Operator
Theory| Operator
Theory |
In General > s.a. measure [operator-valued].
* Operations on operators:
Adjoint, extensions (e.g., Friedrich
extension).
@ General references: Murray & Von Neumann AM(36),
AM(43); Dunford & Schwartz 58, 63, 71; Dixmier
69; Beals 71.
@ Hilbert space: Achiezer & Glazman 61; Jordan 69; Cirelli & Gallone
74; Reed & Simon 72–78; Schechter 81.
@ Related topics: Atiyah 74 [elliptic]; Lahti et al JMP(99)
[operator integrals].
Types of Operators > s.a. Projector; Subnormal; Symmetric.
* Bounded: An operator A: 
1
→ 2 such
that there exists c in R, for all v in 1,
Av 2 < c v 1 (c is
its norm).
* Hermitian: An operator A on
a Hilbert space such that 
,
A
= A,
,
for all and in
(A), i.e.,
complex symmetric; Relationships: This condition is weaker than self-adjointness
(it could
be that A
A),
but it still implies that , A 
R;
However, if A and A2 are
maximal hermitian, then A is
self-adjoint.
* Essentially self-adjoint:
A Hermitian one defined on a dense subspace; It admits a unique self-adjoint
extension.
* Self-adjoint: An operator A such
that A*
= A,
i.e., A is Hermitian and with (A*)
= (A); Useful
because (1) The eigenvalues
are real; (2) There are complete orthonormal sets of eigenvectors; (3) One
can meaningfully define functions f(A), with f Borel
measurable.
* Normal: A matrix/operator A such that A*A = AA*;
It can be diagonalized; The unitary geometry of the rows is the same as that
of
the columns.
* Trace-class: An operator T : X → Y between
Banch spaces which can be written as T(x) = 
i=1infty 
i xi(x)
yi, with xi 
1,
yi 1,
and {i}
in 
1;
one can then define the trace of T by tr T:= inf i=1infty i .
@ General references: Reed & Simon 72 [infinite-dimensional spaces].
@ Self-adjoint: Dubin et al JPA(02) [spectral and semispectral measures]
@ Self-adjoint extensions: Bonneau et al AJP(01)mar-qp [self-adjoint
extension examples];
> s.a. scattering.
@ PT-symmetric: Caliceti et al a0705 [non-selfadjont, with real discrete spectrum].
@ Unbounded: Bagarello RVMP(07),
a0903 [algebras,
intro and applications]; Jorgensen a0904 [duality theory].
Spaces of Operators > s.a. observable
algebras [von Neumann].
* 
(): The space of bounded operators on a (separable) Hilbert
space, a W*-algebra; Its topological dual is the space of trace-class operators.
* 
(): The
space of compact operators on ;
Its topological dual is ().
@ General references: Blackadar 06 [C*-algebras and von Neumann algebras];
Lledó a0901 [operator algebras, informal overview].
@ Maps between operator algebras: Salgado & Sánchez-Gómez mp/04-in
[Jamiolkowski positivity criterion].
Related Concepts > s.a. Boundary-Value
Problems; matrix [determinant];
norm; series [Taylor]; Stone's
Theorem.
@ Spectral theory: Cirelli 72; Müller-Pfeiffer 81; Friedrichs
80; Hislop & Sigal 95; > s.a. Bloch
Theory, Toeplitz.
@ Eigenvalues / eigenvectors: Andrew & Miller PLA(03)
[continuous + point
spectrum, Lanczos algorithm]; Georgescu a0811 [purely discrete].
@ Operations on operators: Gill & Zachary JPA(05) [fractional powers of linear
operators].
Differential Operators > s.a. conformal
structures [invariant]; D'Alambertian; laplacian.
* Zero modes: The eigenfunctions of a differential operator with zero
eigenvalue; They physically correspond to massless excitations of the field.
* Pseudodifferential operators:
Used in quantization, especially in the phase space (Wigner-Moyal) formulation
and in quantum field theory in surved spacetime.
@ Spectrum: Giacomini & Mouchet JPA(07)-a0706 [1D,
finding gaps in spectrum].
In Physics > s.a. observable
algebras.
@ Dynamical systems: Sakai 91.
@ Quantum theory: De Lange & Raab 91; Svozil qp/96 [discrete
operators and observables]; Bonneau et al AJP(01)mar-qp [self-adjoint
extensions]; Kempf PRD(01)
[symmetric operators
and symmetries];
D'Ariano PLA(02)
[universal observables]; Ozorio de Almeida & Brodier qp/05/JPA
[semi-classical evolution]; > s.a. annihilation, creation, formulations and representations
of quantum mechanics.
@ Unitary operators: Accardi & Sabbadini qp/00 [enhancing
specified components].
main page – abbreviations – journals – comments – other
sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 7
aug 2009