developed. The main objective of this series of paper is to give a survey on the development of this theory in the past twenty years. We start with a brief background of the theory of pseudo-differential operators which including its symbolic calculus on $\R^n$. Next we introduce pseudo-differential calculus with operator-valued symbols. This allows us to discuss elliptic boundary value problems on smooth domains in $\R^n$ and elliptic problems on manifolds. This paper is based on the first part of lectures given by the authors while they visited the National Center for Theoretical Sciences in Hsinchu, Taiwan during May-July of 2014.

RENORMALIZED TRACES AS A LOOKING GLASS INTO INFINITE-DIMENSIONAL GEOMETRY

2001 ◽  
Vol 04 (02) ◽  
pp. 221-266 ◽  
Author(s):  
SYLVIE PAYCHA
Keyword(s):  
Vector Bundle ◽  
Vector Bundles ◽  
Differential Operators ◽  
Index Theorem ◽  
Wodzicki Residue ◽  
Infinite Dimensional ◽  
Pseudo Differential Operators ◽  
Geometric Concepts ◽  
The One ◽  
Definition Of

This paper, based on results obtained in recent years with various coauthors,1–3,13,53 presents a proposal to extend some classical geometric concepts to a class of infinite-dimensional manifolds such as current groups and to a class of infinite-dimensional bundles including the ones arising in the family index theorem. The basic idea is to extend the notion of trace underlying many geometric concepts using renormalized traces which are linear functionals on pseudo-differential operators. The definition of "renormalized traces" involves extra data on the manifolds or vector bundles, namely a weight given by a field of elliptic operators which becomes part of the geometric data, leading to the notion of weighted manifold and weighted vector bundle. This weight is a source of anomaly arising typically as a Wodzicki residue of some pseudo-differential operator. We investigate the anomalies that arise when trying to extend to the infinite-dimensional setting classical results of finite-dimensional geometry such as a Weitzenböck formula, Chern–Weil invariants or the relation between the first Chern form on a complex vector bundle and the curvature on the associated determinant bundle. When comparable, we relate our approach to the one adopted for similar problems in noncommutative geometry.

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